Quantum Groups and Their Representations (Theoretical and Mathematical Physics)

Anatoli Klimyk Konrad Schmtidgen

Quantum Groups and Their Representations

Springer

This book starts with an introduction to quantum groups for the beginner and continues as a textbook for graduate students in physics and mathematics. It can also be used as a reference by more advanced readers. The authors cover a large but well-chosen variety of subjects from the theory of quantum groups (quantized universal enveloping algebras, quantized algebras of functions) and q-deformed algebras(q-oscillator algebras), their representations and corepresentations, and noncommutative differential calculus. The book is written with potential applications in physics and mathematics in mind. The basic quantum groups and quantum algebras and their representations are given in detail and accompanied by explicit formulas. A number of topics and recent results from the more advanced general theory are developed and discussed.

Preface

The invention of quantum groups is one of the outstanding achievements of mathematical physics and mathematics in the late twentieth century. The birth of the new theory and its rapid development are results of a strong interrelation between mathematics and physics. Quantum groups arose in the work of L.D. Faddeev and the Leningrad school on the inverse scattering method in order to solve integrable models. The algebra Uq (512 ) appeared first in 1981 in a paper by P.P. Kulish and N.Yu. Reshetikhin on the study of integrable XYZ models with highest spin. Its Hopf algebra structure was discovered later by E.K. Sklyanin. A major event was the discovery by V.G. Drinfeld and M. Jimbo around 1985 of a class of Hopf algebras which can be considered as one-parameter deformations of universal enveloping algebras of semisimple complex Lie algebras. These Hopf algebras will be called Drinfeld—Jimbo algebras in this book. Almost simultaneously, S.L. Woronowicz invented the quantum group SUq (2) and developed his theory of compact quantum matrix groups. An algebraic approach to quantized coordinate algebras was given about this time by Yu.I. Manin. A striking feature of quantum group theory is the surprising connections with many, sometimes at first glance unrelated, branches of mathematics and physics. There are links with mathematical fields such as Lie groups, Lie algebras and their representations, special functions, knot theory, low-dimensional topology, operator algebras, noncommutative geometry, and combinatorics. On the physical side there are interrelations with the quantum inverse scattering method, the theory of integrable models, elementary particle physics, conformal and quantum field theories, and others. It is expected that quantum groups will lead to a deeper understanding of the concept of symmetry in physics. Currently there is no satisfactory general definition of a quantum group. It is commonly accepted that quantum groups are certain "nice" Hopf algebras and that the standard deformations of the enveloping Hopf algebras of semisimple Lie algebras and of coordinate Hopf algebras of the corresponding Lie groups are guiding eiamples. Instead of searching for a rigorous definition of a quantum group it seems to be more fruitful to look for classes of Hopf algebras that give rise to a rich theory with important applications and con-

VI

Preface

tain enough interesting examples. In this book at least three such classes are extensively studied: quasitriangular Hopf algebras, coquasitriangular Hopf algebras, and compact quantum group algebras. The aim of this book is to provide a treatment of the theory of quantum algebras (quantized universal enveloping algebras), quantum groups (quantized algebras of functions), their representations and corepresentations, and the noncommutative differential calculus on quantum groups. The exposition is organized such that different parts of the text can be read and used (almost) independently of others. Sections 1.2 and 1.3 contain the main general definitions and notions on Hopf algebras needed in the text. This book is divided into four parts. Part I serves (among others) as an introduction to the theory of Hopf algebras, to the quantum algebra Uq (s12), the quantum group SL q (2), the q-oscillator algebra, and to their representations. The reader can use the corresponding chapters as first steps in order to learn the theory of quantum groups. A beginner might try to become aquainted with the language of Hopf algebras by reading Sect. 1.1 and portions of Sects. 1.2 and 1.3 and then passing immediately to Chaps. 3 or 4 (or start with Chaps. 3 or 4 and read parallel to them the relevant parts of Chap. 1). The main parts of the material of Chaps. 1, 3, and 4 can also be taken as a basis for an introductory course on quantum groups. Parts II—IV cover some of the more advanced topics of the theory. In Part II (quantized universal enveloping algebras).. and Part III (quantized algebras of functions) both fundamental approaches to quantum groups are developed in detail and as independently as possible, so readers interested in only one of these parts can restrict themselves to the corresponding chapters. Nevertheless the connections between both approaches appear to be very fruitful and instructive (see Sects. 4.4, 4.5.5, 9.4, 11.2.3, 11.5, and 11.6.6). A reader who is interested in only noncommutative differential calculus should pass directly to Part IV of the book and begin with Chaps. 12 or 14 (of course, some knowledge about the corresponding quantum groups from Chap. 9 and the L-functionals from Subsect. 10.1.3 is still required there). Together with Sects. 1.2 and 1.3, Parts II—IV form an advanced text on quantum groups. Selected material from these parts can also be used for graduate courses or seminars on quantum groups. Moreover, a large number of explicit formulas and new material (for instance, in Sects. 8.5, 10.1.3, 10.3.1, 13.2, and 14.3-5) are provided throughout the text, so we hope the book may be useful for experts as well. Let us say a few words about the selected topics and the presentation in the book. Our objective in choosing the material was to cover important and useful tools and methods for (possible) applications in theoretical and mathematical physics (especially in representation theory and in noncommutative differential calculus). Of course, this depends on our personal view of the matter. We have tried to give a comprehensive treatment of the chosen

Preface

VII

topics at the price of not including some concepts (for instance, the quantum Weyl group). Although we develop a number of general concepts too, the emphasis in the book is always placed on the study of concrete quantum groups and quantum algebras and their representations. Most of the results are presented with complete (but sometimes concise) proofs. Often, missing proofs or gaps in the existing literature have been filled. For some rather technical proofs (in particular of advanced algebraic results) readers are referred to the original papers. In many cases we have omitted proofs that are similar to the classical case. Having the potential reader in mind, we have avoided abstract mathematical theories whenever it was possible. For instance, we do not use cohomology theory, category theory (apart from Subsect. 10.3.4), Poisson—Lie groups, deformation theory, and knot theory in the book. We assume, however, that the reader has some standard knowledge of Lie groups and Lie algebras and their representation theory. The book is organized as follows. Formulas, results, definitions, examples, and remarks are numbered and quoted consecutively within the chapters. When a reference to an item in another chapter is made, the number of the chapter is added. For instance, (30) means formula (30) in the same chapter and Propostion 9.7 refers to Proposition 7 in Chap. 9. The end of a proof is marked by 11 and of an example or a remark by A. The reader should also notice that often assumptions are fixed and kept in force throughout the whole chapter, section, or subsection. Bibliographical comments are usually gathered at the end of each chapter. There the sources of some results or notions are cited (as far as the authors are aware) and some related references are listed, but no attempt has been made to report the origins of all items. We want to express our gratitude to A. Schiiler and I. Heckenberger for their indispensible help and valuable suggestions in writing this book and to Mrs. K. Schmidt for typing parts of the manuscript. We also thank Yu. Bespalov, A. Gavrilik, and L. Vainerman for reading parts of the book. Kiev and Leipzig, March 1997

A. U. Klimyk, K. Schmiidg en

Table of Contents

Part L An Introduction to Quantum Groups 1.

2.

Hopf Algebras 1.1 Prolog: Examples of Hopf Algebras of Functions on Groups 1.2 Coalgebras, Bialgebras and Hopf Algebras 1.2.1 Algebras 1.2.2 Coalgebras 1.2.3 Bialgebras 1.2.4 Hopf Algebras 1.2.5* Dual Pairings of Hopf Algebras 1.2.6 Examples of Hopf Algebras 1.2.7 *-Structures 1.2.8* The Dual Hopf Algebra A° 1.2.9* Super Hopf Algebras 1.2.10*h-Adic Hopf Algebras 1.3 Modules and Comodules of Hopf Algebras 1.3.1 Modules and Representations 1.3.2 Comodules and Corepresentations 1.3.3 Comodule Algebras and Related Concepts 1.3.4* Adjoint Actions and Coactions of Hopf Algebras 1.3.5* Corepresentations and Representations of Dually Paired Coalgebras and Algebras 1.4 Notes

3 3 6 6 8 11 13 16 18 20 22 23 25 27 27 29 32 34

q Calculus 2.1 Main Notions on q-Calculus 2.1.1 q-Numbers and q-Factorials 2.1.2 q-Binomial Coefficients 2.1.3 Basic Hypergeometric Functions 2.1.4 The 'Function 1(P0( 4 ; q Z) 2.1.5 The Basic Hypergeometric Function Zoi 2.1.6 Transformation Formulas for 0 2 and 4 (10 3 2.1.7 q-Analog of the Binomial Theorem 2.2 q-Differentiation and q-Integration 2.2.1 q-Differentiation

37 37 37 39 40 41 42 43 44 44 44

-

35 36

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X

2.2.2 g-Integral 2.2.3 q-Analog of the Exponential Function 2.2.4 g-Analog of the Gamma Function 2.3 g-Orthogonal Polynomials 2.3.1 Jacobi Matrices and Orthogonal Polynomials 2.3.2 g-Hermite Polynomials 2.3.3 Little q-Jacobi Polynomials 2.3.4 Big g-Jacobi Polynomials 2.4 Notes

3.

The Quantum Algebra Uq (512 ) and Its Representations 3.1

3.2

3.3

3.4

3.5

3.6

The Quantum Algebras Uq (s12) and Uh(s12) 3.1.1 The Algebra Uq (s12 ) 3.1.2 The Hopf Algebra Uq (s12 ) 3.1.3 The Classical Limit of the Hopf Algebra Uq (512) 3.1.4 Real Forms of the Quantum Algebra Uq (512) 3.1.5 The h-Adic Hopf Algebra Uh(s12) Finite-Dimensional Representations of Uq (512 ) for g not a Root of Unity 3.2.1 The Representations Tw i 3.2.2 Weight Representations and Complete Reducibility 3.2.3 Finite-Dimensional Representations of ("q.( s 1 2 ) and Uh(s12) Representations of Uq (s12 ) for g a Root of Unity 3.3.1 The Center of Uq (512 ) 3.3.2 Representations of Uq (512) 3.3.3 Representations of Uqre8 (s12) Tensor Products of Representations. Clebsch—Gordan Coefficients 3.4.1 Tensor Products of Representations Ti 3.4.2 Clebsch—Gordan Coefficients 3.4.3 Other Expressions for Clebsch—Gordan Coefficients 3.4.4 Symmetries of Clebsch—Gordan Coefficients Racah Coefficients and 6j Symbols of Uq (su2) 3.5.1 Definition of the Racah Coefficients 3.5.2 Relations Between Racah and Clebsch—Gordan Coefficients 3.5.3 Symmetry Relations 3.5.4 Calculation of Racah Coefficients 3.5.5 The Biedenharn—Elliott Identity 3.5.6 The Hexagon Relation 3.5.7 Clebsch—Gordan Coefficients as Limits of Racah Coefficients Tensor Operators and the Wigner—Eckart Theorem 3.6.1 Tensor Operators for Compact Lie Groups

46 47 48 49 49 50 51 52 52 • 53 53 53 55 57 58 60 61 61 • 63 65 66 66 67 71 72 72 74 • 78 81 82 82 84 84 85 88 90 90 92 92

Table of Contents

Tensor Operators and the Wigner—Eckart Theorem for Ù (su 2) Applications 3.7.1 The Uq (512 ) Rotator Model of Deformed Nuclei 3.7.2 Electromagnetic Transitions in the (4012) Model Notes

XI

3.6.2 3.7

3.8 4.

93 94 94 95 96

The Quantum Group SL q (2) and Its Representations 4.1 The Hopf Algebra 0(SL q (2)) 4.1.1 The Bialgebra 0(Mq (2)) 4.1.2 The Hopf Algebra 0(SL q (2)) 4.1.3 A Geometric Approach to SL q (2) 4.1.4 Real Forms of 0(841, (2)) 4.1.5 The Diamond Lemma 4.2 Representations of the Quantum Group SL q (2) 4.2.1 Finite-Dimensional Corepresentations of 0(S.L q (2)):

97 97 97 99 101 102 103 104

Main Results 4.2.2 A Decomposition of 0(S L q (2)) 4.2.3 Finite-Dimensional Subcomodules of 0(SL q (2)) 4.2.4 Calculation of the Matrix Coefficients 4.2.5 The Peter—Weyl Decomposition of 0(S.L q (2)) 4.2.6 The Haar Functional of 0(SL q (2)) 4.3 The Compact Quantum Group SUq (2) and Its Representations 4.3.1 Unitary Representations of the Quantum Group SUq (2) 4.3.2 The Haar State and the Peter—Weyl Theorem for 0(SUq (2)) 4.3.3 The Fourier Transform on SUq (2) 4.3.4 *-Representations and the C4-Algebra of 0(SUg (2)) 4.4 Duality of the Hopf Algebras Uq (512 ) and O(SL q (2)) 4.4.1 Dual Pairing of the Hopf Algebras Uq (s12) and 0(SL q (2)) 4.4.2 Corepresentations of 0(S L q (2)) and Representations of Uq (512) 4.5 Quantum 2-Spheres 4.5.1 A Family of Quantum Spaces for SL q (2) 4.5.2 Decomposition of the Algebra 0(4) 4.5.3 Spherical Functions on 51 4.5.4 An Infinitesimal Characterization of 0(SL) 4.6 Notes \

104 105 106 108 110 111 113 113 114 117 117 119 119 123 124 124 126 129 129 132

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XII

5.

The q-Oscillator Algebras and Their Representations 5.1 The g-Oscillator Algebras Aqc and Aq 5.1.1 Definitions and Algebraic Properties 5.1.2 Other Forms of the g-Oscillator Algebra 5.1.3 The g-Oscillator Algebra and the Quantum Algebra Û q (s12) 5.1.4 The g-Oscillator Algebras and the Quantum Space Mq 2 (2) 5.2

5.3

5.4

133 133 133 136 137 140 140 140

Representations of g-Oscillator Algebras 5.2.1 N-Finite Representations 5.2.2 Irreducible Representations 141 with Highest (Lowest) Weights 5.2.3 Representations Without Highest and Lowest Weights 143 5.2.4 Irreducible Representations of Acq 145 for q a Root of Unity 147 5.2.5 Irreducible *-Representations of and A g 5.2.6 Irreducible *-Representations 148 of Another g-Oscillator Algebra 149 The Fock Representation of the g-Oscillator Algebra 149 5.3.1 The Fock Representation 150 5.3.2 The Bargmann-Fock Realization 152 5.3.3 Coherent States 5.3.4 Bargmann-Fock Space Realization 153 of Irreducible Representations of Oq (s12) 154 Notes

Part II. Quantized Universal Enveloping Algebras 6.

Drinfeld-Jimbo Algebras 6.1 Definitions of Drinfeld-Jimbo Algebras 6.1.1 Semisimple Lie Algebras 6.1.2 The Drinfeld-Jimbo Algebras Uq (g) 6.1.3 The h-Adic Drinfeld-Jimbo Algebras Uh(9) 6.1.4 Some Algebra Autornorphisms of Drinfeld-Jimbo Algebras 6.1.5 Triangular Decomposition of Uq (g) 6.1.6 Hopf Algebra Automorphisms of Uq (g) 6.1.7 Real Forms of Drinfeld-Jimbo Algebras 6.2 Poincaré-Birkhoff-Witt Theorem and Verma Modules 6.2.1 Braid Groups 6.2.2 Action of Braid Groups on Drinfeld-Jimbo Algebras 6.2.3 Root Vectors and Poincaré-Birkhoff-Witt Theorem .. 6.2.4 Representations with Highest Weights 6.2.5 Verma Modules

157 157 157 161 165 167 168 171 172 173 173 174 175 177 179

Table of Contents XIII

6.3

6.4 7.

6.2.6 Irreducible Representations with Highest Weights 6.2.7 The Left Adjoint Action of Uq (g) The Quantum Killing Form and the Center of Uq (g) 6.3.1 A Dual Pairing of the Hopf Algebras Uq (b + ) and Uq (b_) °P 6.3.2 The Quantum Killing Form on Uq (g) 6.3.3 A Quantum Casimir Element 6.3.4 The Center of Uq (9) and the Harish-Chandra Homomorphism 6.3.5 The Center of Uq (9) for q a Root of Unity Notes

180 181 184

184 187 189 192 194 196

Finite-Dimensional Representations 197 of Drinfeld—Jimbo Algebras 7.1 General Properties of Finite-Dimensional Representations 197 of Uq (g) 197 7.1.1 Weight Structure and Classification 200 7.1.2 Properties of Representations 7.1.3 Representations of h-Adic Drinfeld—Jimbo Algebras. 202 7.1.4 Characters of Representations and Multiplicities 203 of Weights 204 7.1.5 Separation of Elements of Uq (g) 7.1.6 The Quantum Trace 205 of Finite-Dimensional Representations 207 7.2 Tensor Products of Representations 7.2.1 Multiplicities in Tensor Products of Representations 208 211 7.2.2 Clebsch—Gordan Coefficients 212 7.3 Representations of Ùq (g1) for q not a Root of Unity 212 7.3.1 The Hopf Algebra (4 (glr,) 213 7.3.2 Finite-Dimensional Representations of ej/q (g1n ) 7.3.3 Gel'fand—Tsetlin Bases and Explicit Formulas 214 for Representations 217 7.3.4 Representations of Class 1 7.3.5 Tensor Products of Representations 218 7.3.6 Tensor Operators and the Wigner—Eckart Theorem 219 7.3.7 Clebsch—Gordan Coefficients for the Tensor Product Tm 0 Ti 220 7.3.8 Clebsch—Gordan Coefficients for the Tensor Product Tm 0 Tp 221 7.3.9 The Tensor Product Tm 0 T1 for q± 1 0 224 7.4 Crystal Bases 225 7.4.1 Crystal Bases of Finite-Dimensional Modules 226 7.4.2 Existence and Uniqueness of Crystal Bases 227 7.4.3 Crystal Bases of Tensor Product Modules 228

XIV

Table of Contents

7.5

7.6 7.7

229 7.4.4 Globalization of Crystal Bases 230 7.4.5 Crystal Bases of U(n_) 232 Representations of Uq (g) for q a Root of Unity 232 7.5.1 General Results 234 7.5.2 Cyclic Representations 235 7.5.3 Cyclic Representations of the Algebra U,(sli+i ) 237 7.5.4 Representations of Minimal Dimensions 7.5.5 Representations of U E (sli +i ) in Gerfand—Tsetlin Bases 238 240 Applications 242 Notes

8. Quasitriangularity and Universal R-Matrices 8.1 Quasitriangular Hopf Algebras 8.1.1 Definition and Basic Properties 8.1.2 R-Matrices for Representations 8.1.3 Square and Inverse of the Antipode 8.2 The Quantum Double and Universal R-Matrices 8.2.1 The Quantum Double of Skew-Paired Bialgebras 8.2.2 Quasitriangularity of Quantum Doubles of Finite-Dimensional Hopf Algebras 8.2.3 The Rosso Form of the Quantum Double 8.2.4 Drinfeld—Jimbo Algebras as Quotients of Quantum Doubles 8.3 Explicit Form of Universal R-Matrices 8.3.1 The Universal R-Matrix for Uh (S12) 8.3.2 The Universal R-Matrix for U h(g) 8.3.3 R-Matrices for Representations of Uq (g) 8.4 Vector Representations and R-Matrices 8.4.1 Vector Representations of Drinfeld—Jimbo Algebras 8.4.2 R-Matrices for Vector Representations 8.4.3 Spectral Decompositions of R-M atr ices for Vector Representations 8.5 L-Operators and L-Functionals 8.5.1 L-Operators and L-Functionals 8.5.2 L-Functionals for Vector Representations 8.5.3 The Extended Hopf Algebras U(1'(9) 8.5.4 L-Functionals for Vector Representations of Uq (9) 8.5.5 The Hopf Algebras 1-1(R) and UilL (2) 8.6 An Analog of the Brauer—Schur—Weyl Duality 8.6.1 The Algebras fig (soN) 8.6.2 Tensor Products of Vector Representations 8.6.3 The Brauer—Schur—Weyl Duality for Drinfeld—Jimbo Algebras 8.6.4 Hecke and Birman—Wenzl—Murakami Algebras 8.7 Applications

243 243 243 246 247 250 250 254 257 258 259 259 261 264 267 267 269 272 275 275 277 281 283 285 288 288 289 291 293 294

Table of Contents

8.8

XV

8.7.1 Baxterization 8.7.2 Elliptic Solutions

295

of the Quantum Yang-Baxter Equation 8.7.3 R-Matrices and Integrable Systems Notes

297 298 300

Part III. Quantized Algebras of Functions 9.

Coordinate Algebras of Quantum Groups and Quantum Vector Spaces 9.1 The Approach of Faddeev-Reshetikhin-Takhtajan 9.1.1 The FRT Bialgebra A(R) 9.1.2 The Quantum Vector Spaces XL(f ; R) and XR(f ;R) 9.2 The Quantum Groups GL q (N) and SL q (N) 9.2.1 The Quantum Matrix Space Mq (N) and the Quantum Vector Space Clq'T 9.2.2 Quantum Determinants 9.2.3 The Quantum Groups GL q (N) and SL q (N) 9.2.4 Real Forms of GL q (N) and SL q (N) and *-Quantum Spaces 9.3 The Quantum Groups 0q (N) and Spq (N) 9.3.1 The Hopf Algebras 0(0q (N)) and 0(Sp q (N)) 9.3.2 The Quantum Vector Space for the Quantum Group 0q (N) 9.3.3 The Quantum Group SO q (N) 9.3.4 The Quantum Vector Space for the Quantum Group Spq (N) 9.3.5 Real Forms of 0q (N) and Spq (N) and *-Quantum Spaces 9.4 Dual Pairings of Drinfeld-Jimbo Algebras and Coordinate Hopf Algebras 9.5 Notes

10. Coquasitriangularity and Crossed Product Constructions 10.1 Coquasitriangular Hopf Algebras 10.1.1 Definition and Basic Properties 10.1.2 Coquasitriangularity of FRT Bialgebras A(R) and Coordinate Hopf Algebras O(C) 10.1.3 L-Functionals of Coquasitriangular Hopf Algebras 10.2 Crossed Product Constructions of Hopf Algebras 10.2.1 Crossed Product Algebras 10.2.2 Crossed Coproduct Coalgebras 10.2.3 Twisting of Algebra Structures by 2-Cocycles and Quantum Doubles

303 303 303 307 309 310 311 313 316 317 318 320 323 324 325 327 330 331 331 331 337 342 349 349 352 354

XVI Table of Contents

10.2.4 Twisting of Coalgebra Structures by 2-Cocycles and Quantum Codoubles 10.2.5 Double Crossed Product Bialgebras and Quantum Doubles 10.2.6 Double Crossed Coproduct Bialgebras and Quantum Codoubles 10.2.7 Realifications of Quantum Groups 10.3 Braided Hopf Algebras 10.3.1 Covariantized Products for Coquasitriangular Bialgebras 10.3.2 Braided Hopf Algebras Associated with Coquasitriangular Hopf Algebras • • 10.3.3 Braided Hopf Algebras Associated with Quasitriangular Hopf Algebras 10.3.4 Braided Tensor Categories and Braided Hopf Algebras 10.3.5 Braided Vector Algebras 10.3.6 Bosonization of Braided Hopf Algebras 10.3.7 *-Structures on Bosonized Hopf Algebras 10.3.8 Inhomogeneous Quantum Groups 10.3.9 *-Structures for Inhomogeneous Quantum Groups 10.4 Notes

357 359 362 363 365 365 370 376 377 380 382 386 388 390 394

11. Corepresentation Theory and Compact Quantum Groups. 395 11.1 Corepresentations of Hopf Algebras 395 11.1.1 Corepresentations 395

11.1.2 Intertwiners 11.1.3 Constructions of New Corepresentations 11.1.4 Irreducible Corepresentations 11.1.5 Unitary Corepresentations 11.2 Cosemisimple Hopf Algebras 11.2.1 Definition and Characterizations 11.2.2 The Haar Functional of a Cosemisimple Hopf Algebra 11.2.3 Peter—Weyl Decomposition of Coordinate Hopf Algebras 11.3 Compact Quantum Group Algebras 11.3.1 Definitions and Characterizations of CQG Algebras . . 11.3.2 The Haar State of a CQG Algebra 11.3.3 C*-Algebra Completions of CQG Algebras 11.3.4 Modular Properties of the Haar State 11.3.5 Polar Decomposition of the Antipode 11.3.6 Multiplicative Unitaries of CQG Algebras 11.4 Compact Quantum Group C*-Algebras 11.4.1 CQG C*-Algebras and Their CQG Algebras 11.4.2 Existence of the Haar State of a CQG C*-Algebra 11.4.3 Proof of Theorem 39

397 397 398 401 402 402 404 408 415 415 419 420 422 426 427 429 429 431 433

Table of Contents XVII

434 11.4.4 Another Definition of CQG C*-Algebras 435 11.5 Finite-Dimensional Representations of GL q (N) 435 11.5.1 Some Quantum Subgroups of GL q (N) 436 11.5.2 Submodules of Relative Invariant Elements 437 11.5.3 Irreducible Representations of GL g (N) 439 11.5.4 Peter—Weyl Decomposition of 0(GL q (N)) 441 11.5.5 Representations of the Quantum Group Uq (N) 442 11.6 Quantum Homogeneous Spaces 442 11.6.1 Definition of a Quantum Homogeneous Space 11.6.2 Quantum Homogeneous Spaces 443 Associated with Quantum Subgroups 445 11.6.3 Quantum Gerfand Pairs 11.6.4 The Quantum Homogeneous Space Uq (N-1)W q (N) . 447 11.6.5 Quantum Homogeneous Spaces 451 of Infinitesimally Invariant Elements 452 11.6.6 Quantum Projective Spaces 454 11.7 Notes

Part IV. Noncommutative Differential Calculus 12. Covariant Differential Calculus on Quantum Spaces 12.1 Covariant First Order Differential Calculus 12.1.1 First Order Differential Calculi on Algebras 12.1.2 Covariant First Order Calculi on Quantum Spaces 12.2 Covariant Higher Order Differential Calculus 12.2.1 Differential Calculi on Algebras 12.2.2 The Differential Envelope of an Algebra 12.2.3 Covariant Differential Calculi on Quantum Spaces. 12.3 Construction of Covariant Differential Calculi on Quantum Spaces 12.3.1 General Method 12.3.2 Covariant Differential Calculi on Quantum Vector Spaces 12.3.3 Covariant Differential Calculus on CqN and the Quantum Weyl Algebra 12.3.4 Covariant Differential Calculi on the Quantum Hyperboloid 12.4 Notes 13. Hopf Bimodules and Exterior Algebras 13.1 Covariant Bimodules 13.1.1 IA-Covariant Bimodules 13.1.2 Right-Covariant Bimodules 13.1.3 Dicovariant Bimodules (Hopf Birnodules)

457 457 457 459 461 461 462 463 464 464 467 468 471 472 473 473 473 477 477

XVIII Table of Contents 13.1.4 Woronowicz' Braiding of Bicovariant Bimodules 13.1.5 Bicovariant Bimodules and Representations

480

of the Quantum Double 13.2 Tensor Algebras and Exterior Algebras of Bicovariant Bimodules 13.2.1 The Tensor Algebra of a Bicovariant Bimodule 13.2.2 The Exterior Algebra of a Bicovariant Bimodule 13.3 Notes

483 485 485 488 490

491 14. Covariant Differential Calculus on Quantum Groups 491 14.1 Left-Covariant First Order Differential Calculi 14.1.1 Left-Covariant First Order Calculi 491 and Their Right Ideals 494 14.1.2 The Quantum Tangent Space 496 14.1.3 An Example: The 3D-Calculus on SL q (2) 14.1.4 Another Left-Covariant Differential Calculus 498 Oil SL q (2) 498 14.2 Bicovariant First Order Differential Calculi 498 14.2.1 Right-Covariant First Order Differential Calculi 499 14.2.2 Bicovariant First Order Differential Calculi 14.2.3 Quantum Lie Algebras 500 of Bicovariant First Order Calculi 504 14.2.4 The 4D+ - and the 4D_-Calculus on SL q (2) 14.2.5 Examples of Bicovariant First Order Calculi 505 on Simple Lie Groups 506 14.3 Higher Order Left-Covariant Differential Calculi 14.3.1 The Maurer-Cartan Formula 506 507 14.3.2 The Differential Envelope of a Hopf Algebra 14.3.3 The Universal DC of a Left-Covariant FODC 508 14.4 Higher Order Bicovariant Differential Calculi 511 14.4.1 Bicovariant Differential Calculi and Differential Hopf Algebras 511 14.4.2 Quantum Lie Derivatives and Contraction Operators 514 14.5 Bicovariant Differential Calculi on Coquasitriangular Hopf Algebras 517 14.6 Bicovariant Differential Calculi on Quantized Simple Lie Groups 521 14.6.1 A Family of Bicovariant First Order 521 Differential Calculi 14.6.2 Braiding and Structure Constants of the FODC F±,z 524 525 14.6.3 A Canonical Basis for the Left-Invariant 1-Forms 14.6.4 Classification of Bicovariant First Order 527 Differential Calculi 528 14.7 Notes

Table of Contents

XIX

Bibliography

529

Index

545

Part I

An Introduction to Quantum Groups

1. Hopf Algebras

The underlying mathematical notion for quantum groups is that of a Hopf algebra. These are associative algebras equipped with additional structures such as a comultiplication, a counit and an antipode. In some appropriate sense, these structures and their axioms reflect the multiplication, the unit element and the inverse elements of a group and their corresponding properties. The purpose of this chapter is twofold. First, the reader unfamiliar with this topic may try to learn the basics of Hopf algebras here and then continue with Chaps. 3 or 4. Subsections marked by * should be omitted by beginners. Secondly, Sects. 1.2 and 1.3 collect general definitions and notation used throughout this book.

1.1 Prolog: Examples of Hopf Algebras of Functions on Groups Probably the best way to take up a new concept and to understand the ideas behind its abstract definition might be to look at well-chosen examples. We shall do so before we enter the path through the wealth of necessary definitions. Our examples will only work with functions on "ordinary" groups, but they provide the motivation for later constructions on quantum groups. To begin with, let G be an arbitrary group and let ..F(G) be the algebra of all complex-valued functions on G with pointwise algebraic operations. The group operations on G allow one to define the following mappings:

- comultiplication A : .F(G) ---) .F(G x G) by (A(f))(gi,g 2 ) := f(gig2), - counit e : ,F(G) ---) C by e( f) := f (e), - antipode S : ..9G) .F(G) by (S(f))(g) := f(g -1 ). Here e denotes the unit element of G. From these definitions it is clear that A and e are algebra hornomorphisms.

(1)

The group axioms lead to the following identities (2)-(4) for the mappings e and S. First, the associativity of the group multiplication implies that (A id) o d = (id ® d) 0 .6,

(2)

4

1. Hopf Algebras

where id is the identity mapping on T(G). Indeed, the definition of d yields (((d 0 id) o d)f)(g i ,g2 ,g3) = f((g i g2)g3) G .F(G x G x G), (((id 0 d) o d)f)(gi,g 2 ,g3) = f (gi(g2g3)) E .9G x G x G).

Since

gi(g2g3) by the associativity law, the two expressions on the right hand sides coincide and (2) follows. Secondly, we have the relation f (eg) = f (ge) = f(g) which can be expressed as (gig2)g3

(E

id) o = (id

E) 0 = id

( 3)

under the usual identifications of CO .F(G) and .F (G) C with .F(G). Define a mapping m : .F(G x G) --* .F(G) by (rnh)(g) := kg, g), h E .F(G x G), and a linear mapping 71: C .F(G) such that T(1) is the unit element of the algebra T(G). Then, thirdly, the relations g-l g — gg -1 =-- e for the inverse g -1 of a group element g yield the identities mo(Soid)od=1)oE=m0(id®,S)04A.

(4)

Indeed, inserting the above definitions of m, d and S we obtain ((mo(idoS)0d)f)(g) = (((idOS)od)f)(g,g) = f(gg -1 ) f(e) = (rloE)(f) which gives the second equality of (4). The first equality follows similarly. Summarizing the preceding, we have seen that the group multiplication, the group unit and the group inverse of G induce mappings 4 E and S of functions on G such that the corresponding group axioms are expressed as identities (2)-(4). On the level of functions the properties of the group G are encoded in the three mappings 4 E and S defined above. One disadvantage is that the algebra .9G x G) of functions on G x G appeared in the definitions of the mappings d and m. In order to work with functions on G alone, we consider the tensor product .F(G) 0 .F(G) as a linear subspace of .F(G x G) by identifying fi O f2 G .F(G) .F(G) with the function (fl O f2 )(gi , g2 ) := fi (gi)f2(g2 ) on G x G. Then we have m(fi O f2)(9) = fi(g)f(g), that is, m(fi f2) is just the product fi f2 of f and f2 in the algebra T(G). After the embedding .F(G) T(G) C .F(G x G) one difficulty still remains: d does not map T(G) to .F(G) .F(G) in general (for example, d(f) .1(G)0.1(G) if f(t) = sin t 2 and G = So it is natural to look for appropriate subalgebras A of .F(G) for which d(A) C A o A and S(A) c A. Such a subalgebra A together with the linear mappings : A -4 A O A, E : A -4 C and S : A A satisfying the above conditions (1)-(4) is called a Hopf algebra (by Definition 4 below). If the Hopf algebra A is sufficiently "large", one can expect that A and its structure maps 4 e and S store enough information about the group G. We consider three important examples of such Hopf algebras A.

1.1 Prolog: Examples of Hopf Algebras of Functions on Groups

5

Example 1 (The Hopf algebra ,F(G) of a finite group). For a finite group G we have .F(G) .F(G) = (G x G), so by the preceding .F(G) is a Hopf algebra. Example 2 (The Hopf algebra Rep (G) of a compact group G). Suppose that G is a compact topological group. Let T be a continuous representation of G on a finite-dimensional complex vector space V. Let C(T) denote the linear subspace of .F(G) spanned by the matrix elements t o (.), i j = 1, 2, • • • , N, of TO relative to some basis of V. Since t o (gig2) = Ek tik(gOtki(g2), gi,g2 G, and t o (e) = 60 , we have ,

d(t o ) =

E k tik t kj

e(tij = .

( 5 )

The vector space C(T) equipped with the linear mappings d : C(T) C(T) 0 C which satisfy the relations (2) and (3) is called a C(T) and e : C(T) coalgebra (by Definition 2 below). Let A = Rep (G) be the linear span of spaces C(T) for all continuous finite-dimensional representations T of G. Since C(Ti ED T2) -= C(Ti) C (T2 ) and C(Ti 0 T2 ) = C(T1 ).C(T2 ) as is easily seen, A is a subalgebra of .F(G). Recall that the contragredient representation Tc of T acts on the dual V' of the space V by (Tc(g)y')(y) = y'(T(g -1 )y), g E G, v E V, y' E V'. If we choose the basis of V' dual to the basis of V, then Tc has the matrix elements qi = S(t). Thus, we get C(Tc) =--- S(C(T)) and hence S(A) C A. Since Zi(C(T)) C C(T) C(T) by (5), we also have d(A) C A® A. Therefore A = Rep (G) is a Hopf algebra. It is called the Hopf algebra of representative functions of G. Since Rep (G) is dense in the algebra C(G) of continuous functions on G by the Peter-Weyl theorem, Rep (G) is sufficiently "large". Let (t 5 )ri_ 1 be a matrix of functions to E .F(G). A simple consideration shows that the functions t o are matrix elements of some continuous finitedimensional representation T of G if and only if they belong to the algebra Rep (G) and satisfy the equations (5). This means that continuous finitedimensional representations of G can be defined completely in terms of the Hopf algebra Rep (G) without referring to the group G itself. Many other concepts of group representation theory can also be formulated in terms which are intrinsic to the Hopf algebra Rep (G) (see Chap. 11). Example 3 (The coordinate Hopf algebras 0(G) of simple matrix Lie groups). Let G denote one of the matrix groups SL(N, C), SO(N, C) or Sp(N, C). Each element g of G is a complex N x N matrix g = (go ). Define the coordinate functions u on G by u(g) gii , g = (go ) E G. For g, h E G we have

L1(u))(g , h) = u ij (g h)

(gh) ii = Ek gikh ki = Ek 4(g)u. 11 (h)

and i4(e) = bo, so that

=

Ek 111

III

and 6(4) =

(6)

6

1. Hopf Algebras

Let A --= 0(G) be the subalgebra of .F(G) generated by the N2 functions j = 1, 2, - - - , N. Since 1 : .F(G) — ■ .9G x G) is an algebra homomorphism, i, we have L\(A) C A 0 A by the first formula of (6). Any element g E G has determinant 1. Hence the function which is a constant equal to 1 on G belongs to A, so A has a unit. Further, it follows that there are polynomials pii in N2 indeterminates such that (g -1 ) ii = pii (g11 , g12 , - • • , gNN ), so that

51 (uii ) (g) = u4(g-1 ) ---- (g -1 ) ii = pii (ul(g) , 74(g) , • - • , u l;,'; (0) = pii (ul, 4 - - • That is, we have S(u i ) e A and hence 51 (A) C A. Therefore, by the discussion preceding Example 1, A = 0(G) is a Hopf algebra. It is called the coordinate Hopf algebra of G. A In all three examples of Hopf algebras the multiplication is commutative (being the ordinary multiplication of functions) and the antipode S is an algebra homomorphism such that S2 = id. These facts no longer remain true for general Hopf algebras and quantum groups. Summarizing and slightly simplifying the above considerations, we have rephrased (certain) groups in terms of Hopf algebras of functions on the groups. This turns out to be the proper perspective that makes it possible to "deform" or to "quantize" the groups by passing to noncommutative algebras. Formally this is done by introducing Hopf algebras depending on say one parameter (usually denoted by q or e h , h being thought of as Planck's constant) which specialize to the Hopf algebra associated with the group at a particular value (usually q = 1, resp. h = 0). The elements of such a deformed Hopf algebra will be viewed as functions on a "quantum group". The language of groups and representations will also be used for the deformed Hopf algebras as an inspiring source of motivation, though there are no groups around.

1.2 Coalgebras, Bialgebras and Hopf Algebras Let us fix some general notation which will be kept throughout the book. The letter K always stands for a commutative ring with unit. If V and W are vector spaces over K, £(V, W) denotes the 1K-linear mappings of V to W and V' is the vector space of all K-linear functionals on V. We set ,C(V) := L(V, V). The symbol 0 means the tensor product OK. We identify K0 V and V 0 K with V by the isomorphisms 1 0 y y, respectively. y and y 0 1 If not specified otherwise, the letter T denotes the flip operator given by T (v 0 w) = w 0 v.

1.2.1 Algebras Recall that an (associative) algebra is a vector space A over K equipped with a mapping (a, b) ab of A x A to A such that

1.2 Coalgebras, Bialgebras and Hopf Algebras

7

a(bc) = (ab)c, (a + b)c = ac +be, a(b + c) = ab + ac, cx(ab) = (aa)b = a(ab)

( 7 )

for all a, b E A and a E K. An element 1 of an algebra A is called a unit of A if 1a = al = a for all a E A. If a unit element exists in A, it is uniquely determined. Throughout this book, by an algebra we always mean an associative algebra with unit element. The following slight reformulation of the algebra definition is the right perspective for its dualization in Subsect. 1.2.2. Definition 1. An algebra (that is, an associative algebra with unit) is a

vector space A over K together with two linear maps m: A A —> A, called A, called the unit, such that the multiplication or the product, and 71 : K id) = 0 (id

m 0 (rn rn (7/

m),

id) = id = m 0 (id

( 8

ri).

)

(9)

Given such an algebra A, ab := rn(a b) is the product of a and b and the mapping n is determined by its value 71(1) E A, which is the unit element of A. Both definitions of an algebra are easily seen to be equivalent. Equation (8) is the associativity law, while (9) says that n(1) is a unit element of A using the identification of IK0 A and A 0 K with A. The conditions in (7) are built into the requirement that rrt:A0A Ais 1K-linear. The associativity (8) of the multiplication m means that the diagram —

m

AoAcgA

id

AoA. (10)

iidom Trt

A 0A

A

is commutative. Likewise, the condition (9) of the unit can be expressed by the commutativity of the following diagram:

IK0A

A 0A

id 0 71

A 0K

jid

KoA

A

.4=>

l id AoK,

where K 0 A A and A 4=> A 0 K denote the canonical identifications. Let A and B be algebras.. A K-linear mapping (i9 : A 13 is called an algebra homornorphism if (p(aa') = (p(a)(p(d) for all a, a' E A and ço(1A) = 16. The two latter conditions can be rewritten as

8

1. Hopf Algebras

There exists a tensor product algebra AO13 whose vector space is the tensor product of vector spaces of A and B and whose multiplication is defined by ms) sc, (id 0 T 0 id). That is, the algebra A 0 13 has the TriA0 B := (mA product (12) (a 0 b)(a' b 1 ) := aal a,al E A, b,b' E B. For each algebra A one can define the opposite algebra A°P. This is an algebra with the same underlying vector space as A, but with the new multiplication m A , := mA 0 T. That is, we have a op b = b • a, where • op and denote the products of A°P and A, respectively. Many algebras are constructed as quotients of free algebras or tensor algebras. Let us recall these notions in the case where K is the complex field C. Let {x I i E II be an indexed set of generators. We denote by I' the union of all sets In, n E No , where 10 consists of a single element o which is not in I. Fort = (i 1 ,i2, • • • ,ik) E /k and j = (ii/i21 . • • I in) E I, E No, we set (i, j) = (i 1 , • • • , ik , j 1 , • - • ,j n ) and xi := x i, • • • x ik • For o E 1 0 and i E Ik, k E No, we put (o, i) = (i,o) := i and s o :--= 1. Then the free algebra C(x i ) with generators x i , i E I, is defined as follows: it is a complex vector space with basis {x I i E /oe } and with the multiplication defined by x i xi E I'. The algebra C(xi ) possesses the following universal property: for any indexed subset {a I i E I} of another algebra A there exists a unique algebra hornomorphism ç :C(x i ) —> A such that ça(xi ) --= ai for all i G I. Next we define the tensor algebra T(V) over a vector space V. Set Vrg° := K and V ®n := V 0 • • V (n times). The direct sum T(V) := @n œ 0 V ®71 of these vector spaces becomes an algebra, called the tensor algebra T(V) over V, with multiplication determined by xn y k := x n yk for x n E V en and yk E V®k. The algebra TM also has a universal property: for every linear map (pi : V -4 A of V to an algebra A, there exists a unique algebra homomorphism ça : T(V) —> A such that ça(v) p 1 (v) for y E V. The universal properties of the algebras C(x i ) and T(V) imply the following fact: for any basis {x j i E II of the vector space V, there is a unique algebra isomorphism ço of the algebras T(V) and C(xi ) such that ço(x) = x i , i E I. Often in this book an algebra A is introduced by stating that it is generated by elements x l , • • • with relations ..,k,x17 f ( 0, k = 1,2, • • ,s. • • • S r) By such a phrase we mean that A is the quotient algebra of the free algebra C(x i ) with generators xi, • • • , Sr by the two-sided ideal I of C(xi ) generated by fk (xi, • • • , xr ), k = 1, 2, • • • , s. For simplicity we shall denote the image of the elements x i E C(xi) in the quotient algebra A = C(xi) also by X. 1.2.2 Coalgebras

Recall that an algebra is a vector space A with two linear mappings m : A® A -4 A and : K —> A such that the diagrams (10) and (11) are commutative. We now dualize this definition by reversing all arrows and replacing all mappings by the corresponding dual ones. In doing so, the multiplication

1.2 Coalgebras, Bialgebras and Hopf Algebras

9

m:A0A >Ais replaced by the comultiplication LI :A >A0A, the unit .71 : IK —> A by the counit E : A —> IK and the diagrams (10) and (11) go into the following diagrams (13) and (14), which have to be commutative: —

—

0 id

A0A0A

(13)

id 0 Zi LI

A0 A K

A c

E

id

A id ®

A 0A

E

A 0K id

id K

A

< >

A

(14)

A 0K

We thus obtain the following "dual" notion to an algebra. Definition 2. A coalgebra is a vector space A over K equipped with two linear mappings 2i : A A 0 A, called the comultiplication or the coproduct, and E : A —> K, called the counit, such that 0 id) (E

= (id

(15)

o

id) 0 = id = (id

E)

(16)

o

Equation (15) is referred to as the coassociativity of the comultiplication 2i, because it dualizes the associativity (8) of the multiplication tn. Proceeding in a similar manner, most concepts on algebras can be dualized to coalgebra notions by giving diagrammatic formulations of the definitions and then reversing all arrows. We only list the outcomes of these dualizations in the following, but the reader unfamiliar with this procedure should carry out a few more examples in order to get used to working with "costructures" . Let A and B be coalgebras. A K-linear mapping cp :A >Bis said to be a coalgebra homomorphism if —

Zi5 0

(cp (,o)

2■ A

and

EA = Es

(p.

The tensor product coalgebra A® B is the coalgebra built on the vector space A0 B with comultiplication ZiA 013 := (id ® T (3) id) o (LIA ® 43) and counit E.A08 EAOEB. The coopposite coalgebra A"P is the coalgebra on the vector space A equipped with the new comultiplication ZIAcop T 0 4A and the counit EA. A coalgebra A is said to be cocommutative if T o LI = 21. A linear subspace B of A is a subcoalgebra if Zi(B) C B 0 B. A K-linear subspace I of A is called a (two—sided) coideal if .A(/) C

/ /

A

and E(/) = {0}.

If I is a coideal of A, the quotient vector space A/I becomes a coalgebra with comultiplication and counit induced from A.

10

1. Hopf Algebras

Next we introduce the so-called Sweedler notation for the comultiplication. If a is an element of a coalgebra A, the element A(a) E A 0 A is a finite sum A(a) =

E aii a2i aii,a2i

(17)

E A.

Moreover, this representation of A(a) is not unique. For notational simplicity, we shall suppress the index i and write the sum (17) symbolically as A(a) =--

E a(i) a(2).

(18)

Here the subscripts (1) and (2) refer to the corresponding tensor factors and the symbol E reminds us that A(a) is actually a sum. Let us define inductively mappings A (n) : A 4 A®n+1 by -

A (n ) := (id 0 A) o A -1 , n> 1,

and A (1) =--

(From the coassociativity (15) it follows that ZA (n) is in fact equal to n 1 compositions of A independently of their order, that is, A (2) = (id0,A) 0,A = (A 0 id) o A, etc.) Then the element A (n)(a) E A®n+1 is denoted by g a( 2) g

(19)

• • • (g) a (n+1) .

One can also replace the sums in (18) and (19) by the symbol E(a) in order to emphasize the element a E A. The reader may check the correct use of this notation by adding the corresponding elements to the sum. The Sweedler notations (18) and (19) turn out to be very useful when working with general Hopf algebras, because they allow one to give elegant shorthand formulations of definitions, formulas and proofs. We strongly recommend the reader to practise its use (for instance, by carrying out proofs in Propositions 1, 5 and others below) in order to become acquainted with this powerful tool. The result will make the effort worthwhile. The coassociativity (15) in terms of the notation in (18) and (19) yields

E(a(1) )(1)

(a( 1 ))( 2)

E a( 1) 0 (a( 2))( 1 ) (a( 2))( 2) = A (2) (a) = E a(1) a(2) a(3) .

a( 2) =

The counit property (16) reads in the Sweedler notation (18) as a=

E ame(a( 2)).

(20)

Proposition 1. Let A be a coalgebra and B an algebra. Then the vector space L(A, 8) of all K-linear mappings of A to 13 equipped with the convolution product defined by

(f * Oa) := (m 8 o (f 0 g)AA )(a) becomes an algebra with unit

77,3

E f (a(i))g(a(2)),

a E A,

(21)

11

1.2 Coalgebras, Bialgebras and Hopf Algebras

Proof. By the associativity of the multiplication in 13 and the coassociativity of the comultiplication in A, we obtain

((f * g) * h)(a) =

E f(a( 1))g(a(2))h(a(3)) = (f * (g * h))(a),

that is, the convolution product (21) is associative. The relation ((r/B 0 6A) * f)(a) =

EA(a(1))f(a(2))

= f (EEA(a(1))a(2)) =

f (a)

by (20) shows that 7p3 o eA is a left unit. It is similarly proved that 716 o EA CI is a right unit. Corollary 2. The dual vector space A' of a coalgebra A is an algebra with

product

(f Oa) := (f * g)(a) =

E f(a( 1))g(a(2)),

Proof. Apply Proposition 1 to the algebra B

a G A,

f,g G A'.

(22) CI

K (over K).

The dual space B' of an algebra B does not become a coalgebra in general by dualizing the multiplication (see the polynomial algebra C[x] for a counterexample), however it does if the algebra 13 is finite-dimensional. The reason is that we always have A' 0 A' C (A A)' and hence mA, : 0 A' A', while B' B' is only a proper subspace of (B BY if 13 is infinite-dimensional, so that z1 (f) f o nit3 may not lie in B' 8'. There are two possibilities to remedy this situation: to replace the dual space B' by a subspace (this will be done in Subsect. 1.2.8) or to enlarge the tensor product (for instance, by taking its completion in some topology). Example 4 (The coalgebra MN (K)' ). Let uii be the linear functional on the algebra MN(K) of N x N matrices over K defined by t4(g) := gii for g = is a coalgebra (gii) G MN(K). Then MN(K)' = Lin fuji I i j = 1, 2, - • - , = E k uik g u.'; and counit 6ii . with comultiplication ,

1.2.3 Bialgebras

A bialgebra is an algebra and a coalgebra, where both structures are compatible in the sense of the following Proposition 3. If A is a vector space which is an algebra and a coalgebra, then the following two conditions are equivalent:

d:A K are algebra homomorphisms. .40 A and e: A (ii) m : A® A A and ïj :DC —) A are coalgebra homomorphisms. Proof. By the above definitions, the assertions of (i) mean that ziom = m.4..4 0(460.6), zion = r1A0A, corn = niKo(e0e),

E077

= r/K 7 (23)

12

1. Hopf Algebras

while the statements of (ii) can be expressed as ,60m = (mom)Q6 A0A , EA0A = Eom, 46on =

( non )046 x , 6K =

eon. (24)

The right hand sides of the first relations of (23) and (24) are both equal to (m m) o (id 0 T 0 id) o (.6 ,6). The second equality of (23) and the third of (24) are equivalent. The two last equations both mean that e(1) = 1. Likewise, the third relation of (23) and the second of (24) are equivalent. (The reader is invited to express the relations of (23) and (24) as diagrams and to check their equivalence in this manner.) LI Definition 3. A bialgebra is a vector space which is an algebra and a coalgebra such that the conditions in Proposition 3 hold. That is, if A is an algebra and a coalgebra, it is a bialgebra if and only if for any a,b E A we have ,6(ab) = ,6(a)46(b), e(ab) = e(a)e(b), 46(1) = 1 01, e(1) = 1.

(25)

Let A and B be bialgebras. By a bialgebra hom,om,orphism of A to B we mean a mapping that is both an algebra and a coalgebra homomorphism. The vector space A 0 B endowed with the tensor product algebra and coalgebra structures becomes a bialgebra. There are three bialgebras A°P, Ac°P and A°P'"P which are obtained from A by taking the opposite of either the algebra or coalgebra structure or of both of them. That is, A°P has the opposite multiplication mAop and the comultiplication of A, A"P has the multiplication of A and the opposite comultiplication .6Acop , and °P'c°P carries the opposite multiplication rnAop and comultiplication .6Acop . (The reader may verify that the above structures are indeed bialgebras.) A linear subspace I of a bialgebra A is called a biideal if it is both a twosided ideal of the algebra A and a coideal of the coalgebra A. The quotient All of a bialgebra A by a biideal I is again a bialgebra with structures inherited from A. We now introduce two concepts of distinguished elements of a bialgebra A. The motivation for this terminology stems from Examples 6 and 7 in Subsect. 1.2.6. A nonzero element g c A is called group-like if A(g) = g 0 g. An element s A is called primitive if .6(x) = 1 + 1 0 x. Proposition 4. Let A be a bialgebra. Then the product of group-like elements is group-like. If x and y are primitive elements of A, then we have e(x) = e(y) = 0 and the element [z, y] := xy — yx is also primitive.

Proof. The first assertion is obvious. Let x and y be primitive elements. Since (id 0 E).6(x) = se(1) + e(x)1 = x by (16) and E (1) = 1, we obtain e(x) = O. Using the fact that .6 is an algebra homomorphism, we get .6(xy) = (x0 1 + 1 0 x)(y 0 1 + 1 0 y)

xy 01+x0y+y0s+ lOsy,,

A(yx)= ys 0 1 -Ey0s-ExOy+10yx, so that Lt ([x, y]) = Li(xy) A(ys) = y] 0 1 + 1 0 [s, y].

1.2 Coalgebras, Bialgebras and Hopf Algebras

13

By Proposition 4, the set of group-like elements of A is a semigroup with unit and the primitive elements of A form a Lie algebra with respect to the

commutator bracket [., • 1. Example 5 (The bialgebras O(IN(K)) and K[x i , x 2 , • • • , xN]). Let A --=-OWN (K)) be the algebra generated by the functions uii from Example 4. That is, A is just the polynomial algebra in N2 indeterminates --= 1,2, • - - , N. Then A becomes a bialgebra with comultiplication and The correspond0 uin; and E(u) counit determined by ,A(u2i ) = ing coordinate algebra of the diagonal matrices in MN (EC) is the polynomial algebra K{x i , x2 , • • • , x N ] in the indeterminates xi = u!. It is a bialgebra with comultiplication = xi 0 xi and counit E(xi ) = 1.

Ek

1.2.4 Hopf Algebras

The following notion is fundamental in quantum group theory. Definition 4. A bialgebra A is called a Hopf algebra if there exists a linear mapping S: A > A, called the antipode or the coinverse of A, such that —

mo(Soid)o

=noe=rrio(idOS)o.

(26)

Clearly, the equations (26) are just the requirement that the diagram A®A lid AOA

A

LI

---- AOA

IS 0 id

177 0 E

S 1m

A

m

A0A

is commutative. In the Sweedler notation (18), the relations (26) say that

E S(a(1))a(2) = e(a)1

(27)

where ,A(a) = a( 1 ) 0 a( 2). The equations (26) can be nicely interpreted by using the convolution product (21) of the algebra £(.4, A). Comparing the formulas (21) and (26) we see that (26) means nothing but that S is the convolution inverse of the identity mapping, that is, S* id = id *S=noe. Thus, a bialgebra A is a Hopf algebra if and only if the identity map of A is invertible in the algebra .C(A,A). Moreover, the preceding characterization implies that an antipode of a Hopf algebra is uniquely determined. The next proposition gives some important properties of the antipode. Proposition 5. The antipode S of a Hopf algebra A is an algebra antihomomorphism and a coalge bra anti-homomorphism of A. This means that S : A > A°P is an algebra homomorphism and S : A Â"P is a coalgebra homomorphism. That is, we have —

14

1. Hopf Algebras

S(ab) = S (b)S (a), a, b E A, 460S=To(S0S)o,6

and S(1) = 1,

(28)

and E0S=E.

(29)

Proof. The proof is a nice exercise in the use of the Sweedler notation. The reader who is unfamiliar with it might first try to carry out the proof itself and then compare it with the detailed presentation below. Using the Hopf algebra axioms and the facts that L and E are algebra homomorphisms, we compute

E S (b(i)e(b(2)))S (a (1)*(2)))

S (b)S (a)

.

s,b(1) s ,a( 1 ),) E _a(2)b(2),

=

E S(b(1)) ,9 (a(1))(a(2)b(2))(1) ,9((a(2)b(2))(2)) E S(b(o)S(a(1))a(2)b(2)S(a(3)b(3)) E S (b (1) )(e(a(1) )1)b (2) S (a( 2)b ( 3)) E a( 1 )

E(b (1) ) S a( 2 )b( 2) = S ab

This proves the first equality of (28). The second one follows at once from the relation mo (Sold) o,6(1) = e(1)1 combined with the facts that 46(1) = 1 01 and E(1) = 1. Similarly, we compute

E S(a) g S(a i ) = = =

E s(a (2) e(a (3) )) ® S(a i ) z_g (S(a(2)) z_g (S(a(2))

E(s(a ( 2) )

0 S(a(1)))(e(a(3))1 0 1) 0 S(a(i)))((a(3)S(a(4))))

s(a( 1 ) ))(a (3) O a(4))(S(a(5)))

E(s(a(2))a(3) E(e(a( 2) )1 0 S(a(i))a(3))((S(a(4))) E(1 0 S(a(1))a(2))G 6 (S(a(3))) E(1 ® E(a(1)) 1 )((S(a(2))) = which is the first relation of (29). The second relation is derived by

E(S (a)) = e (E ame(a( 2)))) = e

S(a( l))a( 2)) = e(e(a)1) = -

LI In the Sweedler notation the first equality of (29) can be rewritten as

E 5(a)( 1) O S(a)(2) = E s(a( 2) ) 0

(30)

1.2 Coalgebras, Bialgebras and Hopf Algebras

15

Let A be a Hopf algebra. Since the bialgebra A°P ,"P has opposite multiplication and comultiplication, the antipode S of A satisfies the relations (26) for A°P'c°P as well. Hence S is also an antipode of A°P'"P and the bialgebra Aop,cop is a Hopf algebra. However, for the bialgebras A°P and Ac°P to be Hopf algebras S needs to be an invertible linear mapping of A. Proposition 6. For any Hopf algebra A the following conditions are equivalent: (i) The antipode S of A is invertible as a linear mapping of A. (ii) The bialgebra A°P is a Hopf algebra. (iii) The bialgebra Ac°P is a Hopf algebra. In this case, the inverse S -1 of S is the antipode of A°P and A'P. Proof. (i)—(ii): The inverse 8 -1 of S is an algebra anti-homomorphism of as S is by Proposition 5. Hence, by formula (28) we have

E S' (a

2 )a( i ) =

=

E s-

A

(a(2))(S -1 o 51)(a(0) = S -1 (E S(a( 1 ))a( 2))

5-1 (e(a)1) = e(a)l

E a( 2)S -1 (a( i)) = e(a)1

for a e A. That is, S' satisfies condition (26) for the bialgebra A°P, so that it is an antipode of A°P. (ii)—qi): Let :5" denote the antipode of A°P . Since S is also an algebra anti-homomorphism of A, we obtain and similarly

,§S(a) =

S (E e(a( 2))a(0) =

e(a(208(a(1))

E(a( 3)(a( 2))),S'51(a( 1)) =

= E(a (2) (e(a(1) )1)

=

E a(3):5"(S(a( 1))a( 2)) =a

and similarly S,§(a)= a for a E A. Therefore, is the inverse of S. The equivalence of (i) and (iii) is proved in a similar manner.

111

Corollary 71 If a Hopf algebra A is commutative or cocommutative, then 52 = id. Proof. In either case we have A = A°P or A == A"P , so A°P or Ac°P is a Hopf algebra and the assertion follows from Proposition 6. 111

The antipode S of a Hopf algebra A is said to be of finite order if Sn = id for some n E N. The smallest such n is called the order of S. Otherwise, one says that S is of infinite order. By a result of D. E. Radford [Radii, the antipode of any finite-dimensional Hopf algebra is of finite order (so, in particular, S is invertible and A°P and Ac°P are Hopf algebras). Further, a result of R. G. Larson and D. E. Radford (see [LaRD says that for any finite-dimensional complex cosemisimple Hopf algebra (see Sect. 11.2 for this notion) we even have 82 = id. The latter solved an old conjecture of I. Kaplansky.

16

1. Hopf Algebras

We shall see later that it is often easy to check the Hopf algebra axioms for a set of generators, so the following simple fact appears to be useful. Proposition 8. Let Ag be a subset of an algebra A which generates A as K be homommphisms and A A and e : A an algebra. Let L : A A an anti-homomorphism of the corresponding algebras. If the S: A coassociativity condition (15) and the counit condition (16) (and the antipode condition (26)) are satisfied for elements in A g , then they are valid on the whole of A and hence A is a bialgebra (resp. a Hopf algebra).

Proof. In the case of zX and e the assertions follow at once from the fact that (46 0 id) 0 4 (id 0 460 0 4 (e 0 id) o 46 and (id 0 e)46i are all algebra homomorphisms. In the case of S it suffices to show that (26) holds for a product ab provided that it is valid for a and for b. In order to do so, we compute E S((ab)( 1 ))(ab)(2) =-- ES(a ( i) b( i ) )a(2) b(2) = ES(b( 1 ) )(S(a ( 1 ) )a (2) )b(2) S(b( 1) )(e(a)1)b( 2)

e(a)e(b)1 = e(ab)1.

E

The other equality of (26) is verified similarly. We close this subsection with a few general definitions and facts. Let A and B be Hopf algebras. A linear mapping (io : A --03 is called a Hopf algebra harnomorphism if it respects the Hopf algebra structures, that is, if (,c) is a bialgebra homomorphism and wo SA = Si30(p. Any bialgebra homomorphism between Hopf algebras is already a Hopf algebra hom.omorphism, that is, the condition (io o SA Sz3 o (io is then automatically fulfilled. A .Hopf ideal of A is a biideal I such that S(I) C I. The quotient A// of a Hopf algebra A by a Hopf ideal I is again a Hopf algebra.

1.2.5* Dual Pairings of Hopf Algebras Let us consider a finite-dimensional Hopf algebra A. Recall that by Corollary 2 the dual vector space A' is an algebra with respect to the multiplication f g(a) (f g g),6(a) E f (a( n )g(a( 2) ). For f C A' we define a functional 461(f) E (A® A)' by ZA(f)(a0b) := (f om)(a0b) f (ab), a, b G A. Since A is finite-dimensional, (A® = A' OA' and so LX(f) E Af ga41 . It is not difficult to show that the algebra A' equipped with the comultiplication IA becomes a Hopf algebra. The antipode, the counit and the unit element of this Hopf algebra A' are given by (S f)(a) = f(S(a)), E (f) == f (1) and 1A' (a) =e(a), respectively. That is, the Hopf algebra A' is obtained by "dualizing" the structure maps of the Hopf algebra A. This situation is generalized in Definition 5. A dual pairing of two bialgeb ras II and A is a bilinear mapping K such that x A (.,.) (Au(f)1a1

a2) = (.f, 040, 2)7 (.6.f2la) =

0 f21 ,6A(a)))

(31)

1.2 Coalgebras, Bialgebras and Hopf Algebras

(f, 1 4) = eu(f),

(1u,a)

17

(32)

e.4(a)

for all f, fi, f2 E LI and a, a l , a2 E A. A dual pairing (•,.) of 14 and A is called nondegenerate if (f, a) = 0 for all f E 14 implies that a = 0 and if (f,a) = 0 for all a E A implies f = O. Proposition 9. If (•,-) is a dual pairing of two Hopf algebras 14 and A, then we have (S(f), a) = (f,S(a)), f E U, a E A. (33)

Proof. Consider the linear functionals F, G and H on the tensor product Hopf algebra Li A defined by F(f 0 a) = (f,a), G(f 0 a) = (S(f),a), H(f 0a) =(f,S(a)), f E 14, a E A. We show that FG = HF =e in the algebra (LI 0 . Indeed, by (31) and (32), we get

FG(f 0 a) =

E(f(1), a(1))(S(f(2))1a(2))

= (E(f)1,a) = e(f

E(f(i)s(f(2)),

a)

a),

that is, FG = e 7 and similarly HF = e. Thus, G = (HF)G = H(FG) = H which yields (33). El

The discussion preceding Definition 5 has shown that for any finitedimensional Hopf algebra A we have a dual pairing of the Hopf algebras := A' and A defined by (f, a) := Pa), a E A, f E A'. Obviously, this pairing is nondegenerate. Let us discuss the above notion with some remarks. First note that in equation (31) the pairing of 14 and A must be extended to one oft' 014 and A® A by setting (fi.o f2, al a2) := (h, ai)(f27a2). Dual pairings of two Hopf algebras are by no means unique. For any couple of Hopf algebras 14 and A there is always the trivial pairing defined by (f, a) := eu (f)eA(a). Further, compositions with Hopf algebra homomorphisms in both variables again give a dual pairing. Let •) be a dual pairing of two Hopf algebras 14 and A Then Iu E (f,a) = 0 for all a E AI and IA := {a E A (f,a) = 0 for all f E /41 are Hopf ideals of 14 resp. A and the pairing (., •) oft' and A induces a nondegenerate dual pairing of the quotient Hopf algebras /4///,/ and Ai/A. From the formulas (31) we see that any dual pairing of two bialgebras is completely determined by the values of the bilinear form (., •) on sets of generators of the algebras 14 and A. This allows one to use shorthand descriptions of pairings by means of matrices. A nondegener ate dual pairing of two Hopf algebras 14 and A stores a lot of information about the links of both structures. (For instance, it follows then from (31) that Li is commutative if and only if A is cocommutative.) In quantum group theory there is a general philosophy which says that then the Hopf algebras LI and A provide dual approaches to the same "quantum object".

18

1. Hopf Algebras

1.2.6 Examples of Hopf Algebras

Three important examples of Hopf algebras related to groups have already been developed in Sect. 1.1. Two others will be described below. The following example is the starting point for our first approach to quantum groups. The quantized universal enveloping algebras studied in Chaps. 3 and 6 are deformations of the Hopf algebras U(g) which are universal enveloping algebras of Lie algebras. Example 6 (The universal enveloping algebra U(g)). Let g be a Lie algebra over K with Lie bracket [., •]. The universal enveloping algebra U(g) is defined as the quotient algebra of the tensor algebra T(g) of the vector space g by the two-sided ideal I generated by the elements x0y—y0x— [x y], where , y E g. That is, U(g) is the free algebra generated by a vector space basis of g subject to the relations [x, y] =-- xy — yx for x, y E g. By its very definition, the algebra U(g) admits the following universal property: given a linear mapping (to : g A of g to another algebra A satisfying

(P(410 (Y)

'10 (Y)40 (x),

xl Y E 91

(34)

there is a unique algebra homomorphism : U(g) A such that (;(3(x) = (p(x) for x E g. This fact is used in proving the following assertion. There is a unique Hopf algebra structure on .6(x) = x 0 1 + 1 0 s,

E(x) = 0,

the algebra U(g) such that S(x) = —x, s E p.

(35)

In order to prove this, we define linear mappings (tc) = 46, E and S of g to the algebras A = U(g) U (g), K, U(g)°P by (35). Condition (34) is obviously fulfilled for the mappings E and S. One verifies (by arguing as in the proof of Proposition 4 above) that d also satisfies (34). Therefore, by the universal property of U(g), these mappings extend to algebra homomorphisms of U(g) to A, denoted still by d, E and S. By Proposition 8, it suffices to check the Hopf algebra axioms on the generators x E g. This is easily done; as an example, we have m o (S 0 id) o d(x) = m(--x 0 1 + 1 0 x) = 0 = e(x) for x E g. By (35), any x E g is a primitive element of the Hopf algebra U(g). In the case K = C it can be shown (see [Bou2]) that the elements of g exhaust all primitive elements of U(g), so the Lie algebra of primitive elements of U(g) is just the Lie algebra g itself. In particular, this gives an intrinsic characterization of the Lie algebra g in terms of the Hopf algebra structure of U(g). From (35) we see that the Hopf algebra U(g) is cocommutative. Let B be either the Hopf algebra Rep (G) of Example 2 for a connected compact Lie group C or one of the Hopf algebras 0(G) of Example 3. Suppose that K = C and g is the Lie algebra of the Lie group G. It is well-known that each element a E U(g) acts as a left-invariant differential operator à on G. If a = x1x2 • - • xn with xl, x2, • • • , x E g, the action of ;a,' on a function f E G'(G) is given by

1.2 Coalgebras, Bialgebras and Hopf Algebras

19

an f)(g) =--

ati

f • • • atn,

exp(tixi) • • • exp(tnxn )).

(36)

ti =•••=tn =o

One can show that there is a nondegenerate dual pairing of the Hopf algebras U(9) and 13 determined by (a, f) := (a n(e), where a C U(g) and f C B. A Example 7 (The group algebra KG). Let KG be the vector space with basis given by the elements of G. It becomes an algebra by extending linearly the product of G. The unit of this algebra is the unit element of G. There is a unique Hopf algebra structure on the algebra KG such that A(g) = g 0 g, e(g) = 1 and 8(g) = g-1 for g E G. The elements of G are obviously group-like. If K = C, it is easily checked that they exhaust all group-like elements of the Hopf algebra KG. The Hopf algebra KG is cocommutative. Let A denote one of the Hopf algebras .F(G), Rep (G) or 0(G) defined in Examples 1, 2 and 3, respectively. Then there is a nondegenerate dual pairing (,•) of the Hopf algebras A and B := KG defined by (f, Ei Ag) f (g i ) , where f E A and Ei Ag iE B. Example 8 (The tensor algebra T ( T)). Let V be a vector space and T(V) the tensor algebra over V defined in Subsect. 1.2.1. Arguing similarly as in Example 6 (replacing thereby the universal property of U(g) by that of VV. )), it follows that there exists a unique Hopf algebra structure on T(V) such that A(v) = v 1 +1 0 v, e(v) = 0 and 8(v) = -v for v E V. This Hopf algebra is cocommutative. For arbitrary elements v1 , v2, - • • , vn E V we have

A(v i • - • Vn ) =

EE

vp ( i ) •••

7)0)

O Vp(k+i)

Vp(n)

7

(37)

k=0 PEPnk 8(V1V2 • • • Vn) = ( - 1) n Vn • " V2V1.

(38)

denotes the set of all permutations of {1, 2, • • • , n} such that 7)(1) < •• • < p(k) and p(k+1) < • • m. There is a close analogy between the usual binomial coefficients and their g-analogs. Many identities of the usual binomial coefficients have their counterparts for the g-binomial coefficients. For example, [n + 1 1 in

q = [m n- 1 ] q

+e

[n rn

= q n- m+1 [ n L m - 1]

Proposition 2. If v and w are noncommuting variables satisfying the relation vw = qwv, then we have (V ±

w)

n

m] m n-m V

[n

E

[ n1 m=0 LM,

V

Tri

W

7/ 77Z

(18)

1

If q is a primitive p-th root of unity and p is odd, then (v + w)P = vP +w 7 .

Proof. Formula (18) can be proved by induction on n using the first identity in (17). The second assertion follows from (18).

40

2. q-Calculus

Proposition 3. For 0 1, then using (12) we can transform (21) into the series zol(a', ; ; q 1 , ) with base q - 1

2.1 Main Notions of q-Calculus

41

The function defined by the series (21) is called the basic hypergeometric function (or q-hypergeometric function) 2 (p 1 • It can be analytically continued outside of its domain of convergence. More general basic hypergeometric functions are defined by the series rçOs(ai, a2, • • •

lat.;

bl, b2, • • •

60 .9

b$ ; q3

b2, • • • , bs '

On(a2;q)n -..( a r ; On (( l)nen-1)/2)1+8

(b1;072(b2;

• • -

n (q Z; q)n,

(22)

;

In particular,

rçor-i(ai,a2, - • • ,ar;b1, b2, " ,br-i; q, z) cc n Z (aii q)n(a2; On • (ar; •• -On (br-1; (q; nt2 -0 (bi; q)n(b2;

(23)

Again, if at least one of the numbers al, a2, • • , a,. is a negative integer, then the series (22) terminates. If 0 < II < 1, then the series (22) converges absolutely for all z if r < s and for all lzi < 1 if r = s + 1. It also converges absolutely if q > 1 and z < Ib1b2 • • • b3 1/laia2 • • • ar k If the series (22) does not terminate, then it diverges for z 0 if 0 < II < 1 and r > s + 1 and if iql > 1 and izi > 1b1b2 • bsi/laia2 • ad. 2.1.4 The Function içoo(a; q, z)

By (23), we have

oc (24) n=0

(TOn

From the expressions for i wo(a; q, z) and i(po(a; q, qz) we obtain ivo (a;q,z) i (po(a;

- vioo (a;

q l qz) = (1 —

a)Z

q, z) - a “po (a; q, qz) = (1 -

viagaq; q, z), i ço0(aq; q, z).

These relations yield the equation (1 - z) i wo(a; q, z) = (1 - az) “po(a; q, qz) and so (az; , ) i(poo,; q, qn z). (25) iwo(a; q, z = (z; qn If Iqi < 1, then i çoo (a, q, qn 1 when n cc. Therefore, by (25), 00 00 (a; q) ri 1 azqr (az, 0 00 z = (26) içoo(a, q, 1 zqr (z; q),, (q; n=0 r=0 when Iql < 1. The latter formula has some interesting corollaries. One is

1.400(a; q l z)vpo(b; q, az) = içoo(ab; q, z).

(27)

Formula (26) implies that 1(10 0( 1 ; q7 z) = 1. Therefore, by (27),

içoo(a -i ; q, az) = liwo(a; q, z)} -1 .

(28)

2.1.5 The Basic Hypergeometric Function 2 (1, 1 For many reasons it is useful to have formulas relating hypergeometric functions with different parameters. For the function 2 cp 1 (a, b; c; q, z) the transformation formula

q)(az; 2(P i(a,b; c; q, z) = (c; q)„(z; q)„

„

(29)

holds when WI < 1, zI 1 and lql < 1. Carrying out the q-integration by parts we get

L

oc

xv -1 Eg (- (1 - q)s)dgs xv = — Eg (- (1 - q)s) -F

[[]] v

r svEq(_(,_ q)s)dqs.

[[v]] o

(59)

From the equality eg (-x)E g (s) = 1 we derive Eg (-x) -) 0 when s +oc. Hence the first summand on the right hand side of (59) vanishes and we get

fg (v + 1) = pErg (v). In particular, if u = n E N, then fq

(n ) =

- 111! = (q; q)n-i

If F(u) is the "ordinary" gamma function, we have rg (v) q / 1. The q-gamma function can also be represented as

Fg (v) = (1 - 0 1-11 (Qq)c°

(q ,q) 00

(60) (I.) when

(61)

where WI < 1. For 1,71> 1 we define fg (v) := For the study of Clebsch-Gordan coefficients of the quantum algebra Ug (512) it is convenient to use the modified q-gamma function given by

î'q(u)

q-(1)--1)(1)-2)/2r42(v).

(62)

This function has the properties î(v) = Aig (v), î'q (n) = [n - 1 ]! , n E N, and Pg (v -I- 1) = [v]fg (v). (63)

2.3 q-Orthogonal Polynomials By means of formula (63) one can show that if y

Pq (v +1)/Pq (v) Consequently, for y

E

Pq (y - n + 1)

E

N, then

-Pq (m + 1)/ 1 q (m).

N, and n

Pq (v + 1)

-m, in

49

E

N we have

( 1) n

Pq (m + n) Fq (m)

(64)

The Stirling formula describes the asymptotic behavior of the classical gamma function. Its analog for the q-gamma function is

pq(v ± 1) q -v(v-i)/2 (1 _ q2\- exp(-Cq )

if

v

+cc,

(65)

where 0 < q < 1 and Cq = EZ`L o ln (1 _ 2(k +1)\) For the quotient of q-gamma functions one has the asymptotic formula

Pq (v

+ a)

q2u-s)/2(1

Pq (y)

2.3 q-Orthogonal Polynomials q-Analogs of classical orthogonal polynomials play an important role for the study of representations of quantum groups and q-oscillator algebras. In this section we consider some important q-orthogonal polynomials in the case when q is a real number. 2.3.1 Jacobi Matrices and Orthogonal Polynomials

Orthogonal polynomials satisfying three-term recurrence relations are connected with linear operators which can be represented by Jacobi matrices. Let {a n c N0 } , { bn I n c No }, {cn I n E No} be complex sequences such that a n, 0 for n c No . To these sequences we associate a linear operator L acting on a Hilbert space with orthonormal basis in), n = 0,1,2, • • by

(66)

Lin) = anin + 1) + bn in) + cn in - 1),

where I - 1) := 0. Suppose that lz) is an eigenvector of L for the eigenvalue z, that is, Liz) = ziz). Then lz) can be written as lz) = E:L 0 pn,(41n) with certain coefficients pn (z). From (66) we get E(pn(z)anin + 1) +p(z)b(n) ±pn(z)enin — 1)) = z Epn( )1 )• n=0

n=0

Equating coefficients of the vector In) we obtain the following three-term recurrence relation for the coefficients pn(z):

cn+1Pn+i(z) + bnpn (z) + an-iPn-i(z) = zpn (z).

(67)

Let us set p_1(z) = 0 and po (z) = 1. Then (67) determines the coefficients Pn(z) uniquely and each p(Z) is a polynomial in z of degree n. Suppose now that the sequences (a n ) and (bn ) are real and en = an _i for n E N. Then L is a symmetric operator. By Favard's theorem, there exists a nonnegative Borel measure t on the real line such that the polynomials pn (z) satisfy the orthogonality relation Pi (A)Pk(A)C 111 (A)

5j k

,

, k = 0 ,1 , 2 , - • -

(68)

One can prove that the symmetric linear operator L always admits a selfadjoint extension, say L, in the Hilbert space .55. If E(.) is the spectral measure of L and WO = (0 I E(A) 10), then (68) holds. If the closure E of the operator L is self-adjoint, then any measure i satisfying (68) is uniquely determined, that is, we have a unique orthogonality relation for the polynomials p72 (z) This is in particular true if the symmetric operator L is bounded. In this case the measure rt has compact support. If the closure of L is not self-adjoint, then L has infinitely many self-adjoint extensions and to every such extension there corresponds a nonnegative Borel measure tt such that (68) is valid. Special cases of the recurrence relation (67) lead to the so-called qorthogonal polynomials. 2.3.2 q-Hermite Polynomials

The recurrence relation 2xp n (x) = Pn+ (x) + (1 — q 7 )Pn_1(x)

(69)

determines the continuous q-Hermite polynomials pn (x) = 117,(xlq). They are given by Hn(Xlq) =

(qi

E (q; q)k(q; q)n-k k=0

e î(n_ -2k)01

where x = cos O. The generating function for these polynomials is Rze io ;

00

001 -2

Ho. -

2x zqn

Hn (x1q) z n

z 2 q 2n ) - 1 = n=0

n=0

7

(q;

where again x = cos 9. If —1 < q < 1, then the polynomials lin (xlq) are orthogonal with orthogonality relation 00

L1 '

1 n (x1q) 11,n(xlq)

(I(1 — 2(2X2 i) qk

q2k)) (1 x2)-1/2dX

k=0

nair(qi

On/(q; Om

•

2.3 q-Orthogonal Polynomials

51

The corresponding self-adjoint operators are related to the position and momentum operators for the q-oscillator algebra (see [ChK] and [K1S]). The polynomials 1-1 7,(xiq) := i- nIIn (ix1q) ,

(70)

i :-= -\/-_,

satisfy the recurrence relation

2xl-t n (siq) = hri+i(xlq) + (qn - 1)h, _ 1 (xlq) .

(71)

For 0 < q < 1, the orthogonality relation for these polynomials is

I

(-1)n(q,q),„a (logq -1 ) 6mn " hn,(sinhulq -1 )h,,(sinhulq -1 ) du --= (q^ 1. q• - 1);. 1 00 (_ qe211. ; 0 00 (_ qe -2u ; 0 00 .

.7

.

The discrete q-Hermite polynomials Hfl (s; q), n c No, are defined by the recurrence relation xpri (x) = Pn+i(x) + q n (1 - qn )pn _ 1 (x).

(72)

In explicit form these polynomials are given by [n/2] ,

,

tq; q),(-1)k q k(k- 1)

I In (x; q) ,-

5

( q2 ; q 2 )k(q;

k=0

n-2 k

7

On-2k

where [n/2] denotes the integral part of n/2. For 0 < q < 1, the orthogonality relation for the polynomials Hn (s; q) is of the form in ( q2j+2 ; 0 00 ( q; q2) 00

00

EIH,n ( qi ; q)11,(q1 ; q) + 11,(-qi ; q) Hn (-qi; OP2

J=1

(q2 ; q2) 00

+1/2 (q; q2 ) 00 11„,(1; q)H„(1; q) = qn(n-1) I 2 (q; On6Trin , that is, these polynomials are orthogonal on the set s = +qi, j E No.

2.3.3 Little q Jacobi Polynomials -

The little q- Jacobi polynomials pfl (x; a, blq) are expressed in terms of the qhypergeometric function

2(P1

by

Pn(x; a, big) =---

2 (P1(q -n ,

abq n+1 ; aq; q, xq).

(73)

One can easily verify that lim q,i pn ((1 - x)12; q&, le lq) = cP4'3) (s), where c is a constant and P4c1'3) (s) is the classical Jacobi polynomial. The polynomials pn (s; a, blq) satisfy the recurrence relation

xpn (s) = AnPn+ 1 (s) where

(An

+ Cn)Pn(S) + CnPn--1(s),

(74)

An, = (—qn )(1 - abqn +1 )(1 - aqn +1 ){(1 - aben +1 )(1 Cn = (-aqn)(1 - q 71 )(1 - bqn ) 1(1 abq 212 )(1 - abq 2n+1 )} -1

The orthogonality relation for the little g-Jacobi polynomials is 00

Epm (qx ; a, blq)p n (qx ; a, bfq)

(bqi q)x

b mn

(aq)s

(q; q)x

o

(75)

hn (a,b; q)'

for 0 < q < 1, where hn (a , b; q) =

(aq\

_n (abq; q) n (1 - abq 2n+ 1 )(aq; q) n,(aq; q) c„,

)

(q; q)(1

abq)(bq; q) n (abq 2 ; q)„.„

That is, these polynomials are orthogonal on a discrete set of points. 2.3.4 Big q-Jacobi Polynomials

Let a,

c, d be fixed complex numbers.

—n 13 7(ler'f3) (X; el dig)

q

32

The functions

a-OA-n-1-1 cx+1 1

_ qa+ldie XI i

c q-07)

n G No,

are polynomials in z , called the big q- Jacobi polynomials. If a, c, d c No, then they satisfy the orthogonality relation

G

(76) R and

fc d

' 3) (x; c, dlq)(qx I c; q),(-qx I d; q)ocl q x = hn6 Tim) 13,,,( a J3) (x; c, dlq)P4" (77)

where 20

hn =

13-1-1. ; q ) a±i ( q; q )n (1

q a-1-13-1-1)( q0-1-1 ; O n (_ q0-i-l ci d;

(

c(1 q)(q; q)

x{(1

( -d/ c; q)

1 (-qc1 d; q) (qa+0+ 1 ;

qa-143-1-2n+1)(qa-1-1; on(_qa-1-1dic; 00-1.

Thus, the big g-Jacobi polynomials are also orthogonal on a discrete infinite

set of points.

2.4 Notes The expressions (a; g) n for q G N have been known since the time of L. Euler. They have been used in combinatorics, Galois theory and finite geometry. The hypergeometric g-series 2 (09 1 was invented in the nineteenth century by E. Heine. The g-integral was introduced and studied by F. H. Jackson [Jki at the beginning of the twentieth century. A good exposition of the theory of basic hypergeometric functions and g-orthogonal polynomials can be found in the book [GR]. A nice survey on q-special functions is given in [Ko5].

3. The Quantum Algebra Uq (512 ) and Its Representations

This chapter is devoted to a detailed study of the Hopf algebra Uq (512 ). It can be considered as a one-parameter deformation of the universal enveloping algebra U(s12) (see Example 1.6) and is the simplest example of the quantized universal enveloping algebras Uq (g) which will be investigated in Chap. 6. Following common terminology in physics, we call Uq (s12 ) a quantum algebra.

3.1 The Quantum Algebras Uq (512 ) and Uh(s12) 3.1.1 The Algebra Uq (512 )

Let q be a fixed complex number such that q O and q2 1. We denote by Uq (s12 ) the algebra (that is, the associative algebra with unit) over C with four generators E, F, K, K -1 satisfying the defining relations 2, —2F7

1

KK -1 = K -1 K =1, KEK -1 =q 2 E, KFK -1

(1) (2)

[E, F] = KK-1

It can be shown by induction that the relations (1) and (2) imply the formulas [E, Fm]

[En , E'2 Fm

EFT'

E = [m]Fm -1 [K ;1 - m ] ,

a En F - FEn = [n]En- 1 [K; n - 1],

( 3) (4)

[n]![mj!

v-•

[r]![n - 7]![m - 71! Fm—rEn—r X [K; n

nt][K; n m 1] - - • [K ; n m

where we set [a] : = [ajg and [K; -

:= (Kqn

-

-

+ 1] n m ,

K -1 q - n)/(q

-

(5)

q- ') for n c Z.

Proposition 1. The set {F1 Km En I m E Z, 1,n E No}, as well as the set {E n K n I m c Z, 1,n E Nob is a vector space basis of Uq (s12). Proof. It follows from (1) and (2) that Uq (512) is spanned by the monomials Fl KmEn , m E Z, 1,n E No. To prove that they constitute a basis of Uq (812) we consider the polynomial ring C[x, y, z, z -1 ) in commuting indeterminates

54

3. The Quantum Algebra Uq (s12) and Its Representations

x, y, z, z -1 satisfying zz -1 =-- z -l z =-- 1. Define linear transformations Liand C —1 of C[x,y,z,z -1 1 by k( x r z S yt) = tj2r xr zs -1- l yt

f'(xr zs y t ) = xr+ 1 z'y t , k( sr z s y t ■) q—28 r s xzy

—1 t-I-1

"

q-1

'J r

z

s yt .

Direct computations show that these transformations satisfy the relations (I) )?, X = E,F,K, K -1 , determines and (2). Therefore, the assignment X a representation of Uq (512) on C[x,y, z, z -1 ]. Since the monomials xrzsyt, s E Z, r, t E No , form a basis of C[x, y, z, z -1 ] and PlfCmkn(1) = xl zmyn , the operators Pi km Pn 771 G Z, 1, n E No, are linearly independent. Thus, the set {Fi Km En m E Z1 1, n E No} is a basis of Uq (512 ). The second assertion is proved similarly. Using Proposition 1 one can show that the algebra Uq (512 ) has no zero divisors, that is, we have ab 0 in Uq (s1 2 ) whenever a 0 and b O. Proposition 2. The quantum Casimir element

Cg := EF +

K' K - ' (q

(21 -

1)2

= FE +

Kq+

(6)

lies in the center of the algebra Ug (s12). If q is not a root of unity, then the center of Ug (s12 ) is generated by Cg . Proof. The second assertion is proved in Sect. 6.3. For the first assertion it is sufficient to verify that Clq commutes with E, F, K and K -1 . Using the relations (1) and (2) this is easily done.

Proposition 3. There exist automorphisms 0 and v = ±1, of the algebra Ug (s1 2 ) such that

a

CVO}, Ti G Z,

0(E) = F, 0(F) = E, 0(K) = K -1 , 11-2n /9,, n ,„(K) = ct,n,v(E) = aKnE, 19,„(F) = va - 1 Proof. First one shows by direct computation that the images of the generators under 0 and 19„, n,, again satisfy the relations (1) and (2). As a sample, we check that t9 t9,, n,„ preserves the relation (2) by computing

19(E)19(F) -19(F)19(E) = v(KnEK - Tiq -2n)F v(K'FKnq -2n)E = v(EF - FE) = (q q -1 ) -1 09(K) -19(K -1 )). Hence there exist algebra homomorphisms 0 and t9 cle ,n,v given by the above formulas on the generators. Since 0 o 0 = id and t9 n, 11 o = id, these homomorphisms are actually automorphisms.

If q is not a. root of unity, it can be shown (using the second assertion of Proposition 2) that the group of automorphisms of the algebra Uq (512) is (see [A1C1). generated by the automorphisms 0 and

3.1 The Quantum Algebras Uq (s12) and Uh(s12)

55

3.1.2 The Hopf Algebra Uq (s12 ) A key property of the algebra Uq (s12 ) is that it carries a Hopf algebra structure. Proposition 4. There exists a unique Hopf algebra structure on Uq (512 ) with comultiplication 4 counit e and antipode S such that

46(E)=E0K+10E, 46(F) F 01 + K-1 0 Fl ,6(K) = K

K7

51(K) =•- K -1 1 8(E) = -EK -1 , S(F)= -KF, e(K) = 1, e(E) = e(F) = O. Proof. First we extend 6 to an algebra homomorphism of the free algebra C(E Fl K 7 K -1 ) to U(s1 2 ) Uq (s12 ). In order to prove that 46 passes to the quotient algebra Uq (s1 2 ) of qE,F,K,K -1 ) it suffices to show that 46 preserves the relations (1) and (2). We shall do this in the case of (2). For the relations (1) the proof is similar and simpler. We have ----- K -1 EF F E

EK -1 ICF + EF ®

-K -1 0FE-K -1 E0FIC-F0E-FE0K K -1 0 (K - K -1 ) ±(K - K -1 ) K

--= {46(K) - 46(K -1 )11(q-

It is proved similarly that 8 extends to an anti-homomorphism of Uq (s12) and

e to an algebra homomorphism of Uq (s1 2 ) to C. By Proposition 1.8, it suffices to verify the Hopf algebra axioms on the generators. For instance, we have (60id)0,6(E)=10 1® E + 10E0K+EOKOK=(id0,6)06(E). The other conditions are checked in a similar manner. El Definition 1. The algebra U(s12) endowed with the Hopf algebra structure from Proposition 4 is called the quantum algebra U q (512)•

Some useful formulas about the Hopf algebra Uq (s1 2 ) are collected in Proposition 5. In the Hopf algebra Uq (s12 ) we have

82 (a) = KaK-1 for all

a E Uq( 51 2),

( 7)

46(E1 KmFn) = I n 1 = E

q r0-8(n-s) []![np(1EI_ rKmn+8F8 0 Er K i-r±m Fn-s

r=0 .9=0

[r]![l - ri![s]![n - s]!

S(Fn K m El ) = —1) 1+n q 1(n-1)-1(1-1) E1 10 -1-m Fn

(8) (9)

Proof. Since 52 is an algebra homomorphism of Uq (s12) by Proposition 1.5, it suffices to check the equality (7) on the generators. We have 82 (E) = --S(EK -1 )= -S(K -1 )8(E)= KEK -1 and similarly 52 (F) = KFK-1.

56

3. The Quantum Algebra

Uq (812)

and Its Representations

By induction one proves the formulas for ,(El) = ,A(E)I and z1(F) = z 1 (F) n using that ,A is an algebra homomorphism. The formula (8) is obtained by multiplying the expressions of 44E1 ) 7 4A(K m ) = K rn ® Km and ,(1'12 ). The relation (9) follows from the expressions for 5(F), S(K) and 5(E) and 0 the fact that S is an algebra anti-homomorphism.

The next proposition deals with Hopf algebra isomorphisms of Uq (s12)• Proposition 6. (i) For any a E CVO}, there is Oa of Uq (512) such that

a Hopf algebra automorphism

V a (E) = aE, IMF) = a-1 F, 19 a (K) = K. Any .Hopf algebra automorphism of Uq (s12 ) is of this form. (ii) Let a be a nonzero complex number. There exist Hopf algebra isomorUq -1(s12) and Oa : Uq (512) phisms (p c, : Uq (s12 ) U_ q (512) such that

çoa (E) = aFK, ço a (F) =

A

, cp c,(K) = K,

zPa(E) = aE, Oa (F) = -a-1 F, 11) 0 (K) = K. Any Hopf algebra isomorphism (io : Uq (512 ) Uq -i(s12 ), resp. Uq (512) U_ q (512 ), is of the form p = cpa, resp. 11) (iii) Two Hopf algebras Uq (512 ) and Up (s12 ) are isomorphic if and only if q is equal to p,-p,p - 1

Proof. That the mappings defined above are indeed Hopf algebra isomorphisms is easily verified on the generators. We now prove the converse assertions and begin with two preliminary results. The first one is that the set of group-like elements of Uq (512 ) is just the set of powers Kn , n E Z. Put K ±1 }. The second one is that .Fq± := E Uq (s12) 0(x) = 1 x x Lin{E,FK,1-K} and = Lin{1-K -1 }. Both results are derived from formula (8) by writing an arbitrary element x E Uq (512 ) as a linear combination of basis elements El K m Fn and comparing coefficients in the expression of ,A(x). It is clear that the set of group-like elements and the sets .Fq± are preserved under Hopf algebra isomorphisms. Now suppose that V) : Uq (s12 ) Up (s12) is a Hopf algebra isomorphism. Since the image CIO of the group-like element K is again group-like, the first result above yields that CK) =--- K n for some n E Z. Similarly, 0 -1-(K) = Km,m E Z. Hence we have CIO = K or 11)(K) = K -1 . If ip(K) = K -1 , then we have C.F:7E) = .F77, . But the latter is impossible, since dim 3 and dim ,Fp- = 1 by the second result. Thus, IP(K) = K. Next we note that the map Ks1C -1 leaves the space .F4,-» invariant and has eigenvalues 1, q 2 , 47 -2 with eigenvectors 1 - K,E,FK, respectively. Therefore, we have either q2 = p2 and CE) --= aE,1,1)(EK) = #FK or q2 =7;1-2 and CE) = aFK,O(FK) = OE for some a, E C. Hence q is equal to p, -p,p -1 or -p-1 . This completes the proof of Now suppose that q = p. Then, as shown by the preceding, tk(K) = K 11;(E) = ûE and

3.1 The Quantum Algebras Uq (512) and Uh(s12)

57

0(FK) = OFK and so 0(F) = OE By (2), 3= eci and hence 1/, = which proves (i). The last assertion of (ii) follows similarly. There is another important variant of the quantum algebra Uq (s12 ) which often occurs in the literature. It is defined as follows. Suppose that q 4 0, 1 and let 6q (512) denote the algebra over C with generators E, F, K K -1 subject to the relations

KK -1 = K -1 K =1, KEK -1 = qE, KFK -1 = q -1F, K2

[E,F]=

(10)

_ K -2

1• g - g-

(11)

Arguing similarly as in the proof of Proposition 4, it follows that this new algebra Oq (512 ) becomes a Hopf algebra with structure maps given by 46(K) = KoK, d(E)=E0K+K -1 0E, d(F)= FOK+K -1 0F, (12)

S(K) = K -1 , 8(E) = -qE, 8(F) = -q -1 F, s(K) =1, E(E) = e(F) = 0. (13) An advantage of the Hopf algebra Ûq (512 ) over Uq (512 ) is that the comultiplications of the generators E and F are given by the same formulas. Using the above formulas and Proposition 1, one can show that there is an injective Hopf algebra homomorphism ço : Uq (s12 ) -> t-71 q (s12 ) determined by cp(E) = EK, (p(F) = KF and AK) = K2 . If we identify Uq (512) with its image under ço, then Uq (512) becomes a Hopf subalgebra of Ùq (512). Note that the algebra t/q (512 ) has four one-dimensional representations (sending K to 1, -1, i, -i and E, F to 0), while Uq (512) has only two such representations. Hence the algebras Uq (512) and 0q (512) are not isomorphic. 3.1.3 The Classical Limit of the Hopf Algebra Uq (512 )

In this subsection we shall discuss how the enveloping algebra U(s12) is related 1 of the Hopf algebra Uq (s12). Recall that U(s12) to the classical limit q is the algebra with generators x+ ,x_,y and defining relations [y, x+ ] = 2x+ , [y, x_1 = -2x_, [x+ , x_} = y and that the Hopf algebra structure of U(s12)

has been described by Example 1.6. Let us formally write q = K = ehH and consider the limit h -> 0. Then the relations (1) and (2) imply (by differentiation at h = 0 resp. by taking the limit h 0) that [H, = 2E, [H, F] = -2F and [E, F] = H. That is, we obtain the relations of U(s12). Similarly, the Hopf algebra structure of U(s12) is recovered from that of Uq (s12) as h -> 0. However, this viewpoint is not quite adequate if we want to compare the theory of Uq (s12) (at least for q not a root of unity) with the classical situation. In order to develop the proper setting we first give a slight reformulation of the definition of Uq (512) that applies also in the case q2 = 1.

58

3. The Quantum Algebra Uq (s12 ) and Its Representations

Clearly, the elements E, F, K, K -1 and the algebra Uq (s12) satisfy the relations

-1(K K-1) o f

G := (q -

[G, = E(qK + q -1 K -1 ) , [G, = -(qK + q -1 K -1 )F, [E,

=G, (q - q -1 )G = K - K -1 .

(14) (15)

Let tIq (s12 ) denote the algebra with generators E, F, K, K -1 , G and relations (1), (14) and (15). The algebras U(s12) and Uq (512), g 2 1, are isomorphic (by mapping E -) E, F -) F, K -) K and G -) (q - q -1 )(K - K -1 )). The Hopf algebra structure of Uq (512) induces a Hopf algebra structure on Oq (s12) which is given by the formulas stated in Proposition 4 and the relations A(G) = G K + K -1 0 G, E(G) = 0 and S(G) ----- -G. It is easy to verify that these formulas endow the algebra Qs12) with a Hopf algebra structure also in the cases q = +1 which were excluded for Uq (s12). We consider the Hopf algebra Oi(s12) as the classical limit of the Hopf algebra tIq (s12)• The two algebras U1 (s12) and U(s12) are closely related to each other. That is, there are canonical algebra isomorphisms

01 (512 ) U(s12 ) CZ2 and U(s12 )

Oi(s12)/(K - 1).

Indeed, putting q = 1 into (1), (14) and (15) we see that K belongs to the center of 01(s12) and that K 2 = 1. Thus the defining relations of the algebra 01 (512) read as [E, = G, [G E] = 2E K and [G, .= -2F K Hence there exists an algebra isomorphism of Oi (s12) and U(s12 ) CZ2 which maps E x + x, F x_ and G -4 yx. Here x is the generator of the two-dimensional algebra CZ2 such that X2 = 1 and we have simply written ab :=- a b for a E U(s12 ) and b E CZ2. That is, the algebra Oi (s12 ) is an extension of the algebra U(s12) by a single central element x satisfying X2 = 1. The appearance of this additional generator x explains why we later have two irreducible representations in any dimension when q is not a - root of unity. Finally, we describe the Hopf algebra structure on U(s12 ) CZ 2 that is induced from 01(s12) by the above isomorphism. Tracing back the corresponding formulas, the structure maps on U(s12 ) CZ2 are given by .

d(x + ) = xs + 01 +10x + , A(x_) = xx_ 01+10x_, d(y) = yO1 +10y,

E(x +) = E(x_) = E(y) = 0, S(x) =

-

xx +, S(x_) = xs_, S(y) = -

-

y.

3.1.4 Real Forms of the Quantum Algebra Uq(512)

Recall from Subsect. 1.2.7 that by a real form of Uq (512 ) we mean this Hopf algebra together with an involution * : U(s12) U(s1) which turns Uq (512)

into a Hopf *-algebra. We do not give proofs in the subsequent discussion. The reader is invited to carry out one or two of them using the following general pattern. One

3.1 The Quantum Algebras Uq (812 ) and (A (s12)

59

first verifies that the corresponding formulas define an involution of the free algebra C(E, F, K, K -1 ). Next one shows that the defining relations (1) and (2) are invariant under this involution, hence it passes to the quotient algebra Uq (512) to define an involution there. Finally, one checks on the generators that Zi and E are *-preserving. Case 1: q E R. For any -y E R, 0, there exists a real form of Uq (s12 ) with involution determined by the equations E* = F* = 1,-1 K -1 E7 K * = K. Using the description of all Hopf algebra automorphisms of Uq (s12 ) from Proposition 6(i) we see that two such real forms with parameters -y and are equivalent if and only if -y-y1 > O. Thus, there are precisely two inequivalent real forms of Uq (s12) in this case. The first one is given by

E* --= FK, F* = K -1 E, K* = K. The corresponding Hopf *-algebra is denoted by Uq (su2) and called the compact real form of Uq (s12). The second one is denoted Uq (su 1 , 1 ) and defined by K* = K. E* = -FK, F* Case 2: Iql= 1. For any fiy C C, = 1, we then have a real form of Uq (512) given by E* =--yE, F* K* = K. But all such real forms are equivalent, so we can assume that -y = 1. This real form is denoted by Uq (s12 (R)). It is clear that the three real forms of Ug (512) defined above correspond to the three real Lie algebras su2 , su i , 1 and s12 (R) of the complex Lie algebra sl(2, C). Note that the classical Lie algebras sui,i and s12 (R) are isomorphic. But their quantum analogs Uq (sui,i) and Uq (s12(R)) are not equivalent, simply because they are defined for different parameter domains. Case 3: q C The following sets of formulas define two inequivalent real forms of Uq (512) which have no classical counterparts:

= Nr-1FK, F* =

Et

K* = K,

= Ft = if* = K. It can be shown that only for the above three parameter domains of q real forms of Uq (s12 ) exist and that the previous list is exhausting. Et

The Hopf algebra 0,7 (512) has, up to equivalence, precisely three real forms. They are denoted by tliq (5112), 0.7 (sui,1 )1 Ûq (s12 (R)) and are determined respectively by the formulas Et

= F, Ft = E, K* = K

for

q E R,

= -F, F* = -E, Kt = K for q E R, Et = -E, F* = -F, K* = K for lqi = 1 Et

3. The Quantum Algebra Uq (s12) and Its Representations

60

3.1.5 The h-Adic Hopf Algebra

U h (s 1 3)

Let h be an indeterminate. We denote by Uh(s12) the (complete) h-adic algebra (see Subsect. 1.2.10) with three generators E, F and H obeying the defining relations

e hH e-hH {H, El = 2E, [H, F] =--- -2F, [E, Fi =

sinh hH sinh h

eh - e -h

(16)

where ehH is the series I hH + OH? 12! + • • • belonging to Uh(s12)• The elements El H" Fn , 1,m, n = 0, 1, 2, • • •, constitute an h-adic vector space basis of Uh(s1 2 ), that is, these elements are linearly independent and the h-adic completion of their linear span coincides with Uh(s12). The element e(H -1) ± e -h(11-1) Ch :=7

EF +

e h(H+1)

= FE +

(eh -e-h)2

e -h(Hd-1)

(17)

(eh - e-h)2

commutes with all elements of Uh(s1 2 ) and generates the center of Uh(s12). There exists a unique algebra automorphism 0 of the algebra Uh(512) such that 0(E) = F, 0(F) = E, 0(H) = -H, 0(h) = -h.

(18)

One can easily prove that there exists a unique h-adic Hopf algebra structure on Uh(s12 ) with comultiplication 4 counit E and antipode S defined on the generating elements by

e(H) e(E) = e(F) = 0,

Ee- hH

s(F)

= _ ehH F.

The h-adic Hopf algebra obtained in this way is called the h-adic quantum algebra Uh(S12). From the preceding formulas we see that the algebra Uh(s12) does not change and that the comultiplication d of Uh(s12) goes into the opposite comultiplication dc°P = o d when h is replaced by -h. We can also define the h-adic analog Ûh(s1 2 ) of the quantum algebra Ûq (s12 ) considered above. It is generated by elements E, F and H satisfying the defining relations

e 2hH [H, = E, [H, = F, [E, = with

eh

e -2hH -

e

h

the h-adic Hopf algebra structure ,A( E) = E e hll

e-hH

,A(F) F (8, e hH

c hH

d(H) = H el+ 1 ® H, e(H) = E(E) = E(F) = 0, S(H) = -H, S(F) = -ehE, S(F) =

F,

3.2 Finite-Dimensional Representations of Uq (s12) for q not a Root of Unity

61

An h-adic Hopf algebra A is called quasitriangular if there exists an invertible element R. in the completion AA (in the meaning of Subsect. 1.2.10) such that 1cc9 (a) = 7ZA(a)R. -1 , a E A, and

(A 0 id)R, = 'R, 13 7Z23 and (id 0 A)R, =

R13R-12

on sAllA(g)A,

where 7Z12 = Ei xi ® yi 0 1, 7Z13 = Ei xi ® 1 (g) yi , 7Z23 — Ei 1 ® xi ® yi for 7?, =-- Ei xi ® yi. The element R. is called a universal R-matrix of A. It is useful for applications in physics, because it yields a solution of the quantum Yang-Baxter equation R12R13R23 = R231'43/712 for any representation of the algebra A. A detailed exposition of this theory will be given in Chap. 8. Here we state only the following proposition which is proved in Subsect. 8.3.1. Proposition 7. The h-adic quantum algebra Uh(512 ) is quasitriangular and admits the universal R-matrix R, = eh ®k

cc , (n+1)/2(1 n -2)n , — v ) En 0 Fri E tin

n=0

q

= eh.

7

[711 q !

(19)

3.2 Finite-Dimensional Representations of Uq (s12 ) for q not a Root of Unity Throughout this section we assume that q is not a root of unity.

3.2.1 The Representations 21,1

If T is a representation of the algebra Uq (512), then the operator T(K) is invertible and we have T(K)T(E)T(K) 1 = q2T(E), T(K)T(F)T(K)

T(E)T(F) - T(F)T(E) =

1 --= q -2 T(F),

T (K) - T (K) -1 1

.

(20) (21)

Conversely, since (1) and (2) are the defining relations of the algebra Uq (512)) any triple of operators T(E), T(F) and T(K) on a vector space V such that T(10 is invertible and (20) and (21) are fulfilled gives rise to a representation T of Uq (512 ) on V. Now let I be an integral or half-integral nonnegative number and let i1J E { +1, -1}. Let 171 be a (21+1)-dimensional complex vector space with basis em , m = -1,-1 + 1, • - • ,l. For notational convenience, we set ei +i = e_i_ i = O. Define operators T1 (E), Tw i(F), Ti (K) acting on V/ by ILA (K)em = wq 2me ni , T,,i(E)e,,, = al - m][1 + m + 1]) 1/2en, + 1,

Twi(F)e,„, = coal + m][/ - in + 1D 112em-11

(22) (23)

62

3.

The Quantum Algebra Uq (s12) and Its Representations

where [n] [n] q (see (2.1)). A simple computation shows that these operators satisfy the relations (20) and (21). Hence they define a representation TA of the algebra Uq (512) on Vi. Apart from the sign factor, the representation TA is built in a similar manner as the (21+ 1)-dimensional representation of the Lie algebra sl(2, C). The same method as in the classical case shows that Ti is irreducible. Therefore, the image Tw i(Cq ) of the Casimir element Cq (see Proposition 2) is a scalar multiple of the identity operator I on V. Since TA(E)ei = 0, we obtain from (6) that TA(Cq ) = co(q1+1 +q -1-1 )(q

(24)

q -1 ) -2 /.

Let Tw ,i and Tw i,i/ be two representations such that (w,1) ( 2,1'). Since q is not a root of unity (!), the spectra of the operators /14 (K) and T,1(K) do not coincide, so the representations T,,i and Tw f,i , are not equivalent. Moreover, by (24), the scalars of T i (Cq ) and T,,/,(C q ) are different. Thus, we have proved Proposition 8. For any I E N0 and co E {1, -1}, Z.,/ is an irreducible representation of the algebra Uq (512 ). If (w, I) (41% the representations and are not equivalent and the values of the operators T„,i(Cq ) and T,,,i,(Cq ) are different.

There is a unique scalar product (a, .) on Vi such that (ern , en ) = 6,n . If q E R, then the operators (22) and (23) satisfy the conditions TA(E)* = TA (F) and T1 (K)* = TA (K ). Therefore, by the definition of the involution of Uq (s112), TA is a *-representation of the *-algebra Uq (su2) on VI. The family TA, w E {-FL -1 } , 1 E N0 , exhausts all finite-dimensional irreducible representations of Uq (512) as shown by Proposition 9. Any irreducible finite-dimensional representation T of (.1q (512 ) is equivalent to one of the representations TA, w E {+1, -1 } , 1 =-- 0, I, • • -.

Proof. Because the underlying vector space V is finite-dimensional, the operator T(K) has at least one eigenvector e, say T (I0e = lie. Since KEr = q 2r Er K by (1) and hence T(K)T(Er)e = perT(Er)e, T(Er)e is either zero or an eigenvector of T(K) with the eigenvalue pq 2r. Since these numbers are pairwise distinct (because q is not a root of unity) and V is finite-dimensional, there exists r E No such that eo := T(Er)e 0 and T(E)eo =-- 0. Let T(K)eo = Aeo. Set F( 8) := F8 /[s]! and e, := T(F (8) )eo, s = 0, 1, 2, • • •. Since KF = q -2 FK , we have T(K)es = A g -28 _e There exists a smallest p E N such that T(F(P))e o 0 and T(F()+ 1) )eo = 0. By (3), for any r E {0,1, 2, • • .} we have

T(E)er =

T(E)T(F (r ) )eo = T(EF (79 )eo /1 7

=

-

qr- -

cir -^ 1

T(F (r—

i)

- (r - 1)

)e0

4711..A1

1

erl•

3.2 Finite-Dimensional Representations of Uq (s12) for q not a Root of Unity

63

Thus, the subspace 171 spanned by the vectors eo , el , • • ,e is invariant under the operators T(E), T(F), T(K). Since the representation T is irreducible, = V and the vectors e0 , e 1 , • • • , ep form a basis of V. Let us find the possible eigenvalues A. Since T(F (P+ 1) )eo = 0, we have {T(E)T(F(P+ 1 )) - T(F (P+ 1) )T(E)}eo = 0, so (q-PA - q pA-1) (47 .7 -1) by (3). Therefore, A = ±qP, and the representation T is given by

(25)

T(E)e, = w[p - r T(F)e, = [r + 1]er-E17 T(K)er wqp-2T

( 26)

1P) 112ei, for w - T([/ - r]!/{l r where w c {±1}. We set l := and er r = -1,-1 +1,- - ,I. Then the formulas (25) and (26) turn into T(E)e's = ([l

s

11[1 _ s ] ) 1i2 e1.9+1,

s +11[1 s]) 112 e's _ i , T(K)e' s = w e e.

T(F)e'3 =

This shows that the representations T and TA are equivalent.

The representations Ti with w = 1 are denoted by T1. They are called representations of type 1. Obviously, the representation TA is the tensor product of T1 and the one-dimensional representation Two . 3.2.2 Weight Representations and Complete Reducibility We begin with a preliminary result.

Lemma 10. Let T be a finite-dimensional representation of Uq (512 ). Then the operators T(E) and T(F) are nilpotent, that is, there is an r E No such that T(E)' = T(FY = 0. Proof. The finite-dimensional representation T takes in some appropriate basis the following matrix form: • ••

O

• •• • ••

(27)

• • •

where TM, T (n) are irreducible representations of Uq (s12) and asterisks mean certain submatrices. By Proposition 9, each representation V ic) in (27) is equivalent to some TA. Therefore, since the operators Tw i(E) and Twi(F) are nilpotent by (22) and (23), there exists in E No such that T(E) t = T (k) (F)m = 0 for k = 1,2, - • • , n. Hence, by (27), T (k) (E) r = T (k) (F)r = 0 for some r. Next we give the analogs for Ug (512) of some notions from classical representation theory of Lie algebras. Let T be a representation of Uq (512) on

3. The Quantum Algebra Uq (512) and Its Representations

64

a finite-dimensional vector space V. For any complex number A, we set VA := { x E V I T(K)x = Ax } . If VA {0 } , then VA is called a weight space and A is a weight of the representation T. The nonzero vectors of V), are the weight vectors. A vector x E V such that T(E)x = 0 and T(K)x = px, is called a highest weight vector and p is called a highest weight of T. If V is the linear span of weight spaces of T, then T is called a weight representation. Then V is the direct sum of weight spaces. In other words, T is a weight representation if and only if the operator T(K) is diagonal with respect to some basis of V. Each representation 71,1 is a weight representation.

Proposition 11. Every finite-dimensional representation of Ug (512) is a weight representation. Proof. Let T be a representation of Uq (s12 ) on a finite-dimensional vector space V. It is known from linear algebra that the operator T(K) is diagonalizable if there exist pairwise distinct numbers a l , a2 , • • • , ak such that

(T(K) a i )(T(K) - a2 ) • • • (T(K) - ak)V = {0}.

By Lemma 10, there exists m E N such that T(F)mV = {0}. Recall that [K; = (K qn K-icn)/(q_

) Put K0 := / and

r-1

Kr =11i=-(r-1) [if; r - m + j],

r = 1, 2, • • • ,m.

We prove by induction on r that T(Fm - r)T(Kr)V = {0 } ,

r = 0, 1,2, • • • ,m.

(28)

For r = 0 this is true because T(F)mV = {0}. Suppose that (28) holds for i = 0,1,2, • • • ,n - 1. Setting B = En Fm 1T1=-11 [K; n - m + j ] , we have T(B)V = {0 } and n-1

B=

E biFm - z (H[K ; i n m + j]) (11[K ; 2i i=o

by (5), where

j=i

-

n

-

m + A) En'

J=1

bi = [n]! [m]!/ [i] ![n i=0

E

il! {m - ij! . Further, we get trri-l-n- 1 [K; - n - m + j]) En-i biFm-i n-i-1

i=0

H 3=0

{K;

- 711 -

ii)

Since T(B)V = {0} as noted above and T(Frn - r)T(Kr )17 = {0 } for r 0,1,2, n 1 by the induction hypothesis, the last equation implies the relation T(Fm - n)T(Kr,)V = {0}. This completes the proof of (28). In the case r = m formula (28) yields ,m-1

T(K m )17 = (II

qiT(IC -1 )T(K 2 q q-1

V = {O}.

3.2 Finite-Dimensional Representations of Uq (512)

for q not a Root of Unity

65

Applying some appropriate power of T(K) we get ( m-1

Hi

{

T(K)

q-j}{T(.10-1-q -3 })V = 101.

The numbers in the set q -1 ,-q -1 , j E Z, are pairwise distinct since q is not CI a root of unity. Proposition 12. Any finite-dimensional representation T of Ul (s12 ) is com-

pletely 7-educible, that is, T is a direct sum of irreducible representations. Proof. Let T be a finite-dimensional representation of Ui7 (512 ) on a space V. We first suppose that T reduces to the matrix form (27) with all T(i) equivalent to a fixed irreducible representation 71,1. Let Vi = {v E V T(E)v = 0 } . From the special form of T it is clear that all v E Vi are highest weight vectors belonging to the same weight. In particular, V1 is a linear subspace of V. Let v4, • • • , vt) be a basis of For each i = 1,2,•• • , k, let Vi be the span of vectors vl := Fav6, s = 0, 1, • • • , 21. Repeating the reasoning of the proof of Proposition 9, it follows that on each space Vi the irreducible representation Twi is realized. We show that the set of vectors via is linearly independent. Indeed, only vectors of the same weight may be linearly dependent. If aivt) = 0 and ai vsi = 0, then the action by the operator T(E) 8 yields so a i = 0 for all i = 1, 2, • • , k. Hence the vectors via are linearly independent Since T is a weight representation by and we have a direct sum 1.7 := Proposition 11, we have V =17. Thus, T is completely reducible. Now let T be an arbitrary representation of the form (27). On the block diagonals the Casimir operator T(q) acts as T(i)(Cq ) = cii. The constants ci are the generalized eigenvalues of the operator T(Cq ). It is known from linear algebra that the space V decomposes into the direct sum V = E V I (T(q) c i )rv = 0 for some r} of of generalized eigenspaces V := TWO corresponding to distinct generalized eigenvalues ej . Since Cq belongs to the center of Uq (512 ) we have

Ei

Ei

ED, V.

ED ; v.;

(T (C q ) - j )T T (a)v = T (a)(T (C - c i ))r v = 0 for a E Uq (512), V E V. That is, each Vi is an invariant subspace for the representation T. By Proposition 8, the restrictions Try; of T are of the form treated in the preceding 1=1 paragraph. Hence Tryi and so T is completely reducible. 3.2.3 Finite Dimensional Representations of 6,7 (512) and Uh(s12) -

The previous considerations carry over with a few modifications to the algebras Oq (512) and Uh(s12) as well. In this subsection we state without proof the corresponding classification results for these algebras. Theorem 13. (i) For each co E { +1, -1, i, -i} and 1 = 1N0 there exists up to equivalence a unique (21+1)-dimensional irreducible representation Tw i of

66

3. The

Quantum Algebra Uq (812) and Its Representations

6,1 (51 2 ). This representation acts on basis vectors ei , j = —1, —1 +1,the representation space by

Tw t(E)em =

— mj[l + in + 1]) 112em+i ,

,l, of

(29)

Twi (F)em = wat + m][1 — m +11) 112 ern _ i , 71,1(10e,, = wren, , (30) where et ±i = e_i_i := O. Any irreducible finite-dimensional representation of ildg (s12 ) is equivalent to one of these representations Tut. (ii) Every finite-dimensional representation of 6,/ (512 ) is completely reducible.

Since Uh (s12 ) is an algebra over C[[h]], we consider representations of Uh(s1 2 ) on linear spaces over the ring C[[h]]. Let I be an integral or halfintegral nonnegative number and let V1 be a complex vector space with a basis ei, i = —1, + 1, • • • ,l. Then the formulas Ti (E )em

Ti(F)em =

=

mi[1+ m+11) 1/2 em+il

+mi[l — m+1])112em _ i , Ti (H)em = 2mem

(31) (32)

define an indecomposable representation T1 of Uh(s1 2 ) on the vector space 11[[h]] (see Subsect. 1.2.10) over the ring C[{111]. (Recall that a representation T is called indecomposable if the underlying space cannot be represented as a direct sum of nontrivial invariant subspaces.) The representation T1 is not irreducible, since any subspace itk Mil is invariant. Proposition 14. Every indecomposable representation of Uh(s12) on a vector space space of the form V[[h]], where V is a finite-dimensional complex vector space, is equivalent to one of the representations

3.3 Representations of Uq (512) for q a Root of Unity

Throughout this section we assume that the parameter q is a primitive p-th root of unity, where p > 3. The representation theory of Uq (s12) is then very different from the case when q is not a root of unity. We set p' = p if p is odd and p' = pI2 if p is even. 3.3.1 The Center of Efq (512 )

Let 3 denote the center of the algebra Uq (512). If q is not a root of unity, we know from Proposition 2 that 3 is generated by the Casimir element Cq defined by (6). However, in the root of unity case the center 3 is much larger. Proposition '15. (0 The elements Ev f , FP' , 10)% IC —P/ belong to 3. (ii) 3 is generated by E", F"', K"', IC —Pf and the Casimir element Cg.

3.3 Representations of Uq (s12) for q a Root of Unity

67

Proof. (j): We give the proof for F"'. By (1), ICFP1 = q-2P1 FP'IC = FP1 K. Since [pi, = 0 by Proposition 2.1, we have EFP1 = FP1 E by (3). Thus, FP' G h. The proof for E"', KP' and 1C -121 is similar. (ii): A proof can be found, for example, in [DCK]. Corollary 16. Any irreducible representation of Uq (512) is finite-dimensional.

Proof. Since an irreducible representation T maps central elements into scalar operators, it follows from Proposition 15(ii) that T(Uq (s12 )) coincides with - 1. Therefore, if y is the linear span of operators T(Er1C8 P), r, s,t < a nonzero vector of the representation space V, then T (Uq (s12 ))v is a finitedimensional invariant subspace, so that T(U q (512 ))v = V. Let 3 0 be the subalgebra of 3 generated by the elements E"', F"', K"', and let J be the two-sided ideal of Uq (s12 ) generated by E"', FP' , 101 - 1. Using the assumption qP = 1 it is easy to calculate that .6(1CP1 ) = 101 1CP', 46(E"') = A(FP) =-

0 1+

0 101 + 1 0 El'', 0 FP'.

Since 8(E"') and S(FP') are multiples of Ell if - P' and 1.01 F/1 , respectively, it is clear from these formulas that 3 0 is a Hopf subalgebra and J is a Hopf ideal of Uq (s12 ). Therefore, the quotient Ugres(512) Uq (s12)/J is a finitedimensional Hopf algebra. Using Proposition 15 it can be shown that the algebra Uq (512) is a 3obimodule with free left (and right) 30-module basis

ErKsF t ,

r, s, t = 0, 1, 2, • • • ,p' -1.

(33)

Hence the elements (33) also constitute a vector space basis of Ur(s12)• 3.3.2 Representations of Uq (51 2 )

In this subsection T denotes an irreducible representation of Uq (512 ) on a vector space V. Then the images T(EP ) and T(FP ) of the central elements EP' and FP' are scalar multiples of the identity operator. Concerning the values of these scalars we distinguish three possible cases. Case 1: T(EP') = T(F/91 ) = O.

In order to find such representations T let us first look at the representations TA from Subsect. 3.2.1. For any (.0 E +1, —1} and 1 E 1N0 the operators TA(E), TA(F) and TA(K) given by (22) and (23) also define a representation Tw i of the algebra Uq (s12) in the root of unity case. Next we construct a three-parameter family of representations Tab), which plays a crucial role in the following. Let a, b, and A be complex numbers such that A 0 and let V denote a p'-dimensional vector space with basis ei , = 0, 1, 2, • • • , p' - 1. One immediately checks that the operators {

68

3. The Quantum Algebra

Uq (512)

and Its Representations

TabA(K)ei = q -2j Aei , Tab),(F)ep f_i = beo, TabA(F)ei = ei+17 2
0 1

(35) satisfy the defining relations (1) and (2) of the algebra Uq (512 ). Hence these operators determine a representation Tab), of Uq (512 ) on V. Clearly, we have for any A O. 7-100A (FP ) 7-100A (EP Proposition 17. (0 The representation KA is irreducible if and only if 21 < p' . The representations 71,1, (.4) c { +1, 1}, 1 = 0, 1, - - (p' - 1), are pairwise nonequivalent and satisfy the condition Twi(EP ) = Twi(FP) = O. (ii) The representation Tao, is irreducible if and only if A ±qn for n = 0, 1, • - p' -2. The irreducible representations Ton are pairwise inequivalent. (Hi) The p' -dimensional irreducible representations Too, A = wgP' -1 , and 71 (p,_0 12 for ci) = ±1 are equivalent. Any two other irreducible representations of the form Tw1 and T00), are not equivalent. (iv) Each irreducible representation T of the algebra Uq (s12) such that T(EP') = T(FP) = 0 is equivalent to one of the irreducible representations from (j ) or (ii). -

,

Proof. (0: By the formulas (25) and (26), the action of the representation Tw 1 on the generators E, F, K can be written as

Twi(K)er'

wq21-2reri

ji(E)ert =

T,I(F)er = w{r+11er' ±i ,

where e' 1 = e'21+ := 0 and e ------- 0, 1, • • , 2/, is a basis of the representation space 1/1. First suppose that 21 > p'. Since [pl = 0, Twi(F)ep' _ 1 = [plep' , = O. Hence the subspace W spanned by the vectors e i = 0,1,2, • • • , p' -1, is invariant for the representation Twi . Since W V Twi is not irreducible. Now assume that 21 < p'. It is obvious that Tw i(EP') = T1(F) = O. Since q is a primitive p-th root of unity, all q-numbers [1], [2], • • - [p' - 1] are nonzero. Using this fact the remaining assertions of (i) follow completely similar to the classical case. (ii): The reducibility of the representations Too„ A = ±qn, 0 < n < p' - 2, and the irreducibility of the others are proved as in (i). Suppose that two irreducible representations T00, and Toot, are equivalent. Then there exists an invertible operator B such that BT00),(a) = Toot,(a)B for all a E Uq (s12)Let fen and {e} be the corresponding bases in the representation spaces of T00,\ and Toot,, respectively. Since ker Too),(F) = Ce 'lland ker Tool,(F) Cet, it follows that cet; = Be4' for some c E C. But then we have A./3e1)3 = BT00),(104 = Too p (K)B4 = cT00,..,(104 = citet: = pBe4 so that A = p. (iii): The equivalence of T00A, A = -1 , and Tumpt_ 1 )/2 can be seen at once from the above formulas. Other irreducible representations Too, and Twi cannot be equivalent, because they act on spaces of different dimensions. (iv) is proved in the same way as Proposition 9. ,

,

,

3.3 Representations of Uq (s12) for q a Root of Unity

69

Note that any irreducible representation from Proposition 17 is a representation with highest weight and with lowest weight. Case 2: T(EP 1 )

0 and T(F 1 )

O.

Such irreducible representations are called cyclic. The reason for this name will be seen below. Proposition 18. (i ) The representation Tab), is irreducible and cyclic if and

only if all coefficients ab+[i]q(41-i_A-igi-i)(q_q-i)-1, _ 0,1, ... ,p/_i , (in particular, the parameters a and b) are nonzero. (II) Two irreducible cyclic representations Tab), and Teb , A , are equivalent if A' = q -201/4 and only if a' = a +b-l [i] b' = b and for some i E {0, 1, 2, • • • , p' - 1}. (iii) Any irreducible cyclic representation of Uq (s12) is equivalent to a representation Tab.\ Proof. The assertion of (i) is obvious. (ii): Let Tab), and Ta tbiv be equivalent irreducible cyclic representations. Then the spectra of the operators Tab),(K) and Tat biv(K) coincide, so A' = CI q -2i A for some i E {0, 1, 2, • - • , p' - 1}. For the central elements F' wehav Tab),(FP' ) =

bI, TabA(Cq ) =

+ A -1 q -1 )(q - q -1 ) -2 + ab)I.

Since these values must be the same for the equivalent representation Tatbw we get b= b' and Aq + (q

1)2

+ ab =

(q

q -- 1)2

+ a'b,

that is, a' = a + A -4 qi-1 )(q c l ) -1 . Conversely, a direct calculation shows that for such values of a', b', A' the representations Tab), and Ta t b9,1 are indeed equivalent. (iii): Let T be an irreducible cyclic representation of Uq (512 ) on a vector space V. By Corollary 16, V is finite-dimensional. Hence the operator T(10 has an eigenvector e0 E V, say T(K)e0 = Aeo. We set ei T(Fi )eol i = 0, 1„ •, p' - 1. Since T is cyclic, T(FP') = bI for some nonzero b E C. Then we have T(Firi - i)e i = T(FP)e o = beo 0 and so ei 0, The relations (1) imply that T(K)e 1 = q 2 Ae, i = 0, 1, 2, • • • , p' - 1. Because the numbers q -2i A, i = 0,1,2, • • • ,p' - 1, are pairwise distinct, the vectors ei , i = 0, 1, 2, • - • ,p -1, are linearly independent. Since the Casimir operator (6) acts on V as a scalar operator, there is an a' E C such that T(FE)eo = T(q)e0

11(10q+T(K-1)q-l g g -1

eo = deo.

Therefore, bT(E)e o = T(FP')T(E)e o = T(FP 1-1 )de o = de pt _1, that is, T(E)eo = aépi_i, where a := db -1 . Applying relation (3) we find that

70

3. The Quantum Algebra Uq (s12) and Its Representations

_ A -l qi-i) T(E)e = T(E)T(P)eo = (ab + [i]q

q

_ (7 - 1.

ei-1, > O.

Thus we see that W := Lin{e Ii = 0,1, • • • , p' - 1} is an invariant subspace. Since T is irreducible, W = V. The preceding formulas show that T is Li equivalent to the representation TabÀ• Let us see what is special about the cyclic representations Tao,. The above formulas show that one can move from an arbitrary weight vector to a nonzero complex multiple of any other by repeated application of the operator Tab. (E) (or of Ta bA(F)). This explains why these representations are called cyclic. Recall that in contrast to this property, for the representations T 1.,1, 21 < p', the action of the operator 71„1(E) (or of 71,1(F)) defines a linear order on the weight vectors.

Case 3: T(EP) = 0 and T(FP')

0 or T(EP') 0 and T(FP') = O.

In this case the irreducible representation T will be called semicyclic. We first treat the case where T(EP') = 0 and T(FP') O. Examples of such semicyclic representations are obtained by setting a = 0 in the three parameter family Tao,. More precisely, we have Proposition 19. (i) Too, is an irreducible representation such that Tob),(EP 0 if and only if b = 0 and Tobx(FP') 0 and AP' ±1,0. Two such representations Too, and Tobix, are equivalent if and only if b = b' and A = A'. (ii) Every irreducible representation T of Uq (512) such that T (EP' ) = 0 and T (FP ) 0 is equivalent to one of the representations Tobx b 0, A O. Proof The proof of (0 is straightforward. In order to prove (ii), we repeat the reasoning of the proof of Proposition 18(ii). Since T(EP) = 0 by assumption, the constant a must be zero, that is, T is equivalent to Too,. 0

From the formulas (34) and (35) we see that e o is a highest weight vector for the representation Too, and that any representation Too, with b 0 and AP' ±1,0 has no lowest weight vector. Thus a semicyclic representation T such that T(EP') = 0 and T(FP') 0 always admits a highest weight vector, but no lowest weight vector. Similarly, every semicyclic representation T for which T(EP') 0 and T(FP) = 0 has a lowest weight vector, but no highest weight vector. It is clear from their definition that cyclic representations have neither highest nor lowest weights. Now we turn to the semicyclic representations T such that T(EP1 ) # 0 and T(FP') = 0. Let 0 be the algebra automorphism of Uq (512 ) determined by 0(E) = F, 0(F) = E and 0(K) = K -1 . Then 71' := T 00 is also an irreducible representation. The representation 711:j o, := Too, o 0 on V acts as

1- 1,x (K)ei

= ei+ i, i < p'

1-10,(E)epf_1 = beo ,

71(bx (F)eo =0, T b),(F)ei =[i]q

q q

e,_ 1 , i > O.

-

3.3 Representations of Uq (s12 ) for q a Root of Unity

71

0 if and only if T'(EP') 4 0 Clearly, we have T(EP 1 ) = 0 and T(FP') and r(FP') = 0. Therefore, the assertions of Proposition 19 remain valid if and if the elements Ell and FP' are interchanged. In Tob), is replaced by particular, this shows that any semicyclic irreducible representation T such that T(EP1 ) 0 and T(FP') = 0 is equivalent to one of the representations +1,0. b 0, AP'

To' tIA

not ,

Since the Cases 1-3 exhaust all possible cases for the irreducible representation T, the preceding results can be summarized in the following Theorem 20. Every irreducible representation of Uq (s12 ) has dimension < p' and is a weight representation. The irreducible representations T of dimension < p' are up to equivalence the representations Tw 1, 21 < p' - 1, from Proposition 17. Any irreducible representation of dimension p' is equivalent to one of the representations Ta,b), or T6bA from Propositions 17-19.

Finally, let us return once more to the representation Too), for A := we , w = +1, where n is one of the numbers 0, 1, • - - ,p' - 2. Recall that these representations are not irreducible and the linear span of basis elements e i , n < j < p' - 1, is a nontrivial invariant subspace. Comparing the corresponding formulas we see that the restriction of T00), to this subspace is equivalent to the irreducible representation Tw 1, 21 -= p'-n-1, from Proposition 17. A closer look at the operators Too),(E), To0),(F) and To0),(K) shows that the representation Too), is not completely reducible. That is, the assertion of Proposition 12 is no longer true if q is a root of unity. The existence of finitedimensional representations that are not completely reducible is another new and interesting phenomenon of the root of unity case.

3.3.3 Representations of U(s12 ) In this subsection we determine the irrreducible representations of the finitedimensional quotient algebra U(s1 2 ) = Uq(s12)/ J defined in Subsect. 3.3.1. Clearly, the irreducible representations of Ur (s12 ) are in one-to-one correspondence to those irreducible representations of Uq (512 ) which annihilate the elements EP' , FP KP' -1. Thus, only the representations from Case 1 can give rise to irreducible representations of Up(51 2 ). For notational simplicity let us denote the corresponding representations of Uqr"(s12) and Uq (s12) by the same symbol. From the results developed above we obtain Proposition 21. The algebra Uqres(s12) has the following irreducible repre-

sentations: (i) T1,1 with 0 < 21 + 1 < p if p is odd,(ii) T±1,1, 21 + 1 < p', with 21 even if p and p' are even; (iii) T1,1, 21 + 1 < p' , with 21 even and T_1,1, 21 + 1 < p', with 21 odd if I) is even and p' is odd. Any irreducible representation of U(s12) is equivalent to one of these representations.

72

3. The Quantum Algebra Uq (s12) and Its Representations

3.4 Tensor Products of Representations. Clebsch Gordan Coefficients During the study of representations in the two previous sections only the

algebra structure of Uq (512) was needed. In the subsequent sections we essentially use the comultiplication Uq (s12) by considering tensor products of representations. Throughout Sects. 3.4-3.6 we assume that q is not a root of unity. 3.4.1 Tensor Products of Representations Ti

Let T and T' be representations of the quantum algebra Uq (512) resp. 6,7 ( 1 2 ) on vector spaces V and W, respectively. Recall from Subsect. 1.3.1 that the tensor product TOT' is the representation on the vector space VOW defined by T(a (i) )

(T r)(a) :==

na(2))•

(36)

In what follows, let Th and T1 2 11 1 12 E No , be two irreducible representations of Uq (s12) (resp. Û(s12)). The next result is the Clebsch—Gordan formula on the decomposition of the tensor product into a direct sum of irreducible representations. ,

Proposition 22. If q is not a root of unity, then we have T11

7112D-2 7111-12i e 7111 —121+1 e • - • e T111+121.

(37)

Proof. We carry the proof for Uq (512 ). By Propositions 9 and 12, Th i 1712 is equivalent to a direct sum T = ED m w iTw i of multiples of irreducible representations Ti. The operator (Ti i T/ 2 )(K) = Th (K) T1 2 (K) has precisely the eigenvectors em ®e with eigenvalues q 2 (m+n), where m = —11 , • • • , / 1 and n = —12,• • • ,12 . Comparing with the eigenvalues of T1 (K) and their corresponding multiplicities, it follows that mli = lift -= Ili-121,11i —121+1, • - • '1 1 1+ 121 and m,/ = 0 otherwise.

Note that formula (37) is also valid for the h-adic algebra Uh(s12)In the classical case, the flip operator 7" : VI 1 ® Vi 2 given by r(vi y2 ) = V2 0 v i provides an equivalence of the representations 71 1 T1 2 and 712 0 Th of U(s1 2 ). This is no longer true for Uq (51 2 ), but there is another "natural" operator .h1 1 1 2 that realizes this equivalence at least if q is not a root of unity. We first consider the quasitriangular h-adic Hopf algebra Uh(512) which is equipped with the universal R-matrix R. from Proposition 7. Let 111 1 12

(T1 1 T1 2 )(7?)

and

0 T/2 )(1Z).

to Vi 2 OVi l and, by the relation "P(a)7Z = R.46(a) Then f11 1 12 maps Vt i for a E Uh(s1 2 ), we have

3.4 Tensor Products of Representations. Clebsch-Gordan Coefficients

14 1112( (T11

73

0 T12)(a)) = T(1 0 1112) (R)( 711 1 ® T1 2 )((a))

= T(Tii 0 T/2)( 1R(A(a)) = T(Tii 0 Ti2)(A" P (a)) (Ti 1 0 Ti2) ( 1Z) = (T12 ® 7111)0(0 7- (T11 ® T12 )( R) = ( T12 ® Tii )(41112.

(38)

Since k 1112 is invertible with inverse (Ti i 0 7712)( 7Z -1 ) o 7 - , the operator k1112 realizes the equivalence of the representations 7111 0T12 and 71,0T11 of Uh(S 12) • Using (19) we compute the matrix elements (R1 1 12 )27, of the operator R1 1 12 with respect to the basis {em 0 en } of Vi i 0 Vi 2 . For this reason we abbreviate q = eh. By (31) and (32), we obtain (Ri i i 2 r 1 '2 ni+n,n2-n •.= (e n1+n 0 en 2-n 1 R111 2 I e ni 0 en2) = C(en l -Fn 0 e in 2 -n

I

e WI-10H) /2 En

0 Fn I e 1 0 e in2 ) 7

where c = qn(n+1) I 2 (1 q-2 )([n] q !) -1 . The other matrix elements of R1 1 1 2 vanish. A direct calculation yields -

(Riii2 r n 1,17,n2 ,n2-n =

( 1 _ q-2)n [ t,1 _ niNti + n i + np k -[n]! [11 ± ni]! l i — n i — ni!

1 2 + n2 ]![1 2 - n2 + n]! ([ x

1/2

1/2

q 2 ( n 1 ± n)(n2 -n) qn(n+1)/ 2

[12 - n2 ]![/ 2 + n2 - ni!)

(39)

The expression in (39) is a formal power series in the indeterminate h. But it is also well-defined for fixed q E C, when q is not a root of unity. The corresponding matrix k1112 realizes an equivalence of the representations T1 1 0 T1 2 and 771 2 0 Ti, of the algebra Uq (s12 ) as seen by the following reasoning. Equation (38) holds in particular for the elements e ±hi I 7 E and F of the algebra Uh (s12 ) and for vectors in the complex linear span of the basis elements em 0 en . Now we put q = e h in the matrix entries of it1 1 1 2 65) . Then, by the definitions of the representations and K±1 := e ±hH in ( 3 ,-,‘ Ti for Uh (s12 ) and Uq (512 ) and by the formulas of the comultiplications, (38) is just the intertwining property for the generators of Uq (s12). Hence k1 1 12 intertwines the representations Ti, 0 711 2 and 71/ 2 0 711 1 of Uq (s12 ) and gives their equivalence. A similar result holds for the algebra Oq (s12 ). Let us turn to the special case 1 1 = 1 2 = and denote the operators R1/2,112 and k12,1/2 by R and k, respectively. With respect to the basis e 112 0 e_ 1 / 2 , e_ 1 1 2 0e 1/ 2 , e112 0 e 112) e1 1 2 0 e 1 1 2 these matrices take the form -

-

q

z--

LR

0 0. 1 0 q — q -1 1 0 0

0) 0 0

q

0 1

00 '

0 q - q-1 1 "4 =(q 0 0

1 0

0 001 ' 0 q

(40)

3. The Quantum Algebra Uq (s12 ) and Its Representations

74

3.4.2 Clebsch-Gordan Coefficients Let 1-.7'i denote the subspace of V1 1 0 on which the representation Ti in the decomposition (37) is realized. Let fej l, tek}, {eini } be bases of the vector spaces 14, , Vi2) 1:-// such that the corresponding representations of Uq (s1 2) (resp. Ûq (s12)) are given by the formulas (22) and (23) (resp. (29) and (30)). Since both sets fei

ek I

.= I

{

=- -11,-11 + 1, • • • ,11, k = - 127 - 12 + 11 - /2111/1 - /21+ 1 , - • ,/1 + /2, rn =

1211

-/ +1,—,/}

are bases of VI , 0 1,712 , there exists an invertible matrix C with entries Cq (i i , /2, i; j,k, m) such that (41) The numbers Cq (1 1 ,1 2 ,1; j,k,m) are called the Clebsch-Gordan coefficients (abbreviated, CGC's) of the tensor product T1 1 0 T1 2 . We often write Cq (?,j)

Cq (11,12,1; j,k,m).

In the subsequent treatment we shall see that the main properties of the CGC's known for the classical group SU (2 ) (as described, for instance, in [VKl], Chap. 8) remain valid for the CGC's of td/q (su2 ) as well. In particular, Cq (e,j) -= 0

if j+ k74 m.

an,

m, n = -1, -1 ± 1 , • • • ,1, denote the matrix coefficients of the representation 711 of Uq (512) (resp. Ùq (s12)) with respect to the basis {em } of V/. That is, an are linear functionals on Uq (512) (resp. 6(1 (512)) such that Let

7110e,, =

rn .

(42)

We apply the representation T1 1 0 T1 2 to both sides of (41). On the left hand side, we take into account that acts on eim as Ti does and use the formulas (41) and (42). On the right hind side, we use (42) and the definition of the product in the dual algebra Ug (s12)' (resp. Ùq (512)'). Equating the coefficients of basis elements, we obtain the identity Cq(11 7 12 7 1; j, j7 771 )-tTn

=E

cq ui,12,1;r, s, n)t 7.11) t.(11.92)

for i = -/i, + 1, • • • = -12, -1 2 + 1, - • - , /2. Since the matrix Cq of CGC's is invertible, we derive from this identity that t17.11) t 1( .92) = E

j

cg 17 27 /.r (1) (ii /2 bi Is,n )t mrt• 7 7 7 m)-i(/ 7

(43)

75

3.4 Tensor Products of Representations. Clebsch—Gordan Coefficients

Before we continue the general treatment of CGC`s, let us illustrate the preceding in the case 1 1 = 12 = We abbreviate t11 := t (1/2) -1/2,-1/2' t12 := t (1/2) 4 1/2 t -1/2,1/2' t21 := 1/2,-1/2/ t(22) := 1'1/2,1/2. ,„1 e(i)

= e112

In the above notation, we have

e 1 12 , e 1 = e_ 1 12 e_ 1 12 ,

_ (q -1/2/[2]1/2 )e112

(q 1/2/[211/2\,

1/2 (q 1/2/[2]1/2) e112 0e 1/2

eg

iu 1/2

® e112,

(q -1/2/[2]1/2 \,_ iu 1/2 0

el12.

In order to prove these formulas, it suffices to check that the actions of the representations T1, I = 0, 1, and T112 0 To on both sides are the same. Comparing with (41), we see that Cg @7

= Cq (il

Cq (il i71; Cq

Pf3) = - 1 / 2 / [2] 1 / 2

ou,

1 7 1; 1 1 A 1 0) =

40) = q1/2 /pp./2

1();

1) = Cfq (11110; AI

(11111;

= 1.

Inserting this into (43), we obtain

tiit22 = q(6 0

q -2 t8 0 ) [21,

tg0)/[ 2 ].

ti2t21 =

If we denote the unit element 40 = e of the dual algebra by 1, the two previous equations yield t 11 t22 qt 1 2t2 i = 1. Proceeding in a similar manner we derive from (43) at 1 1 = 1 2 = 1/2 that tlit12 == qt l2 t ll ,

tllt21

t12t21 == t21t12, t11t22

t12t22 == qt22t12,

qt12t21 == 1,

t21t22 == qt22t217 (44) == 1.

t22t11

(45)

These relations define the coordinate algebra of the quantum group SL q (2) which will be extensively studied in the next chapter. In the remainder of this chapter we assume that q is a real number such that q 0, ±1 and we consider the compact real form tig (5u2) of the quantum algebra Ûq (512 ). Then, as noted in Subsect. 3.2.1, any 711 is a *-representation of the *-algebra Ûq (51.12), where the scalar product of the representation space V/ is defined by requiring that the basis {e rn } is orthonormal. Since tig (su2) is a Hopf *-algebra, Th 0 T1 2 is also a *-representation of 6q (5u2 ). Therefore, (37) gives an orthogonal decomposition of the Hilbert space V1 1 01/12 and we can choose the basis {elm } of 171 to be orthonormal. Hence we have CA,1 2 ,1;j,k,m) = (ei

ek , ern ' ).

The unitarity of the matrix Cfq leads to the orthogonality relations for the CGCs: cg (i i, 1 2, 1 ; i7 m

m)Cti(11, 127 II ;

=

bii1

(46)

76

3.

The Quantum Algebra

Uq (512)

and Its Representations (47)

As in the case of SU(2), the CGC's for the quantum algebra (4(5u 2 ) can be calculated by the method of highest weights. Applying the operator T1 (E) to both sides of (41) at m = 1 and taking into account that Tj(E)e = 0 and

E c,(F;i)(ei ® ek) = E

Ti (E)

j±k=m

Cq (ej)Ti(E)(e i ek)

j+k=m

ek + Ciej T1 2 (E)e k ),

Cq (ej){q k (11,(E)ei )

we derive a recurrence relation for Cq (ej), j + k calculations this leads to the solution Cq (.ej)

(-1) 1 1 -i q -(i+1)(11-i)+_1(1+1)/2

1. After some simple

1/2

([11 + j]![12 + - Id!)

-

a'

where j k -= 1 and a is a constant independent of j and k. By the orthogonality relation Ei±k _i ICje;j)1 2 = 1, we find a -2 =

il![12 ±

il2(j+1)(11j)+i(1±1)/2 (11

j±k=1

0112

ki! .

The right hand side can be summed by means of the basic hypergeometric functions 2(101. Indeed, by the relations (2.14) and (2.15), we have [21114/ ± 12 - /11![/1 ± 1 2 ____1g2/10+1) 2451(1 ± 1 2 1 1 ± 1, l -I -12; - 21 1; q27 q2(21+1)), where the notation (2.35) is used. We apply the formula (2.33) to this bypergeometric series. As in the classical case (see [VI(11, Chap. 8), the CGC's Cg (t; j) are uniquely determined only up to a number c(1) of modulus 1, which does not depend on j and k. Hence we can assume that a> O. Thus, a` l = q - ( 1 1+ 1 2 -1 )( 1 + 12 -1 1+ 1 )

(

[21+ 1]![/1 +12 - 11!q211(1+1) + 12 + + 11 - 121![/ - Ii ± 12 ] ! )

1/2

and for j k = 1 we obtain Cq (ii,/2,/;j, k, 1) = (-1)11-i qi{12(12+1)-11(11+0-1(t+1)+2.0+111

[21 + 1P[11 +12 - 1]![11 +j]![12 + k ] ! x

Eli

+ 12 + 1 + 4[1 +11 - 121q1 - 11 + 12P[11 0[12 - kP)

From the relation Ti(E)* =TI(F) and the equality = ([21 ]! [i

m]ql +

i)1/2 dm

112

(48)

77

3.4 Tensor Products of Representations. Clebsch-Gordan Coefficients we derive

where

ek),eb, (49)

ek, TI(F) l- meb = Ni, m (Ti(E) i- m(ei

Cg (t; j) =

j + k m, Ti(E) = (E) Ti,(K) + (.1‘ -1 ) 07112 (E) and Ni on =

Ti i (K), we Cl + mP) 1 / 2 ({2lP[1- 77 ]!) -1 /2 . Setting E1 2 := T1(E) and K1 2 have (E1 1 0 Ki2)((-Kii) -1 E12) = q 2 ((lCi i ) -1 E12 )(E1 1 0 K 12 ). Therefore, the ooperator Erm Ti(E)i - m can be represented in the form -

q -2

(E11

K12 ) k

( ( KO'

(50)

0 Et 2 ) 1-m-k

by Proposition 2.2. Combining (49) and (50) we get r,

1-m

=E

(

raj. [r]![1 - 7• - r]!

r=o

[1 1

[12 + 1 - - rj![11 + + ri! ) 1 / 2 x

(

-

-

[12 - 1 + + r]![11 j - rp

+ ml! k]! [21]![l -

1/2

Cq(11,12,1; + r,k +1- m r, 1 ),

where j k = m and f = rk - j(1 m r). From the latter equation and (48) we obtain the following expression for the CGG's of 0q (su2): (_1)/i - jq0]-([2/ Cq(t

+1][t4) 1 / 2 [/1 + 1 2 — 1]! (t)[1 1 + 12 + 1 + + 0[12 + Id!

' i) x

E

where m --= + k, fi ,âi(t)

(-1)rqr(i+m+ 1)[1 1 + j + rp[1 2 +1 j [r]![1 - m - r]![11 - j - r ]! [12 - 1 + j + r ]!

= 0 2(12 + 1 ) + 12 -

,A(11,1271)

(51)

+1) - 1(1 + 1 ) + 2i(m + 1 )b - 12

± 1]![1

[1 1 + /2 +

1]!

-

/1

+ /0) 1/2

(52)

m]!, (53) + Trtjql j]![il 0[12 ± k]![12 and the summation is over all integral values of r for which the numbers in the q-factorials are nonnegative. Equation (51) is a q-analog of the Racah formula for the CGC's of the group SU (2 ). = [il

By (2.14) and (2.15), the sum in (51) reduces to the hypergeometric series x) := 3 112(a, b, c; d, e; q, x) we 3ç02. Using the notation 3(p2 (g a lqb , e ;q d 7q e ; have

(-1) 11 -j q0146(t)[l +12 - jPgj][21 + Cg(t;61) — [11 — /2 +1]![/ + .12 — 10112 — 1 +

X34P2 12-1+

where fi is as in (51).

+ 1, -/ + rn

11) 1 /2

0[12 + k ]![ 1 — MP

n 2 n2

3 +1 1 —1-12+i ; ‘1 " ) 7

(54)

3.

78

The Quantum Algebra U(s12) and Its Representations

3.4.3 Other Expressions for Clebsch-Gordan Coefficients

A number of other expressions for the CGC's of Oq (5u2) can be derived from (54) by means of the relations (2.36) and (2.37). Setting n

11

j, a

= q2(i+ii+1) , b

q2(j - i+k) , c

q2(i - t -F/2+1),

d = q 2(-1-12)

in (2.36) it follows from (54) that

cq(q ) =-- [1 - 12 + ( X3‘1'2

q /32

()[l + 1 2 - j]! (

[2 1 + 11) 1 / 2

+1 2 - 11! 1 — li ± 12]![/1 11 + 1, — — 12 k —12 1 1 — 12 ± + 1

kJ! — TrZ]

jj![12

]

2 2(k-12)) 7 qq

(55)

where 02 = { 11 ( l l + 1 ) + 1 2( 1 2 ± 1) - 1(1+ 1)+2,114. Inserting here n = 11 - j, a = q

c = q2(i-t i -k+i) , d q 2u i 2 +3 +0

-2(k+1 2 ) , b

-

and using (2.36) we get a q-analog of the van der Waerden formula for the CGC's of SU(2):

go3d(i )( V; j] [2 1 + 1]) h /2 Cg(t;i) = [1- 1 1 - k]! { 11 + /2 — 1 1 11[1 12 4– j }! [11 j _ 11 e

where 03 =

n

12

1-1 1

12 ,

12 -k

3]! [12 + Id!

,2

(56)

+ 12 — 1 )( 1 1 + 12 ± I + 1) + 2(k11 — j12)}. Putting here

k, a = q 2(11 1 4 2) , b =__

d = q 2(1-12+i+1) ,

c.

and applying again (2.36) we obtain

q134 (.0[212]!at, j] [2/ + 11) 1 /2 Cg (t;i) = [1 X 3v-'2

where 04 = n

j, a

12 +

± /2 — 11!1 [1 ± 12 — /11!til

(-k - 1 2 , 1-11 - 12, 1+ 11 - 12 + 1 1-1 + j + -212 2;

jP[12

k]![12 -F kl!

2 2(rn—l)

(57) )

(l 1 — 1+12)(11 + 1-12 + 1) +2(k1 k1 2 j12)}. Setting q - 2(1 - m) , b

q 2(11-pi+i) , c

q 2(12-i+i+i) d

q2(i-1-12)

into formula (54) and using relation (2.36) we derive a q - analog of the JusysBandzaitis formula for

the CGC's of SU(2):

+ :Mt + mi!([21 + 1 ] ) 1 /2 (-1) 11 -iqP5[1 + /2 - j]![11 +12 Cg (t; j) 1]! 1 A(i)(rejj) 1/2 [11 + I2 +

79

3.4 Tensor Products of Representations. Clebsch—Gordan Coefficients X 3 f12 ( M — 1, —/ — li — 1 2 — 1 1 i — 1 1 ; q2 1 q 2(1-1-il —/2-1-1)) 1 -1 - /2 + j, -11 - /2 + m

(58)

where 05 -= 1 1(12 — 11 — 1)(12 + 1 ± 1) + 2i (1 + li + 1) + 2k/11. Letting .41 1 d

n -- = 1 + 11 + 12 ± 1, a --.-. q2(7n -1) , b --= q2(11-12-1) , c =

= q2(-/-/ 2 )

and applying (2.36) we derive

Cq (E;i)

(-1) /1-1 +k q436 [/ + 12 —

=

[4 ± 1 2 — /]![2/P[11

]

AMU-UP/ 2 [1 1 + 12 + 1 + M—

X 3w2

-1 - - /2 — 1, 11 -1 - 12 + j, -21

1 2l 2 2

1 ± 12

2k1

11, a = q -2(ii-F12+/-F1) b = q 2(m—i) , c

(59) )

; q

where 06 = 1(1 + 12 — 11) (1 -I- 11 + 12 + 1) - 212m n

j !( [2l ± 1 ]) 1/2

2k}. If we set here

47 2(-1-12)

d

= 17 -41

and use (2.36), we get

Cq (e; j) =

(-1)/ 2 +k eLl(t)[/ + 12 + j]![21]![1 1 - j]!([21 + 11)112

[11 - 12 ± 11![12 — 1

2

1]!Ge;

-m - 1, -1 -li - 12 — 1, 1 — 12 — 1 2(12+k+1)) 1/

9

where m = k= . This is the assertion (100) for b = .1. In order to oc in the recurrence derive (100) in the general case we take the limit r c = 1iMr_400 Rcjicm (r), s = j +2k, t = c—b, p = b+ c relation (95). Setting 6.1Pcni and using (101), we obtain

6y1;bccni (t--.9+1)/ 4 (

=

=

[a + b

c][a — b + c+ ii[b k][c + [2b] 2 [2c ± 1.][2c + 2 1

+ 1]

TT/

qC

2

— b + c + 11[b + k][c m +

Ci, 'k+1/2:m+1/2

a b-1/2 c-1/2

j,k+1/2,m+1/2

[2b] 2 [2C + 1 ] {2d

= q (t— s -1)1 4 ( [a + b — c] [a

)

11

2 ,--TA a b-1/2,C+1/2

[2b] 2 [2c + 1J[2c + 2] (P_3)/4 =

q

{b

a + ci [a + b + c + 1 ] [b + k][c + m]

rf- a b-1/2 c-1/2 "ij:k-1/2:m-1/2-

[260] 2 [2C + 11[2C]

(103) On the other hand, formula (86) leads to the identity

R q (abd 7 cf / e)Cc! . b c ryc d

—

ryb

df

Substituting d = b =f — and using the expressions (68), (69), (92) and (93) for special values of CGC's and Racah coefficients, we conclude that (103) remains valid if all qbcm c are replaced by the CGC's qapbcm c . That is, both families of coefficients satisfy the same recurrence relation (103). Since = abc 3km %-/) ,k,rn jkm as shown by (102), it follows that 6'9bc '-'jkm for all values of indices. This completes the proof of formula (100). Relation (100) yields the following asymptotic formula for large r:

f a b ld+r e+r

c

f+r 1 .7

_ir-Fb±cl+eq-112 aoc

([2r + 1][2c + 1]) 1/2 Cf —e,d— f,d—el

iql > 1 1

If iqj k/U' 0 lit 0 14 = ((id 0 A) 0 A)(uii ) and ((e 0 id) o A)(uij ) = Ek E(U)U 1 / = Iiij = ((id 0 E) 0 A)(uii ). El Definition 1. The bialgebra O (Mq P)) is called the coordinate algebra of the quantum matrix space Mq (2). The elements fa i bi ck di I id, k,1 E Nol form a vector space basis of 0(Mq (2)). The formulas (3) and (4) give the comultiplication A only on the generators. The action of A on products of generators can be calculated by taking into account that A is an algebra homomorphism. We find that 1

=__ \---• [ i 2+1 i jl 2 az-i bt+i 0 at-ici+i A(a21 ) = (a 0 a + b 0 c)21 z__./ q-

7

(6)

/

=

i= - 1

/+ij2

ci-i di±i 0 a\ l- iCi+i 7

( 7)

q-

-I

where l E No and the summation is over integral (resp. half-integral) values of i if I is integral (resp. half-integral). We also have .

1

j =-1

A>0

[1 - 21 2 [ P j q- L i

1

+i -pi g _2

xq-1464+i-i) al-i-ih eci-j+ti ce+i-t4 0, a1-ic1+-1 .

(8)

4.1 The Hopf Algebra 0(SL g (2))

99

We conclude this subsection by giving a reformulation of the six quadratic relations (1) and (2). For this purpose we take the matrix R = (Rtki )i,j,k4=1,2 = 1, Rj = q — q -1 and Rid/ = 0 otherwise. with R11 = RH = q, R 19 =-That is, R is just the matrix (3.40) derived in Subsect. 3.4.1 from the universal R-matrix of the h-adic Hopf algebra Uh(s12). Let us consider the following equations k

041. 111rn =

k

winik .R7/1/77, 7

, n, rn = 1, 2.

( 9)

Inserting the values of R these 16 equations reduce precisely to the 6 relations (1) and (2). The equations (9) are the starting point for the FRT approach to quantum groups which will be elaborated in Chap. 9. ,

4.1.2 The Hopf Algebra 0(SL q (2))

By relation (2), we have ad — qbc da —

(10)

This element of 0(Mq (2)) is denoted Dg and called the quantum determinant. Proposition 2. The quantum determinant A i is a group-like element (that iS, 4i(D,i ) = Dg 0; and e(;) =1) belonging to the center of the algebra 0 (Mq( 2 )).

Proof. The proof is given by direct computation using the relations (1) and 1=1 (2).

If q is not a root of unity, then it can be shown that the center of 0(Mq (2)) is generated by Dg . Since Dg is group-like, the two-sided ideal (Dg — 1) generated by the element Dg — 1 is a biideal of 0(Mq (2)). Hence the quotient 0(SL g (2)) := 0(Mg (2))1(Dg —1) is again a bialgebra. Proposition 3. The bialgebra 0(SL q (2)) is a Hopf algebra. The antipode S of 0(SL q (2)) is determined by

S(a) = d, S(b) = —q -l b, S(c)= —qc, S(d)=. a.

(11)

Proof. Put a' := d, := —q - lb, := —qc, d' := a. One verifies that the elements a', Lo', , d' satisfy the defining relations of the algebra 0 (S L q- (2)). But O(SLq-1(2)) is the opposite algebra of 0(SL g (2)). Hence there exists an algebra anti-homomorphism S: 0(S.L q (2)) 0(SL q (2)) such that S(a) = a', S(b) = b', (c) = c' and (d) = d'. To prove that S is an antipode, by Proposition 1.8 it is enough to cheek the antipode axiom (1.26) on the four generators. We omit this straightforward verification. 0 A direct computation shows that for the algebra 0(SL 9 (2)) the matrices

MO

4. The Quantum Group SL 9 (2) and Its Representations

u

(a

c

b) d

and S(u)

( d -q -l b ) a )

are inverse to each other. This fact is actually equivalent to the validity of the antipode condition (1.26) for the generators a, b, c, d. Definition 2. The Hopf algebra 0(SL q (2)) is called the coordinate algebra of the quantum group SL q (2).

As motivated at the end of Sect. 1.1, we shall think of and treat elements of the Hopf algebra C3(SL q (2)) as functions on the "quantum group" SL q (2). However, this Hopf algebra is not a group Hopf algebra (it is neither commutative nor cocommutative if q 1) and the quantum group SL q (2) exists only in terms of the Hopf algebra 0(SL q (2)) and its structure. Recall that the defining relations of the algebra 0(SL q (2)) are the five equations (1) and the two equations (12)

ad - qbc= da q -l bc = 1.

These are precisely the relations (3.44) and (3.45) which were derived in Subsect. 3.4.1 for the matrix elements of the representation T112 of the quantum algebra Ug (512). A geometric approach to these relations will be given in Subsect. 4.1.3. We now begin to study algebraic properties of the Hopf algebra 0(SL q (2)). Using the formulas (2.17) and (12) one proves by induction on in that

a7724M

7T1

7T1

E

k=0

Li k ]

2 q2km-k2

(bC) k Cram

q-

E k=--0

771 [

q -k2 (bC) k .

k

(13)

q-2

The following result is often used in the subsequent sections. Proposition 4. The set fanbmer, bmcrds space basis of 0(SL q (2)).

I m,r,s E No , n E NI is a vector

Proof. By the relations (1) and (2), this set obviously spans A(SL q (2)). Using

one of the representations ruce° from Corollary 20 below, it is not difficult to show that they are linearly independent. In Subsect. 4.1.5 we use the diamond lemma in order to give another proof of the linear independence. Proposition 5. (i) There are algebra automorphisms 0 and 12, a, fi E CVO}, of the algebra 0(SL q (2)) such that

0(a) = a, 0(b) = c, 0(c) = b, 0(d) d, = aa, V a,p(b) = Ob, V,,, a (c) = 0-1 c, V a,i3(d) = (ii) For fi E CVO}, 191,fi is a Hopf algebra autornorphism of 0(SL q (2)). (iii) There is a Hopf algebra isomorphism p 0(SL q (2)) 0(547 -1(2)) = 0(S L q (2))°P such that p(a) = d, p(b) = c, p(c) = b, and p(d) =-- a.

Proof. These results follow by direct verification of the defining relations.

101

4.1 The Hopf Algebra 0(SL,(2)) 4.1.3 A Geometric Approach to SL q (2)

Matrices with complex entries act as linear transformations on complex spaces. We shall see now that the defining relations (I) and (2) of 0(Mq (2)) occur quite naturally if we think of quantum matrices of Mq (2) as transformations of the quantum plane q Let 0(C.72 ) be the algebra with generators x and y satisfying the relation

xy = qyx.

(14)

We call O(C) the coordinate algebra of the quantum plane C. Let a, b, c, d be elements of an algebra A. We "transform" the vectors

) and (x, y) by the matrix ( a b ) from the left resp. from the right, c d Y that is, we set s

(

(

(x,y) 0 (

a b\ xha Ox+b® y) ( x , ) c d) (

a b (x,/,y,/). ) = (x0a+y0c, x0b+y0d)=: c d

(15) (16)

Proposition 6. Suppose q2 +1 O. The couples (s', y') and (s", y") satisfy (14) if and only if the elements a, b, c,d fulfill the relations (1) and (2). Proof. First suppose that s'y' = qy's' and x"y" = qy"x". The first equation says (a0x+b0y)(e0x+d0y)= q(cOx+d0y)(a0x+bOy), that is, (ac - qca)

0 X 2 + (ad - da + q-l bc - qcb) 0 xy + (bd - qdb)) 0 y2 =

O.

Since the elements X2 7 xy and y2 are linearly independent in O(C), we get ac = qca, bd = qdb, ad - da - qcb + q-l bc = 0.

(17)

Similarly, the relation x"y" = qy"x" leads to ab = Oa, cd = qdc, ad - da - qbc+ q-l cb = O.

(18)

The last relations in (17) and (18) imply that bc = cb. Inserting the latter into (17) and (18) we obtain (1) and (2). The same computations read in reversed order prove the opposite implication. CI In Hopf algebra terminology, the second implication of Proposition 6 yields the following (see also Example 1.19). Proposition 7. 0(Cq2 ) is a left and right comodule algebra of the bialgebra 0(Mq (2)) with left coaction (pL and right coaction WR determined by 'PL(x) =a0s+bOy,

(pL(y)=c0x+d0y,

(19)

wii(x)=x0a+y0c,

(pE(y)=x0b+yod.

(20)

102

4. The Quantum Group SL(2) and Its Representations

Proof. It follows at once from the converse implication of Proposition 6 that O(C) and there are algebra homomorphisms Ç0L : O(C) —) 0(Mg (2))

(PR 0(C O32 ) —) 0(C q2 )00(Mq (2)) such that (19) and (20) hold. To show that (Pi, and çoR are coactions, it suffices to verify the conditions (1.51) and (1.52) for the generators x and y which is easily done. El Since 0(SL q (2)) is a quotient bialgebra of 0(Mg (2)), the assertions of Proposition 7 remain valid for C9(SL q (2)). In order to express the quantum determinant also in geometric terms, we introduce the exterier algebra A(C) of the quantum plane. It is the algebra generated by elements and 77 and relations 0, 772 = 0 1

=

Setting := adO+bOri and i' := c®-Edon, we have eV = (ad—qbc)®&7. In the classical case q = 1, the determinant appears in the first factor, so it is natural to take Dg := ad — qbc as the quantum determinant. 4.1.4 Real Forms of 0(SL q (2)) In the following we list three real forms of 0(SL q (2)). That these formulas give well-defined Hopf *-algebra structures on 0(Slig (2)) can be proved by the method sketched in Subsect. 3.1.3. Case 1: q E R. There is an involution of the algebra 0(SLq (2)) determined by . a. , d ib r „ * = c (21) a* = d, b* = —qc, The corresponding Hopf *-algebra is called the coordinate algebra of the real quantum group SUg (2) and is denoted by 0(SUg (2)). It is called the compact real form of SLq (2). Inserting the formulas (21) for the involution into the relations (1) and (12) for 0(SL g (2)) we see that 0(SUg (2)) is the *-algebra generated by two elements a and c subject to the relations

ac = qca, ac* = qc* a, cc* = c*c, a* a

c* c = 1, aa* q 2 c*c = 1.

(22)

The Hopf *-algebra 0(SUq (1, 1)) defined by the involution a* = d, b* = qc, c* = q

1 b, d* --= a

on 0(SL q (2)) is the coordinate algebra of the real quantum group SUg (1, 1). Case 2: WI = 1. There exists a real form 0(Sli g (2,R)) of 0(SL q (2)) with *-structure given by a* = a, b* = b, c* = c and d* = d.

4.1 The Hopf Algebra O(SL 9 (2))

103

4.1.5 The Diamond Lemma The diamond lemma [Ber] provides a general method in order to prove that certain sets are bases of algebras which are defined in terms of generators and relations. For instance, the Poincaré-Birkhoff-Witt theorem for enveloping algebras U(g) can be derived from it. In this subsection we develop a simplified variant of this lemma. Let C(x i ) be the free algebra with generators x i , x2 , • • • , x,. Setting I := {1, 2, • r}, the set X := Isi i E In of all monomials (see Subsect. 1.2.1 for the corresponding notation) is a basis of C(x i ). We define an ordering " j

07 j >

j > 07

j

> j.

(42)

(43) (44)

110

4. The Quantum Group SL,q (2) and Its Representations

4.2.5 The Peter-Weyl Decomposition of 0(S.L q (2))

Let C(T/R) denote the linear span of matrix elements t, i, j = -I, -I + I,. • • ,I. It is clear from Proposition 1.13 that C(T/R) is a subcoalgebra of 0(SL q (2)). Theorem 13. The set ft,g) 1 E N07 i,j =----- -1 1 -1 + 1,. ,I1 is a vector space basis of 0(SL q (2)). The Hopf algebra 0(Sid q (2)) is a direct sum of the coalgebras Mr), that is, 0(SLq(2)) =

ED/E C(71/R).

(45)

Proof. To prove the first assertion, we use essentially the results on the matrix elvments t i( i) established in the previous subsection. From the expressions for

t 7,3(1.) derived there we get t!ii) =

= fri

e-2i,-2i7

(46)

where Fi(39 and Pi(39 are elements of A[0, 01 = CM. It follows from (32), Proposition 10(iv) and (46) that for fixed I the matrix elements 4j1) ,

< j < I, are linearly independent. From (40)-(44) we conclude that the function Fi(39 (4") is a polynomial of degree I - max ( ) • Therefore, the functions Fi(39, = max (lik Iii) + n, n = 0, 1, 2, • • form a basis of A[0, = C[C]. It follows from Proposition 10(iv) and (46) that 00

A[ -2i, -2j] =

C t i(j1) .

Using once more formula (32) the first assertion follows. The second assertion is an obvious consequence of the first one.

(47)

El

The direct sum decomposition (45) is called the Peter-Weyl decomposition of the Hopf algebra 0(S.L q (2)). The coalgebra C(T/R ) carries the full information about the corepresentation TIR T1. All basis elements e l) = t!1) 1 and fi(1) = t(1)/,i (by Lemma 11(i)) belong to C(TIR ). Hence the left coaction Tiri and the right coaction TIR are realized on the coalgebra C(TIR) and are given there by the comultiplication. Theorem 13 is a basic result in the representation theory of the quantum group Sid q (2). A first application of it is the Proof of Theorem 9. We show that TiR is irreducible. Let V 5é {0} be an invariant subspace for TIR and let f = Ei aifi(1) be a nonzero element of V. linearly 0. Since the matrix elements t There is an index r with a,. independent, there are linear functionals Fn on A such that F(t) = binbjr-

111

4.2 Representations of the Quantum Group SL,(2)

We have (id 0 Fn)Till(f) ar f41) E V. Thus, fn(1) E V for each n and hence V = V11 . This proves the irreducibility of T/R . It is obvious that two corepresentations TIR and 71/1,?, I l', are not equivalent, because they act on spaces of different dimensions. Now let T: V V 0 A be an arbitrary irreducible corepresentation on a vector space V. Let F E V', F O. Then the linear mapping A := (Foid)T : V A intertwines the right coactions T and 46. Let P be the projection of

A onto the irreducible invariant subspace Ci(TIR ) = Span { t. i) ,.j = 1, 1+ 1, • • • ,1} of A, given by the decomposition (45). Since 6 - o A = F 0, A is 0 for some 1 and i. Since P o A E not the null operator. Thus, Pj o A Mor (T, TIR ), then ker PI o A and im P 0 A are invariant subspaces for T and respectively. But both corepresentations T and TIR are irreducible, so VI R is that ker I 0 A = {0 } and im PI o A = VIR . Hence 1 o A : V 0 bijective and Ti and TIR are equivalent. —

t dim Mr)

Since for fixed 1 the matrix elements

-

linearly independent and

= (21 + 1) 2 and the right (and .6(t 1) ) = E k ti( ki) t k(ii) we have also the left) coaction of .6 on C(TIR ) is the direct sum of 21+ 1 copies of the irreducible corepresentation TIR (resp. ,

4.2.6 The Haar Functional of 0(SL q (2))

In order to motivate the subsequent definition, we begin with an example.

Example 1. Let A be a Hopf algebra of functions on a group G. Let /I be a measure on G. We set h(f) := f f(g)dp,(g), f E A. (For simplicity we suppress questions on the existence of integrals, but everything works if A = Rep G for some compact group G and i is a Borel measure.) The linear functional h on A is called left- and right-invariant if h(f) f f(g)dm(g)z ---- f f(gog)d(g) = f f(ggo)dtt(g), f E A)

(48 )

for all go E G. In terms of the comultiplication on A these equalities read as h(f)1 = ((id h) o 2)(f) = ((h 0 id) o 2)(f), where 1 is the unit of A. Note that (48) holds if i is the Haar measure of a compact group G. Definition 3. A linear functional h on a Hopf algebra A is called invariant

if it satisfies the condition ((id 0 h) o 20(s) = h(x)I = ((h 0 id) o 20(x), X E A.

(49)

As a second application of Theorem 13 we get a simple proof for the existence and uniqueness of an invariant functional h on the Hopf algebra A = 0(841 (2)) such that h(1) = 1. Since the matrix elements form a vector space basis of A, we can define a linear functional h on A by setting

tly

112

4. The Quantum Group SL, q (2) and Its Representations

h(t;)°0) )

= 1 . and h(t 1) ) = 0,

1 > 0.

(50)

Since il(0) = E k t!kl) t Proposition 1.13, one immediately sees that h is invariant. Conversely, if h is an invariant linear functional on A, then condition (49), applied to x = implies that h(ta) = 0 for I > 0. This proves that there is a unique invariant linear functional h on A satisfying h(1) = 1. This functional will be called the Haar functional of A. Next we want to find the explicit form of the Haar functional h. Let x E A[m, r]. From the definition (24) of LK and RK and from the invariance of h we obtain zmh(x) = h(s)1 =-- h(x)z'. Therefore, h(x) = 0 if (m, n) (0,0). It remains to compute h on A[0,0] = C{(]. We will show that

ta,

1 — q -2 h(C n)-

Let P: A position A

(id

1

n = 0, 1, 2, •

q -2(n+1)

.

(51)

* A[0, 0) be the projection determined by the direct sum decom= EIL ,n A[m, nj. By (3), (4), (13) and (2.17), we have

—

13)

A(Cn ) =

E

2

L Z J q_ 2

eijci ((; q 2) i

Ci( 17 -2( ; q - 2) »

where the expressions (a; O m are given by (2.3). Hence, the first equality in (49) yields

NCI

)

• 1. =

E [1 2 j=--n

/(; q2) i h(v( q -2( ; 2 (1

This is an equality of polynomials in ( we obtain the recurrence formula q -2n

—

qbc. Equating the coefficients of

n

h(( )

n > 1.

Since h(/) = 1 by assumption, the latter implies (51). Formula (51) can be used for the calculation of h(f), where nomial in C. For example, we have

f is a poly-

h(cr ((; q 2) 8 ) = q-2(r+1) (q -2 ; q -2 )r(q -2 ; q -2 )8(q -2 i q -2 )1

(q -2 ; q -2) r±8±1

h(cr(q-2(; 47 —2) s ) =

(52)

q -2) r (q -2 ; 42 -2) 8 (q -2 ; 47 -2 ) I. k"/

(q2; cr2)r±s±i

It follows from (11) that S : A[m, n] —0 A[—n, we get h(S(x)) = h(x), s E 0(SL q (2)). We summarize some of the preceding results in

(5 21)

and S( ( ) = Ç Hence

4.3 The Compact Quantum Group SUq (2) and Its Representations

113

Theorem 14. There exists a unique invariant linear functional h on the Hopf algebra 0(SL q (2)) such that h(1) = 1. It is called the Haar functional of 0(84(2)). The Haar functional vanishes on the space Afm, n] if (m, n) 0 or if r = 0 and (0, 0) and on the basis elements arbkcl and bkddr if r k 1. On the space A[0,0] and on the elements bkck it is given by (51) and by h(b k ck ) = (-1) k q k+1q qg --(k+1)

k E N.

Remark 1. It can be shown (see Theorem 11.13 in Subsect. 11.2.1) that if a Hopf algebra A possesses a Haar functional (that is, an invariant linear functional h satisfying h(1) = 1), then any corepresentation of A is completely reducible. Taking this result for granted, it follows from Theorem 14 that corepresentations of 0(84(2)) are always completely reducible. A

The Haar functional h on 0(84(2)) is not central, that is, we do not have h(xy) = h(yx) in general. (For instance, since q is not a root of unity, O.) The next proposition is a substitute for the h(ad da) = (q q -1 )h(bc) missing centrality. Let 19 be the algebra automorphism 190, 1 from Proposition 5, that is, 19 is determined by the equations I9(a) = = b, = c, I9(d) = q -2 d. In particular, by (27), we have 19(x) = q M+71 X for x E A[m, r]. -

-

Proposition 15. For all x,y E 0(5'4(2)), we have h(xy) = 40(y)x). Proof Since 19 is an algebra homomorphism, it is clear that the assertion holds for y 1 y2 (and all x E 0(SL q (2))) if it is true for Yi and for 1i2. Thus, it suffices to treat the cases y = a, b, c, d. Taking the basis elements for x and using the explicit description of h given in Theorem 14, this reduces to a number of straighforward verifications. We omit the details. 111

4.3 The Compact Quantum Group SUq (2) and Its Representations Recall that the quantum group SUq (2) is described by the Hopf *-algebra 0(SUg (2)) which is just the Hopf algebra C9(SL q (2)) for real q equipped with the *-structure defined by (21). Therefore, the theory developed in Sect. 4.2 applies in particular to SUq (2). But it still remains to deal with all questions and properties related to the *-structure. Throughout this section q is a real number such that q 0, ±1 and A denotes the Hopf * algebra 0(SUq (2)). -

4.3.1 Unitary Representations of the Quantum Group SUq (2) Let us say that a matrix corepresentation y = (vii

A is unitary if

of a Hopf *-algebra

114

4. The Quantum Group SL(2) and Its Representations

E k vLuki = E k Vikt/.4 = bij • 1,

== 1, 2, • • •

d.

(53)

Setting v* = ((v*) ii ) := (v i ), (53) is equivalent to the equations vv = vv* = /. In what follows let TO ) denote the matrix corepresentation (t .i1) ) of A. Proposition 16. For any I E N0 TM is a unitary matrix corepresentation of 0(SUq (2)). ,

Proof. We shall use the algebra automorphism 0 of A which keeps a and d fixed and interchanges b and c. By the formulas (11) for the antipode S and (21) for the involution of 0(SUg (2)), for any n, in, r, s E No we have (anlf-ncrds)* = as (-q -1 (-qc)n dn = (S o 0)(anbm crds).

(54)

Since q is real, it is clear from the construction (compare (8), (33) and (38)) that each matrix element 471) is a real linear combination of terms anbmcrds . Therefore, by (54) and the first relation in Proposition 12, we have (4)* (S o 0) (t 1) ) = S(t). This implies the equations (53), since, by (1 .27),

Ek AS1 (t i( i

= Ek t iç ik) Set k(12).

= bij 1.

A corepresentation T : V -> V A of a Hopf *-algebra A on a finitedimensional Hilbert space V is said to be unitary if there is an orthonormal basis of V such that the corresponding matrix corepresentation of T (see Proposition 1.13) is unitary. A number of equivalent conditions will be given later in Proposition 11.11. Let us define a scalar product on the vector space VIR by requiring that the basis {fi l) } of ViR from (34) is orthonormal. Then, by the previous definition and by Proposition 16, each corepresentation TIR of 0(SUg (2)) on the Hilbert space Vi R in unitary.

4.3.2 The Haar State and the Peter-Weyl Theorem for 0(SUq (2))

Let h be the Haar functional of the Hopf algebra 0(SL q (2)). We introduce two Hermitian forms on A= 0(SUq (2)) by setting (x,

= h(xy*), (x, y) = h(s* y),

x,y E

(SUq (2)) .

(55)

Note that in our terminology scalar products are conjugate linear in the first variable and linear in the second one. We shall see below (see Theorem 17(1)) that (•, and (., .)R are both scalar products on the vector space A. In particular, we have h(ex) = (x,x)/, > 0 for s E A and h(1) = 1. Since a linear functional on a *-algebra with the two latter properties is usually called a state, h is said to be the Haar state of the quantum group SUq (2). From the definition (55) it is obvious that (xz,y)R = (s,yz"`)R and (zs, y)L = z* OL. Moreover, by Proposition 15, (s/ Y)L == (19(1)7x)R.

4.3 The Compact Quantum Group SUg,(2) E;..nd Its Representations

115

The next theorem can be considered as an analog of the classical PeterWeyl theorem for the quantum group SUg (2).

Theorem 17. (i) (-, . )L and (.,.)R are scalar products on the vector space

0(SUq (2)). The Peter-Weyl decomposition (45) of 0(SUq (2)) is an orthogonal decomposition with respect to both scalar products. (ii) The matrix elements tZ),,, of the irreducible core presentations T (1) satisfy the orthogonality relations (a, a) n ,) R = [21 +11-q-lq2n6w6mm/bnn,

tm (1',)n,) L = [2/ + 14-1 Q -2m 611"5mmi 15nn'

(56) (57)

In the proof of this theorem we require the following lemma.

Lemma 18. Let k,1 E

No and let M be a (21 + 1) x (2k + 1) matrix with complex entries. Let At := h(T (1) MT ( k) *) and 'AP := h(T (I) * MT () ). Then - = 0, M' 0 if 1 k and rlt;1 = aI , we have 11;1 = &I for some a, a' E C if 1 =-- k.

Proof. We prove the assertion for the matrix 11//. The proof for .1 is analogous. We denote the matrix T (1) g I by Tr and the matrix I 0 T (I) by T. Using the invariance condition (49) of the Haar functional h, we obtain

T (1) /17/T (k) *

=

(id 0 h)(711) TP MT k) *e ) *)

=

(id 0 h)(T 1) TP M(T I(. k) T4 k) )*)

=

((id 0 h) o L1)(T (1) MT (k) *) = h(T (I) MT (k) *) — .

Since T ( k ) is unitary by Proposition 16 and hence T(k) *T(k ) = I, we have T (1)11;1 - = .11-1T (k ) . Therefore, ti-/-f defines a linear mapping, still denoted by , of 11R to VAR which intertwines the irreducible corepresentations T (1 ) and T (k ) . Hence the invariant subspaces ker it-/-1 and im M are either 101 or the whole space. Thus, we get ker /17/ = 0 if 1 k and fl;/ = al- if 1 = k. (The latter reasoning is just Schur's lemma for Hopf algebras.) El Proof of Theorem 17. (ii): Let Eii be the (2 1 + 1) x (2k + 1) matrix having 1 in the (i, j)-position and 0 elsewhere. Let kij be the corresponding matrix

from Lemma 18. Then (iiii )„ = h(t,(12t s(ki ) *) = (41 2 , t s( )) R . By Lemma 18, we have (412,0)R = 0 if 1 k. Let 1 = k. Since t r(i2t.(1) * E — 2r, 2j — 2i 1 and h(a) = 0 for all a E dti[m,n), (m, n) (0,0), then (t,„(1 2 , t (. ) ) R = 0 for (r,i) 3L (s, j). Since a E AH2r, —2s1, we have 19(4 ) ) = q -2( r+s)4) an d so (t

Hence

,

4,)4

= (3(t)), tT) R = q -2(r+ 8) (47sn) t) R. = 0 if I

k

d if I = k, (i ?

(58)

4. The Quantum Group SL 9 (2) and Its Representations

116

Let now

= h(T(1) Eii T(I) *). By Lemma 18, there exists a constant ai E C such that (kii)ii= h(4t)*) = (tçl) (9)R = ai for all and j (-1 < 2,3 t i3 j < 1). In just the same way, considering gi = h(T(')*Eii T(1)), we find that there exists a constant c

(t

, t)/, = c for all i and j. By (58), we have q -2(i+i ) ai. Consequently, there exists a constant a E C such that a = q2 ia'i = q -2jai for all j and j. Since t!1) e!1) , then by the formulas (33), (13), (2.17) and (52) we have oc_i = q -41 (1-q -2 )/(1-q -41-2 ). Therefore, c = q 21 (1 q2)/(1 q-41-2\) and so E C such that

[21 + 1 ] -1 q2.3 ,

a"z = [21+ 1] -1 q-22 .

(59)

(i ): By (56) and (57), the numbers (a),„ an )L and (tai , tZ)n )R are positive for all real q 0, ±1. Since the matrix elements ta form a vector space basis of A, the equations (56) and (57) imply that (x,x)L > 0 and (x,x) R > 0 for any nonzero x E A. Hence (., -)L and (•, .)R are scalar products on A. The orthogonality of the Peter-Weyl decomposition follows at once from (ii). Let us turn once more to the Haar state. Formula (51) enables us to write the Haar state on C[C] as a q-integral (see Subsect. 2.2.2). For PC) E C[C], we have 1 h(f) =

o

00

f ()d,7-2C = (1 -

1

q -2)

q -2jf( q -2i)

if q

> 11


0 7 m < n, n > 0, m > n,

m + n 0, m n,

m-n c(n-m)/2 (q-2 (;11-2) (m+n)/27 m + n 0, m n. Ç mn(() q In terms of these functions Omn the scalar product (-, .)R for elements of A[m, nj can be derived from the formulas o

(s) Y)R = hth(()Omii(()f2(() * ),

x = fi(Oemn, y = f2(()emn.

(62)

The expression on the right hand side of (62) can be computed explicitly by means of the formulas (52) and (52').

4.3 The Compact Quantum Group SE/q (2) and Its Representations

117

4.3.3 The Fourier Transform on SUq (2)

a complete orthogonal system By Theorem 17, the matrix elements t in 0(SUq (2)) with respect to the scalar product (., •)R. To each element f E 0(SUq (2)) we associate a matrix Po = ( by ja h(ftaii ), E N0, m, n = —1, —1+1, • • • ,1. The mapping f + f( 1) is called the Fourier transform of the Hopf *-algebra 0(SU q (2)) or the Fourier transform on the quantum group SUq (2). Using the formula (56), it is easy to show that the inverse transform is

fa) —

f =E

/E i N.

[21 +

1],1

El

and that the following Plancherel formula holds: [2/ + 11 q

Cf, g)R

1

E. .

— 2i

q f10-M ii

where [21 + 11q is given by (2.1). 4.3.4 *-Representations and the C*-Algebra of 0(SUq (2)) 0(SUq (2)) is in particular a *-algebra, so it is The Hopf *-algebra A natural to look for its *-representations. Let fr be a *-representation of A on a vector space V equipped with a scalar product (-, •). By (22), we have

11*(a)vli 2 + lifr(c)v11 2 = (fr(a) * *(a)v5 v)

(fr (c) * fr(c)v v)

= (fr (a* a + c* c)v v) = (fr(1)v,v) = E V 7 so the operators fr(a) and fr(c) are bounded on V with norms not exceeding 1. This implies that the operators ir. (a*) and 'Fr(c*) are also bounded and have norms less than or equal to 1. Since any x E A is a linear combination of elements ai (c*).1 ck (a*) 1 , it follows that 71-(x) is bounded on V and that there is a constant Cx not depending on the *-representation such that Ilfr(x)II 1, the relations (72)-(74) mean nothing but that the generators xi mutually commute and the equations (71) and (81) read as 4+ 2s i sl = (1+ p2 )1. Taking hermitian variables by setting Yi := xo , V42 x i +x1' and := -xn, the latter becomes the equation y? +0+0 =---That is, in the limit q 1 the *-algebra 0(514L) is just the coordinate *algebra of a 2-sphere in R3 . -

-

4.5.2 Decomposition of the Algebra Q(S) First we proceed in a similar manner as in Subsect. 4.2.2 and decompose the right quantum space 0(Sq2p ) into a direct sum with respect to the coaction of the quantum subgroup K. For this reason, we define an algebra homomorphism RK := (id 0.0(pR : 0(Sq2p ) 0(Sq2p ) 0(K), where (pR is the right coaction of 0(SL q (2)) on 0(St,) and OK is the Hopf algebra homomorphism of 0(Sl q (2)) to 0(K) given by (23). For n E 2Z we set

0(Sq2p )[n]:= Ix E 0 (S:p)

RK(X) = X g znl.

By (76), (75) and (23), we get xo E 0( 51q2p)[0]) xi E 0 ( 51)[2 ] and x i E 0(Sq2p )[-2]. Because RK is an algebra homomorphism, the latter yields that X

i

k

G 0(S2qP)[2i — 2k1 for i, j, k E No.

(82)

As noted in Subsect. 4.5.1, the set {xi i x-4, 44} is a basis of 0(SiL). There-

fore (82) implies that 0 (s:p )[2 n]

el i c [x o ]

= c[x o ixTn

if if

n > 0, n < O.

In particular, thé space 0(Sq2p)[0] of K-invariant elements of 0(S427p) coincides with the algebra C[xol of polynomials in the generator xo. Further, by (82) we have the decomposition

4.5 Quantum 2-Spheres

()(sp )

127

= @,„, ez 0( sp )[2 m 1

Our next result determines the decomposition of the coaction (pR into irreducible components. Theorem 26. The corepresentationçoR of 0(SL q (2)) on O(S) is equivalent to the direct sum of irreducible corepresentationsT1, 1 E N o , with multiplicity 1. That is, we have ÇOR2G

teN0 T1

072 C 9(5 q2 p ) (131 1.0 V1.

(83)

Proof. Let Tirn be the linear span of elements x 1 xx i, j, k E N0 + + k < n. We prove by induction on n that the restriction (,oR,,, of çoR to Wn is equivalent to a direct sum of corepresentations Th 1 = 0,1, • • • n. Obviously, this in turn implies Theorem 26. For n = 0 the assertion is trivially true. Suppose that it holds for some n E No . Since corepresentations of 0(SLq (2)) are completely reducible (see Remark 1 in Subsect. 4.2.6) and any irreducible corepresentation of 0(Sli g (2)) is equivalent to some T1 there exists a çoR-invariant subspace Wn+i of Wn+ 1 such that Wn+ 1 = 14771 Wn+1 and ( 30 FtrWn-F1 at TV . On the other hand, ,

,

i

,

since the basis elements fi(i) (see (34)) of the representation space of Ti TIR are in A[21, —2i] by Lemma 11, we have (id 0 O K )Ti(fi ) = fi(1) Z -2i and Since çoR,n, ED711_ 0 Ti by the induction hypothesis, it so OK (tY)) = 0(Sq2p )[2i1. follows that Ric(14771 ) = (id 0 oic)(pR,n(wn) g By (82), Xnti c Wn-F1 belongs to 0(Sq2p )[2n + 2 ] and up to multiples it is the only element of Wn+ 1 in 0(Sq2p )[2n + 2 ] . Hence there occurs at least one corepresentation Tv in the decomposition (pRI-Wn-F1 ED,, Tv such that 1' > n. But the latter is possible only when çoR [W + 1 Tn+i , because we have dim Wn+i = dim W n+1 _ dim yvn < dim Lin {xi 1 44I j +j ---=m+k= n + 1} = 2n + 1. This completes the induction proof. In the preceding reasoning only corepresentation theory of 0(SL q (2)) was used. Another proof can be given by decomposing the representation gOR of the algebra Uq (512 ) which is induced by the dual pairing from Theorem 21. The key observation in this proof is the fact that the highest weight vectors of (ijii are precisely the complex multiples of xn 1 , n E N0 . El

Remark 2. The following fact from the preceding proof will be used later. Up to multiples, xl 1 and f (1)i are the only elements in the corresponding carrier z 21 spaces of the corepresentation Ti such that (id 0 OK)(pR(x l 1 ) = x i 1 and (id 0 OK )/1(f) = f 0 Z21 , respectively. Moreover, both elements are highest weight vectors for the irreducible representation ti of Uq (512). A The quantum spaces O(S) X q ,p, i+p2 and 0(Sq2 ,_p ) Xq ,_ p, 1±p2 are obviously isomorphic for p E C. As a first application of Theorem 26, we show that these are the only isomorphisms of the quantum spheres O(S).

128

4. The Quantum Group SL q (2) and Its Representations

Proposition 27. Let p, p' E CU{00}. The quantum spaces 0(8,72 ,p ) and 0(Sq2 ,p,) are isomorphic if and only if p = p' or p = -p'.

Proof. Suppose that O is an isomorphism of the quantum spaces 0(Sq2,p ) and Since 9 intertwines the coactions of 0(84(2)), Theorem 26 implies Lin {aci}, where {x i } and {x'i } denote the that 19 maps Vi Lin {xi } to VI! generators of 0(8422 ,p ) and 0(Sq2,p,), respectively. Since the corepresentation T1 is irreducible, it follows that 9(x i ) = Ax for some A E C. From the relations (71)-(74) we see that the one-dimensional representations 7r of the algebra 0 (51472, p) are given by numbers 7r(s i ), i -1,0, 1, such that 7r(s i )7(s_ i ) = 0, 7(x 0 ) = 1 if p = (Do and (q + q -1 )71- (x1)7(x_i) = -1, 7(x0) = p if p E C. Being an algebra isomorphism, 9 preserves the onedimensional representations. Therefore, the previous description implies that 0 (8472 ,00 ) is not isomorphic to 0(8217)P,) for p' E C and that A 2 = 1 and hence 0 = p or p' -p for p, p' E C. As a second application of Theorem 26 we obtain the existence of an ) A linear functional h on the algebra 0(S qP 2 invariant functional on 0(8247P , is called right-invariant if

((h 0 id) o (pR)(x) h(x)1, x E 0(SIL) Proposition 28. There exists a unique right-invariant functional h on 0(4) ) such that h(1) = 1. The functional h vanishes on all subspaces 0(S)[2m], m O.

Proof. The proof proceeds similarly to the case of the Haar functional in Subsect. 4.2.6. By Theorem 26, we can define a linear functional h on 0(Sq2p ) by h(1) = 1 and h(x) = 0 if x E Vi 7 I 0. As in Subsect. 4.2.6, h is rightinvariant and any right-invariant linear functional on 0(8q2p ) with h(1) = 1 is of this form. For x E 0(8,72p )[2n], we have ((h OK)(pR)(x) =-- h(x)z 21 = h(x)OK (1) and so h(x) = 0 if n 0. 0 By Proposition 31 below the algebra 0(Sq2,p ) can be identified with a subalgebra of 0(84(2)) such that the right coaction (pR corresponds to the comultiplication. Then, by the uniqueness assertion of Proposition 28, the right-invariant functional h must be the restriction of the Haar functional of 0(84(2)) to Finally, let us suppose that q is real. We define a Hermitian form (., -) on 0(Sq2p ) by setting

(x, y) = h(xy*), x, y

e

0(S)

.

Because h is the restriction of the Haar state of 0(SUq (2)) to its *-subalgebra 0(Sq2, p ), it follows from Theorem 17 that (-, -) is a scalar product on 0(S,) and that the subspaces 171 are mutually orthogonal with respect to (-, -).

4.5 Quantum 2-Spheres

129

4.5.3 Spherical Functions on Sq2p There exists a basis ei(1) , i = -1,-1 + 1, • • ,l, in the carrier space V/ of the corepresentation T1 in the decomposition (83) such that

E

(e i(i) ) =

e(-1) g t

j

=

-1,-1 +1,•

where t içji) are the matrix elements of the corepresentation 21 of 0(SUg (2)) from Subsect. 4.2.4. By Remark 2 after Theorem 26, we can assume without si 1 . If q is real, then the elements e? ) E O(S) loss of generality that e are called spherical functions on the quantum 2-sphere S. Proposition 29. (i) If O < i < 1, then et(i)

Ni( _ q -20.-Fiw a; g -2) 1

y;

o i ci -2) xL ,

and if -1 <j < O, then = Ar_i _ q 2(i-1) ,(3/a; q -2) 1+ixr pt(;ii,-i) (y; a,01,7 -2 ), (

where 1)4a ' b) (x ; a, OW) are big q-Jacobi polynomials (see Subsect. 2.3.4) and =__ (_ a )l-itp-i)(/+3i+3)/2 [ 2/ 1-

-1/2 _2

1 1

q •-2

a = ( (q ± C 1 y 2 ± p 2 / 4) 1/ 2 ± p/2 , 0 = a ____ p.

(ii) If q is real, the spherical functions e l) satisfy the orthogonality relation 1

(e i( m) , e.(in) ) _ 67nni52 3

q2

[2M 1 _ q -2(2m+1) m -

1 flq( ct ± -273) (0 ± q -2r a ) . q -2 r=1

Proof The proof can be found in [NM].

LI

4.5.4 An Infinitesimal Characterization of 0(S 2 In this subsection we give an elegant description of O(S) as the subalgebra of elements of O(Sli g (2)) which are infinitesimally invariant with respect to a certain twisted primitive element of 6,1 (512)An element z of a bialgebra A is called twisted primitive with respect to a group-like element g of A if A(x) = g -1- gz-i-sOg. If s is twisted primitive with respect to g, then we have S(x)= -g - ixg and E(x) = O. From the formula (12) for the comultiplication of the Hopf algebra Ûq ( 51 2) we see that any linear combination of K -- K -1 , E and F is twisted primitive with respect to the group-like element K of Ûq (s12). We put •

130

4. The Quantum Group SL q (2) and Its Representations -1

q X p := pq 1/ 2,

(K

(84)

IC -1 ) + qE + F, p E C,

XŒ, := K K -1 .

(85)

In the notation of Subsea. 1.3.5, we write x.f = x(2) (f, x ( 1 ) ) for x E 0(SL q (2)) and f E 6(1(512), where (., .) is the dual pairing from Theorem 21. As noted in Subsect. 1.3.5, this defines a right action of the algebra Ûq (512 ) on 0(SL q (2)). Using the formulas (40)-(44) for the matrix coefficients and (67)-(68) for the pairing (., .)' we calculate

t z(j1) .E = V[1+ i + 1] [1 -ijtY1

tg) .F = -,/[1 + i][1

(86)

,,

+ l]tçl) t-1,j •

(87)

Further, let

(88)

fx E 0(SL q (2)) x.X p = 0}

denote the set of right Xp-invariant elements of 0(84(2)). Using the fact that Xp is twisted primitive one immediately checks that 6(19q2p) is a subalgebra of 0(SL q (2)) and 46(6(8L)) C 6(4) 0 0(SL q (2)). Lemma 30. dim fo l for E + No.

(SL) nc(TiR) = 21+1 for 1

E No and 6(4) n c(TiR) =

Proof. Let Ci (T R denote the linear span of matrix elements t )

,

i = -1,-1+

1, • ,l. Clearly, C(T/R ) = - E k t ik (1) t kj i3 =-t C-(T 3 /R )• Since .6(tçl) map z -) x.X p leaves each subspace Ci (T/R) invariant. Moreover, from (86) and (87) we see that Xp acts on the subspaces Ci (TiR), j = -1,-1+ 1, • • • ,l, by the same formulas. In the case p --= co the assertion follows immediately from (86) and (87). Suppose now that p < co. We consider the equation x.Xp = 0 for x ,

3

E. ct•t 1.3(. 1.) E

C (71R ) where ai E C. Inserting the formulas (84), (86) and (87) and comparing the coefficients of tg) , it follows that this equation is equivalent to the recurrence relations (q -2 ql)-y +

with q ---1)1( q -1

i][1 + j + 11 ±

-01 - j + 1] [l + i] = 0 (89)

= ai+1 := 0 for the coefficients di , where we set 1, = q). If ct_i is given, then (89) determines uniquely the coefficients

cti ±i • This procedure yields a solution of the equation a-1+17 ct-t+2) • • • , x.Xp = 0 if and only if ai +i = O. Therefore, since the recurrence relation (89) does not depend on j, it follows that either dim (8q2 p ) nC(TIR) = 21 + 1 or

6(4) nc(TiR) = { 0}.

We find by a direct computation that dim C3(4) n C (T1R ) = 3 and ei(Sq2p) nC(4,2 ) = {0}. In fact, the solutions for the corepresentation are precisely the elements i-19i02ii from (77)-(79), that is,

4.5 Quantum 2-Spheres

(1)

=

.(1) + PO t O,i

131

P lt

(1 + ey112 00 = p an d = q(1 + (12) -1/2 . where 0_ 1 Now we prove by induction on 1 that dim e3(4p) nc(TiR) = 21+1 for integral 1. For 1= 0 this is trivially true. Suppose that it holds for a fixed 1 E N. Take a nonzero element x = E, ceit E 6(8q2 p) nC(T/R ) . By the discussion after Proposition 24 in Subsect. 4.4.2, the product tt„ E 0(5'4(2)) can be computed by means of formula (3.43). In this way we obtain

=

E

"ml =

i,m

Ea

. 0 ,/,1,/±1 m i,m,i+m

1,1,1+1 i+m,1-1-1•

i,m

Thus, xi). E Ci-f-i(Ta i ). Since (5(4p) is an algebra and x E (5(Sq2p) by ashsumption and i E e5(Sq2p), we have x1 1. c (5(Sq2p) nC(T1F+11 ). Because the algebra 0(84(2)) has no zero divisors, x i z4 0 and so ( ( S q2 p) 1 C 1 -FL) {0}. Hence we have dim ô(S) nC(Ta i ) = 2(1 + 1) + 1, which completes the induction proof. Next we show that 6(4p) nc(TiR) = {O} for half-integral 1. Assume the contrary and let 1 be the smallest half-integer such that (5(Sq2p ) n c (TiR) Let x = Ei ai 6,1) E 6(SL)nC(T IR ). {0}. As noted above, 1 1, so l > From formula (3.43) it follows that for k + j 1 the relations of the algebra A q reduce to (1) and the relation N = a+ a. Hence the q-oscillator algebra A q can be considered as a q-deformation of the oscillator algebra A. the By definition, the algebra A q is the quotient of the algebra A two-sided ideal J generated by the element aa+ - q - la+a - qN . Since the second condition of (7) can be replaced by the first of (9), the ideal J is also generated by the central element cq . Thus the element cq given by (6) is zero in the algebra A q and the algebra A q has only the trivial center C.1 if q is not a root of unity. Proposition 1. The following sets are bases of the complex vector spaces:

{( a+)k an qmN k n E N, in E Z} fan (a+)k e N I k,n E N, (0, o)}. A q : {(a+)kgmN q m N an fk,n E N, m G Z, (m n)

Tri

Proof. All assertions can be derived from the diamond Lemma 4.8. Proposition 2. (i ) There exist automorphisms 190, n E Z, of the algebras Acq and A q such that

190(a) = Oa,

190 (a+) =

0n (a) = qnN a,

On(a+) =

1a ,

E CI 3

(a) = q r a

ip er,r (a+ )

On N)

aqra+ ,

(iii) There exists an automorphism Ca) = a+ , 0(a+) =a,

0, and On ,

( (r N) = (11\1 7 7- N .

1

0, of Acq deter-

(ii) There are algebra automorphisms (p a,r, a,r E C, a mined by (70 t ,r

D

aq2r q - N

of the algebra A 7 such that

0(q -N ) = -qqN .

Proof. One checks that the images of the generators under i9p, 9„, ço a,, and //) satisfy the defining relations of the algebras Aqc and A q , respectively. Hence the above formulas indeed define algebra homomorphisms. Since the mappings are invertible, these homomorphisms are actually automorphisms. D

If q is not a root of unity, it can be shown that the automorphisms from Proposition 2 generate the groups of algebra automorphisms of Ag and Acq , respectively. We close this subsection by looking at *-structures on the q-oscillator algebras. If the parameter q is real, then both algebras Acq and A g become *-algebras with involutions determined by a* := a+ and (qN )* := qN . In the case iqi = 1 the algebra A is a *-algebra with involution such that a* := a+ and (qN )* := q - N

136

5. The q-Oscillator Algebras and Their Representations

5.1.2 Other Forms of the q-Oscillator Algebra

In this subsection we list some other variants of the q-oscillator algebra that occur in the literature. They are derived by simple formal algebraic manipulations from the algebra A. As abstract algebras they are of course different. But for N-finite representations of the algebra Aqc these formal replacements lead to representations of the corresponding algebras. First, inserting the formal substitution b := q --N/2 a7

b+

a + q - N/2

into (3) and using (4) we obtain the relation bb+ bb = q-2N Replacing formally tr 2 by q in this equation and in (4), we obtain the relations •

[b,b+] bb+

bb _ q N

(I A b+ = qb + giv 7

eb = q - be

Let Ab denote the algebra with generators b, b+ q N N = NqN = 1 . relations (10) and Similarly, putting A := q N/ 2a,

1

(10)

N subject to the

A+ := a+q N/ 2

and then replacing q2 by q, the relations (3) and (4) lead formally to the equations [A, A] q AA + - qA+ A = 1, q N A+

qA+q N q N A

Aq N .

(11)

The algebra with generators A, A+ , qN , q –N and defining relations (11) and q N q -N = q -Nq N = 1 is denoted by AA. Some authors use the name q-oscillator algebra for the algebra with generators A, A+ subject to the relation AA + - qA+ A = 1. A two-parameter deformation of the harmonic oscillator algebra is derived from the q-oscillator algebra Aqc in the following way. First we replace the parameter q by r. Setting

a := r ctN a,

:= a±ra N

(12)

for a E C and q = r2a+1 1 p = r -2+1 in the defining equations (3) and (4) of the algebra A rc we formally obtain the relations N ^ –N , p N a -+ =pa-+p — ± qa-±a=p aa , p N. a=p –1ap N .

(13)

The algebra Apc q with generators a, a+, p N , p -N and relations (13) is called the (p, q) oscillator algebra. Of course, the inverse of the formal substitution (12), that is, a = r - aNii. and a+ = "+ - crN with r 2 = pq, r 4ar = qp - i transforms the two-parameter deformation A back to the algebra Arc . -

, q

5.1 The q-Oscillator Algebras botq and Ag

137

5.1.3 The q-Oscillator Algebra and the Quantum Algebra 6,7 (s12 )

The quantum algebra Ûq (s12) admits various homomorphisms into the qoscillator algebra A. More precisely, we have the following Proposition 3. (i ) For any a E C I there exists the algebra homomorphism A g such that Ta Ùq (s12)

Ta (E) = a, Ta (F) = a±[N 2a]q , T„CK (ii) There is an algebra homomorphism T: (42(s12)

T(E) =

(a+)2 q

a2 , T(F) = q+q-1 7 -

q N-a .

A q determined by

7- (K) . q N+1/2 .

Proof. (i ): As usual, it suffices to check the defining relations (3.10) and (3.11) of the algebra (4(512). As a sample, we carry out this for the relation (3.11). By (8) and (9), we have

[T,r (E),T,,(F)] = aa+ [N 2ajq — a+ [N 2a]q a aa+ [N — 2a]q — a+a[N — 2a 1 ] q [N + 1.]q [N — 2a] q [N]q [N — 2a l] q (q _ q -1)-21 (q N-Fi q -N-1) (qN-2a _ q -N+2a) (q N _ q-N)(Q,N-2a-i _ q -N+201+1)} —

—

2(N-a)

q

-2(N-a)

TcK (K)2 za (K)-2

q

g -1

(ii): The proof follows by similar calculations using the relations (a) 2 a2 = [N]q [N

—

1 ] g , a2 (a)2 = [N + l]q [N + 2] q .

111

Our next aim is to realize the algebra [ji g (s12) in terms of two commuting q-oscillator algebras. In order to do this we need an algebra Aeqxt which is obtained by adjoining formally elements qN/2 and C IV/2 to A g . More precisely, At is the associative unital algebra with generators a, a+, erl 2 and q-N/2 subject to the relations [a, a±jg

q -N :=

-N/2 q N12 = q /V/2 47 -N/2 =

( q -N/2‘2 1

)

[a, a± l q-i

=

17 qN/2a+ = q 1/2 a -F qN 1'2 1

:= (qN/2)2 7 qN12 a = q -112 aq N/2 .

It is clear that the algebra Ag from Definition 2 is isomorphic to the subalgebra of AV generated by the elements a, a+, q N and q -N Let Aeqxt '2 be the tensor product of two q-oscillator algebras A. The corresponding generators of the algebra .A.eqxt' 2 are denoted by a17 e2/2q-N212. q -N1/2 , a2, Note that every element of the set a l , at q± N1 12 4 commutes with any element from a2,4, q± N2/2 ,

138

5. The q-Oscillator Algebras and Their Representations

Proposition 4. There exists a unique algebra homomorphism c,o : Ûq (512) —› Ar t ' 2 such that p(E) = ata2 ,

c,o(F) = 4a1 ,

-. c,o(K) = q(N1N2)12

(14)

Proof. It is enough to verify that the elements (,,o(E, ço(F), (p(K) and ço(IC -1 ) satisfy the defining relations (3.10) and (3.11) of Uq (s12). As an example, we check the relation (3.11). By (9), we obtain (p(E)c,o(F) = ata2a -21- a1 = [N g [N2 + 1] q .

Similarly, (p(F)(p(E) = [Ni + 1]q [N2] q • A straightforward computation shows that [Ni q [N2 + 1] q - [N1 + 1],AN2] q = [N1 N21J q . Hence c,o(E)c,o(F) - ço(F)cp(E) = (v (K)2 _ p(K1)2)/(q _ q --1) The algebra homomorphism ço from Proposition 4 is called the JordanSchwinger realization of the algebra Oq (512)• An algebra homomorphism7,b of a Hopf algebra 7-1 into another algebra X might be a useful tool for the study of the Hopf algebra. First the composition of IP with representations of X provides us with representations of For the Jordan-Schwinger homomorphism c,o : 0,1 (812) —› Aeqx t '2 this will be carried out in Subsect. 5.3.4 where the irreducible representations of Ûq (S12) are realized on the q-analog of the Bargmann-Fock space. Another application is given by the following lemma. It is a slight generalization of Proposition 1.14 and shows that such a homomorphism turns the algebra X into an 7-1-module algebra. Lemma 5. Let 11) : e / be an algebra homomorphism of a Hopf algebra 7-1 into an algebra X. Then X is a left 7-1-module algebra with respect to the left action of 7-1 on X defined by

a> s

adip, L (a)x :=

E 11)(a(0)•s.11)(S(a( 2 ))),

a C 7t, s E X.

(15)

Proof. Since //) is an algebra homomorphism, it is clear that there are left and right actions of 71 on X given by as := 0(a).s and sa := s-7,1)(a), respectively, where the dot means the multiplication of X. By the associativity of this multiplication, X becomes then an 7-1-bimodule. Therefore, as noted in Remark 2 in Subsect. 1.3.4, ado,/, is a left action of 7-1 on X. Since a r> 1 = (a)1 and

(a i ) > s)(a(2)r> y) = E 0(a(1) )s1/0(a(2) )) 0(a(3) )y0 (S(a (4) ))

E 0(a( 1 ))stP(S(a( 2))a( 3))00(a( 4))) = a t> (sy), X is indeed an Ii-module algebra.

0

Lemma 5 applies in particular to the algebra homomorphisrns from Propositions 3 _ and 4. Let us write down the corresponding action in the case of the

5.1 The q- Oscillator Algebras ..Aq and Aq

139

Jordan-Schwinger homomorphism cp. Setting V) = ça in (15) and using (14), (3.12) and (3.13), the left action (15) for the generators of Ûq (512) yields ado,L(E)x = ata2xq(N2

qq(N2

IN11)/2

N1) 12 xat a21

ado,L(F)x = aixq(N2- Ni)/ 2 _ q -l q (N2 _ N1) 12 xa E Aeq xt,2 .

ado,L(K)x = e1-N2)12 5e2-N1)/2 ,

Our next example describes a few irreducible submodules of the module Aext' 2• q

Û (S12)

Example 1. A direct computation shows that the elements 1/2 r 1/2

:=

1/2 r -112 := a2 q

± -N2/2

q

of Aeqxt ,2 satisfy the relations , E,\ 1/2

r 1/2

ado,LkE ir -1/2 - 1/2' adtp,L(E)r1// 2 = O ' ado,L (A

1/2

1/2

1/2

)r i/2. = r_ 1127 ado, L (.1. )r 1/2

This means that r il/i22 and r

=

1/2

07 adtp,L(21.)r +1/2

q

+1 1/2

r+1/2 •

2 form a basis of an irreducible submodule 71112

of the Ûq (s12)-module Aeqxt ' 2 . Similarly, 4/2

r 1/2 .= -a

(N1+1)/2

-1/2 :=

aiq —(N2+1)/2

are basis elements of another irreducible Ûq (s12 )-submodule 71112 of Using the Clebsch-Gordan coefficients of the algebra ei/q (s12 ) from Sect. 3.4 one can find basis elements of polynomials of r 2 and 7' 11 /12/2 transforming

under any representation 11, t E -1N0 , of Ùq (512). For instance, the element 1/2,.1/2 ro := q1/2 r 112 r _ 1/ 2

q -1/2 7,1/2 4/2 1/2 r 1/2

- LIV

.1 V2]

is invariant under the representation ad o,L , that is, ado , L (a)rg = E(a)r8 for all a E Ùq (512). Further, the three elements rI

1 /2 7111/2 / 2 — 1 ii(N1-N2V2a±a2 r 1/2• 1 7 1/24/2 := r 112 r 71/2 =

1 TO

—

(,-1/2 ,1/2 4,4/2 1/2 1 -1/2

1

0-21q

„I/2..1/12/241 11/ 2 ) -

(Ni-N2V2 a + 1 a2

=

[2]

frr-N2-i

form a basis of an irreducible Ûq (s12)-submodule T1 of Airt12 .

+.3/4 [N2])7 A

140

5. The q-Oscillator Algebras and Their Representations

5.1.4 The q-Oscillator Algebras and the Quantum Space

Mo (2)

Let Aext ,4 denote the 4-fold tensor product of the q-oscillator algebra At defined in the preceding subsection. The generators of this algebra Ar,4 are denoted by aii ,ati ,q Nii/ 2 7 q -Nii/ 2 7 where i, j = 1,2. Recall that the algebra 0(Mq 2 (2)) is generated by elements ul, ul,u?,14 satisfying the relations (4.1) and (4.2) with q replaced by q2 .

Proposition 6. There exists an algebra automorphism Aeqxt,4 such that

: 0(Mq2(2))

-y(u1) = a4-0 (N12-N11+N21+N22)1 2 , ,01) = ail-2q (N21-3Ni1-N12+N22)/2

'y(u)

/-y(74) = a-21-0.(N12- 3N,1 - N21+N22)/ 2 a2+2q (N11-3N12-3N21-N22)/2 ± / 1 _ (N22-N11-N12-N20/2 k

-y (Dq - q 3/2 q -(Ni2+N2i+1),,+,,,+ '11'22

q2 )4a2allq

)

,3/2,(Nii+N22+1),+,+

'21'12'

Proof. The proof is given by direct computation. We omit the details.

0

There is also an algebra homomorphism : 0(S.L q (2)) —> Ag such that u(u1) --= a+,

v(14) = q -N

v(u 21) = —q- N-1 , v(4) = (q —q- 1)q-N-la.

5.2 Representations of q-Oscillator Algebras 5.2.1 N-Finite Representations

We shall use the bra and ket notation as is common in physics. If T(N) is an operator on a vector space V and w E C, then the symbol 1w) denotes a vector from V such that T(N)lw) = wiw). Suppose that V is a vector space and T(a), T(a) and T(N) are three operators on V such that the eigenvectors of T(N) span V and the relations [T(a),T(a+)Jg = q-T(N)

T(N)T(a) = T(a+)(T(N)+ I) ,

(16)

T(a)T(N) = (7 1 (N. ) + I) T (a)

(17) are fulfilled on V. Here the operators q±T(N) are defined by q±T(N)1w) q±wIw) when T(N)Iw) = wiw). Then it is clear that the operators T(a), T(a), and T(q±N ) := q±T(N) satisfy the relations (3) and (4) of the algebra A. Hence these operators define a unique algebra homomorphism T of Ai' to the algebra L(V) of linear operators on V, that is, T is a representation of sit: on the vector space V. Definition 3. A representation T of the algebra deq on a vector space V of the above form is called N finite. -

5.2 Representations of q-Oscillator Algebras

141

The following observations are crucial in the following. Proposition 7. Let T be an N -finite representation of Aq and let 1w) be an eigenvector of T(N) with eigenvalue w.

(0 Either T(a+)lw) = 0 or T(a+)jw) is an eigenvector of T(N) with eigenvalue w +1. (ii) Either T(a)lw) = 0 or T(a)Iw) is an eigenvector of T(N) with eigenvalue w —1. (Hi) If T is irreducible, then 1w) is an eigenvector of T(a)T(a) and T(a)T(a). Proof. It follows from (16) that

T(N)T(a)w) = T(a+)T(N)Iw) + T(a)1w) = (w +1)T(a)1w), which proves (i). The proof of (ii) is similar. If T is irreducible, then T(cq ) oil for some a c C and hence

T(cq )1w) = q l- w[w]lw) T(a)T(a)1w) (q i-w {to ]

q -2w )1w )

(q - N )T (a)T (a+ )1w) = ajw).

This implies that 1w) is an eigenvector of T(a)T(a) and T(a)T(a).

El

Now let T be an irreducible N-finite representation of Al,' and let 1w) be an eigenvector of T(N) with eigenvalue w. Concerning the actions of the operators T(a) and T(a±)n on 1w), there are three possible cases:

Case 1: There exists a natural number n such that T(a)n Iw) = O. Then the representation T is called a representation with lowest weight. Case 2: There is a natural number n such that T(a+)fliw) = O. Then T is called a representation with highest weight. Case 3: T(a)njw) # 0 and T(a) w) 0 for any natural number n. 5.2.2 Irreducible Representations with Highest (Lowest) Weights

Throughout this subsection we suppose that q is not a root of unity. The next proposition says that the irreducible N-finite representations of A' with highest weights are parametrized by complex numbers. Proposition 8. To every complex number w there corresponds an irreducible N-finite representation Tit of Aqc with lowest weight. It acts on a vector space V with basis elements 1W + m), M E No , and the operators T(N), T u and Tw+ (a) are given by

T.(N)lw + m) = (w +m)iw + rn), n(a+)1w + m) =1

+ m + 1),

(18)

142

5. The g-Oscillator Algebras and Their Representations

where we have set 1w 1) := 0 and [m] denotes the q-number (2.1). Every irreducible N -finite representation of A ge with lowest weight is equivaient to one of the representations T. Two representations T and Ti-DE, are equivalent if and only if qv" = qw . —

n

Proof. One immediately checks that the operators (N), Tit (a+) and T: (a) defined by (18) and (19) satisfy (16) and (17), hence they indeed determine an N-finite representation T: of Age . We have Tut (a) w) = 0, so T: has a lowest weight. Since q is not a root of unity, one easily verifies that the representation T: has no nontrivial invariant subspace. Thus, is irreducible. Since the formulas for the operators 21: (e), T (at) and 21: (a) depend only on qw, Tit and T, are equivalent if qw = e l . Conversely, if qw qw , then the operators Tw+(qN) and T,(qN) have different spectra, because q is not a root of unity. Hence the representations 71 and Tt-DE, are not equivalent. Let T be an arbitrary irreducible N-finite representation of Age with lowest weight on a vector space V . Then there exist a number w G C and a nonzero vector 1w) E V such that T(N)(w) = w I w) and T(a)lw) = 0. We denote the vector T(a±)miw) by 1w + m), m E N. Since aa+ = qa+ a + q - N , we have

n

T(a)lw + m) = T(aa+)* + m — 1) = qT (a+

(20)

+ m 1) + q - (w -E rn -1) lw + m 1) .

In particular, T(a)lw + 1) = q - luiw). We show by induction on m that

T(a)lw + m) = q - u) [m]fw + m

-

1).

(21)

Suppose that (21) holds for m =-- 2, 3, • • • , n 1. Then, by (20), we obtain T(a)(w + n)

q

n-1

,

qW-1. ( 17

-n+1

r , ) T(a)1w + n 2) +

i

( "u1 -En — 1) + n 1)

= q - w[n]lw + n 1),

so (21) holds for any m E N. Thus, the operators T (N), T(a) and T(a) act on the vectors ( w + m) as described by (18) and (19). Each vector ( w + m) is nonzero. Indeed, otherwise Vm := Lin fw+k) k>m)- is an invariant subspace for the irreducible representation T by (21) and Proposition 7(i), so that Vm = V. This is impossible, since then 1w) is not in V. Similarly, we get V = Lin {1w+ k) I k E No l. Since the (nonzero!) vectors 1w +m) are eigenvectors of T(N) for pairwise distinct eigenvalues (by Proposition 7(i)), they are linearly independent. Thus we have shown that T is equivalent to T. {

Proposition 9. To any complex number w there corresponds an irreducible N -finite representation T; of Acq with highest weight. It acts on a vector space V with basis elements iw m), 1TZ E No, by the formulas Tt; (N)I — m) = (tv m)lw m), Tt;

(Ow -

m

5.2 Representations of g-Oscillator Algebras

(a+)Iw - m) =

143

- m 1),

where 1w ± 1) := 0. Every irreducible N -finite representation of A ge with highest weight is equivalent to one of these representations T;. Two such representations Tu, and Tu7, are equivalent if and only if qw = qwl Note that the action of the central element cq of Age is given by

Ti-DE (cq ) =

[wil. and Tv, (eq ) =

[w + 1]./".

(22)

5.2.3 Representations Without Highest and Lowest Weights

In this subsection we assume that q is not a root of unity. Let a and w be complex numbers such that 0 < Re w < 1. Let V be a vector space with basis lw m), m E Z. A straightforward verification shows that the operators T(N), T(a) and T„,,(a) on V defined by

T,w(N)lw + m) = (w + m)iw + m) T, w (a+)lw T, w (a) jw + in) = (aqm + w [m])lw

in) =1w ± in ± 1) , (23) in - 1)

(24)

satisfy the relations (16) and (17). Hence these operators define an N-finite representation Tc,,„ of the algebra Age on V. For the central element cq we get

T„„,(cq ) = Proposition 10. If ar ±q- w[m]

- a)1".

(25)

0 for all in E Z, then the representation

is irreducible and has neither a highest weight nor a lowest weight. Two such representations Tc,w and Tc'' are equivalent if and only if a = a' and qw = qw . Every irreducible N -finite representation of the algebra A qc without highest and lowest weights is equivalent to one of these representations Tctw Proof. The properties of the representations TQw asserted above follow from the formulas (23) and (24). In order to prove the last assertion, let T be an N-finite irreducible representation of Age without lowest weight and without highest weight. If lx) is an eigenvector of T(N) with eigenvalue x, then it follows from the assertions (i) and (ii) of Proposition 7 that the vectors T (a)T Ix) , T(a)Ix), r E No , s E N, are linearly independent. Since T is irreducible, their linear span coincides with V by Proposition 7(iii). Thus, the spectrum of the operator T(N) is simple and formed by the eigenvalues x rrt, where m E Z. There is an eigenvalue w of T(N) such that 0 < Re w < 1. Let 1w) be a corresponding eigenvector. Let lw -1), 1w), » fw±rn), - be eigenvectors for T(N) such that T(N)lw i) (w i)lw i) and T(a+)lw + i) = iw + +1 )1 i = -1,0,1, 2, • -. By Proposition 7(i), there is a complex number a such that T(a)Iw) = afw - 1). Since T has no lowest weight, a 0. Then we get T(Ow +1) = T(a)T(a+)lw) -= 71(qa+ a + q -N )liv) = (qa +

144

5. The q-Oscillator Algebras and Their Representations

Using similar equalities, we prove by induction on m that

11 (a)ftv m)

=

Oar + 11 (el-

= (aqm

r

+

w EmDlw m

1 ),

W

(771-1) ) ) lw

m

1)

for m E No . That is, the operators T (N), T (a+) and T(a) act on the vectors iw m), m = —1,0, 1, • • •, by the formulas (23) and (24). If 1w)' lw) and lw — 1), then T(a+)Iw 1)' := aiw)' and T(a)17D)' := 1w —1Y. Setting lw —m-1)' := T(a)lw —mY m E No , and using the relation a±a = claa+— N we find by a simple induction argument that

T(a+)Iw mY = („q -m+1 = v/q mi - +1

( q -m+2

c m+4

q m-2))1 w m 1)/

q - w[m — 1 ])1w — m + 1Y.

By assumption, the representation T has no lowest weight. Therefore, the preceding implies that iw — m)' 0 for m E Nci and that aqm q- w[m] L 0 for any m E Z. Setting — m)

:= H 2= 0

-w r •

11 - 1

MY,

it is straightforward to check that the operators T(N), T(a) and T(a) act

on the vectors lw — m), m = —1, —2, •••, by the formulas (23) and (24).

111

Summarizing the preceding, if q is not a root of unity, then the above Propositions 8-10 classify all irreducible N-finite representations of the algebra A& up to equivalence. Let us briefly turn to the symmetric q-oscillator algebra Ag . As noted in Subsect. 5.1.1, the algebra Ag is the quotient of the algebra Aqc by the two-sided ideal generated by the central element eq . Thus, the (irreducible) representations of Ag are in one-to-one correspondence to those (irreducible) representations of ..,4& that annihilate eq . A representation of Ag is called N-finite if the corresponding representation of Aqc is so. Proposition 11. (0 Let w + and w_ denote fixed complex numbers such that qw+ = 1 and qw - ±1 = 1. Then the representations I'd- , T.++ , T -1 and Tu7_ are pairwise inequivalent irreducible N-finite representations of the algebra A g with highest or lowest weight. (II) Let w E C be such that 0 < Re w < 1 and [w]qm q- w[m] 0 for all in E Z. Then T[w ],w is an irreducible N-finite representation of A g . Two such —

—

representations TH, w and Ti.lb., are equivalent if and only if qw = qv) . (iii) Any irreducible N-finite representation of A g is equivalent to one of the representations from (i ) or (ii).

Proof. (i): From (22) we see that the representation Tid„- (resp. T;) of Acq maps cq into the zero operator and so passes to a representation of Ag if and only if 1 (res p. q2(w+1) = 1). Since [w] = 0 (resp. [w + 1] = 0) or equivalently

ew =

5.2 Representations of q-Oscillator Algebras

145

two representations Tit and T:, (resp. Ttr, and T.,) are equivalent if and only if qw = qw1 , there are precisely two equivalence classes with representatives V and T:,+ (resp. Tfi. and T;). Two representations Tit and Ty-7, cannot be equivalent, because Tit has a lowest weight and Tuj, does not. (ii): If T„,,(cg ) = 0, then a = [w] by (25). The other assertions of (ii) follow from Proposition 10. (iii): Each irreducible N-finite representation of A g yields an irreducible N-finite representation of A ge that annihilates the element eq . Hence the assertion follows from the fact that the representations of (0 and (ii) exhaust all representations T from Propositions 8-10 such that T(cq ) = 0. CI 5.2.4 Irreducible Representations of ,..47 for q a Root of Unity

In this subsection q is a root of unity. More precisely, we assume that qP = 1, where p E N is odd and minimal. Lemma 12. The elements (a+)P , (a)P , (qN )P and (q -N )P belong to the center

of A. Proof. It follows from the relations (3) and (4) that a(a + r -=-- [n](a+r -l- q -N + q-n(a+ra,

(a + ) 72,q-N _.,_ eq -N( a +)n .

Putting n = p and using the r6lations qP = 1 and [p] = 0, we see that (a+ )P commutes with the generators and so with all elements of A. The proof for aP, (q N )P and (q -N )P is similar. 0 Definition 4. Let T be an irreducible representation of Al' . T is said to be cyclic if T(a) 0 and T((a+)P) 0. If T(a) = 0 and T((a+)P) 0 or if T((a+)P) = 0 and T(aP) 0, then T is called semicyclic.

Let ,u, and be complex numbers and 0. Let V be a p-dimensional vector space with basis 1m), in = 0,1, • • • ,p - 1. We define operators on V by the formulas T(a+)Im) = Im+ 1), 0 < m < p-- 2, T(a+ )lp - 1) = ej0),

T(a)im) = [m + it]Im - 1), 1 < m < p - 1,

T(a)0) = [ith]Cl iP - 1 ),

TAC(q ±i )im) = q ±(m+A) im) and Ti (a)1M) = IM — 1) , .1 < m < p — 1,

T(a)I0) =

II) — 1) ,

T(a)IM) = [M+ ti+1]1M+1) , 0 in Ç p-2, Tt (a)p-1) = [ IAC 1 10 ), Ti/g (q ±)i m ) = q±(m+o)im).

146

5.

The

q-Oscillator Algebras

and Their Representations

Both sets of operators fulfill the relations (3) and (4), so they define representations Ti.h c and Ti'hc of the algebra A. One easily checks that the representations Tlic and Ti.",c are irreducible. Moreover, if ii, 0 and 0, then the representation Ti..,/ c is equivalent to T[mic -1. The formulas (18) and (19) also define a representation of .Aqc on V when q is a root of unity. But this representation is not irreducible. The subspace Vp spanned by the basis vectors ( w + m), m = p, p ± 1, • - • , is invariant. The quotient space V/Vp is irreducible. The corresponding representation operators Tip (q±N ), Tw (a+) and Tip (a) are then given by Till (q±N )(w + m) = q±("+m ) kv + m), T„(allui + m) = iw + m +1) ,

T(a)1w ± m) = q' [m]lw ± m — 1), where we set I w

—

1) 7---- 1w + p) :---= O. -

Proposition 13. (0 Suppose that qP = 1 for an odd nonnegative integer p. Then every irreducible representation of the algebra Acq is p-dimensional and equivalent to one of the representations T4c for p,, E Cy Oy T4 ,c for E C y e 0 1 or T, w E C. The representations Tm , p. 0, e 0, are cyclic. The representations To ,c , # 0, are semicyclic with lowest weight and the representations 7',: e e 0, are semicyclic with highest weight. The representations Tv, have highest and lowest weights. (ii) The representations 2-1,a and 71/,11, 1.1,11! 0, are equivalent if and only if = and q4 --= qia'±ic for some k = 0, 1, 2, • - • ,p — 1. The representations To ,c (and the representations Ti:,c) are pairwise nonequivalent. The , representations Tip and Tie are equivalent if and only if qw = qw . Proof. (i): The stated properties of the representations T, Tck and Ti, are

immediately verified. Let T be an arbitrary irreducible representation of the algebra .,4 7 on a vector space V. Since T maps elements of the center to scalar operators, it follows from Lemma 12 that T(A) is spanned by the operators T(ai(a+)i (e)k), 1. In particular, we conclude that dim T(A) 0 and q 1. Then, as noted in Subsect. 5.1.1, the algebras A and A g are *-algebras with involutions such that a* = a+ and (qN )* = qN . The next proposition decides which of the irreducible N-finite representations of Acq introduced in Subsects. 5.2.2 and 5.2.3 are *-representations. Recall that a *-representation of a *-algebra A is a representation T on a vector space V, equipped with a scalar product (., .), such that (26) (T(s)v,v') = (v,T(e)v 1 ), s E A, v, E V. Proposition 15. (i) T (resp. T) is a *-representation of d4fi if and only if le > 0 (resp. qw < (ii) If 0 < q < 1, then Tay, is a *-representation of Act? if and only if qw > 0 and a(q q-1 ) + q- w 5_ O. If q > 1, then T,„, is a *-representation if and only if qv' < O and ctqw + (q - q-1)-1 O .

Proof. We carry out the proof of (ii). The proof of (i) is similar. Suppose that is a *-representation. Applied to s = qN , (26) means that the operator T(q N ) is symmetric, so that its eigenvalues qw+m, m c Z, are real. If qw > 0, then applying (26) to s = a, v = 1w + m), = lw + m - 1), we find that aqm + q - w [m] > 0 for all m e Z. This is equivalent to the inequality

+

(q

1 ) l trW > W cr 27n (q

q' ) 1 for all

m E Z.

Since q-- w > 0, this is impossible if q > 1. If 0 < q < 1, it leads to the condition ct(q-q -1 )+q- " < O. If qw 1 and ctqw + (q - q7 1 ) -1 < O. Conversely, suppose that 0 < q < 1, qv' > 0 and ct(q - q -1 ) + q- w < O. Then one easily checks that there is a scalar product (., .) on the space V defined by (w + m + 11w + m + 1) = (ctqm + q - w[rn])(w + mlw + m) and (w + nlw + m) = 0 for m n such that (26) holds for the generators of A.

148

5. The q-Oscillator Algebras and Their Representations

Hence (26) is true for all s E Acq and Ta„ is a *-representation. The case q> 1 is treated similarly. Comparing Propositions 11 and 15 we obtain Corollary 16. Tit and Tj_ are the only N -finite irreducible *-representations of the *-algebra Aq .

The operator T(N) is usually considered as a q-analog of the number operator. Hence it is natural to require that it has a real spectrum. Let us call an N-finite *-representation T of Acq or Ag a physical *-representation if there exists an operator T(N) as in Subsect. 5.2.1 with real eigenvalues. and Taw , 0 11-y between elements from ffR and 4R. We define a symmetric bilinear form (7, 7') = cB(H-y , 7,7/ E FYR , on IYR x WR , where c is a fixed constant. If a E 4 then the formula 2(7, a)

(a, a)

a,

E

4 /R7

(5)

defines a reflection of the space ifR . Clearly, wa = w_ a . The group W generated by all reflections wa , a E 4 is called the Weyl group of the Lie algebra g with respect to 4. W is a finite group which acts transitively on L. A root a E is said to be simple if it is not the sum of two other positive roots. We state some properties of simple roots: (i) Any positive (negative) root is a linear combination of simple roots with nonnegative (nonpositive) integral coefficients. (ii) The number of simple roots is equal to the rank of the Lie algebra Simple roots are g, that is, it is the dimension I of the Cartan subalgebra linearly independent.

6.1 Definitions of Drinteld Jimbo Algebras

159

-

(iii) If a l , a2, • - at are simple roots of g, then the reflections wai , j = 1, 2, • - • , 1, generate the Weyl group W of g. is not uniquely determined. If Note that the set of simple roots of a l , a2, • • • , ai are simple roots, then wai, wa2, - - • , wai for any fixed w E W are also simple roots. In particular, the set -a i -a 2 , • • • , -at (which corresponds to the basis -I/1 , -H2, • - • , -I11 of 4) is a set of simple roots. e n_, where Formula (1) gives a direct sum decomposition g = n+ ga and n_

Proposition 1. The subalgebras n+ and n_ are maximal nilpotent subalgebras of g. Every maximal nilpotent subalgebra of g can be mapped by an inner n+ and + n_ are maximal autamorphisrn onto n+ . The subalgebras solvable subalgebras in g.

Let us fix an ordered sequence a l , • - • , at of simple roots. Let Eai , • • - , be the corresponding root elements of g. By (4), successive application of the commutator [-, -] gives all root elements Ea , a E A + • That is, the elements Ea, , • • , Ea, generate the Lie subalgebra n + . Analogously, the root elements E_ a„ • • • , E_ a, generate the Lie subalgebra n_. The elements i 1, 2, • • • , I, form a basis of the Cartan subalgebra Thus, the complex semisimple Lie algebra g is generated by the root elements Eai , • • • , Ea,, E,,, • , E„, corresponding to the simple roots al , • - • , at. The 1 x 1 matrix A = (a ii ) with entries

= is called the Cartan matrix of g. It determines the semisimple Lie algebra g uniquely up to isomorphisms. It can be proved that (i) det A 0; (ii) aii are integers, aii E r IC-y Fr/ gives an isomorphism of the vector spaces Uq (n+) Uq (b) Uq (n_) and Uq (g).

Proof. Since the algebra Uq (g) is spanned by the elements Er Ky Fri, r, r' E I, C Q, it suffices to prove that the elements Er FriKy are linearly independent. Suppose that a finite sum a E arr / Er Fr/Ify is zero, where a rr., / E C. If Er and Fr/ are of Q-degrees A r and —AT!, respectively, then Er Fri.K.7 is of Q-degree — A ra. Without loss of generality we may assume that all summands of a are of fixed Q-degree — A r,. We introduce an order in Q. If a = ni ai and a' = then we write a < a' if E ni n and ni < n'i for the smallest index i such that n i n'i . Let J be the set of r C I for which the degree 7, of the term E r in the sum a is maximal with respect to this order. Clearly, the sum of all terms in ,(a) which have maximal Q-degree in the first tensor factor and minimal Q-degree in the second one must vanish, that is, -y

E EJ,r'EJ'

IC-r,,Fra)(K-1

= 0,

where 7, and -yr, are fixed. Hence Ery,rEJ,r / E.» arr'/( E r iCy 0 Fr' K y ) 0. Since the sets {FriKy } and {E r Ky } are linearly independent, we get a„ , ,y = 0 for r C J and r' E J'. Replacing a by a — arri-t (ErFrIK-y) and repeating this procedure it follows that all coefficients arri,y are zero. D Applying the automorphism 9 to the basis elements in Theorem 14 we obtain the following assertion: The elements Fr ifEr,, r, r' E I, C Q, are also a basis of the vector space Uq (g). The mapping Fr oKy ®Erf —> Fr Ky Erd defines a vector space isomorphism of U q (n_) Uq (1)) Uq (n+) and Uq (9). Remark 6. Recall that Uq (n+ ), Uq (n_) and Uq (b) have been defined as subalgebras of Uq (g). Using the approach sketched in Remark 1 it can be shown that Uq (n+ ) (resp. Uq (n_)) is isomorphic to the algebra with generators E 1 ,. ,E1 (resp. F1 , ... ,F1) and defining relations (15) (resp. (16)) and that the elements Ky , -y E Q 2 form a vector space basis of Uq (1)). These facts hold for any q E C such that q? 1. For such q C C the multiplication map gives vector space isomorphisms of Uq (n+ ) Uq (i)) Uq (n_) and Uq (n_) Uq (b) Uq (n+ ) to Uq (13). However, if q is a root of unity, then the decompositions (51) and (52) are no longer direct sums.

6.1 Definitions of Drinteld-Jimbo Algebras

6.1.6 Hopf Algebra Automorphisms of

171

Uq (e)

In this subsection we suppose that q is not a root of unity. The classification of Hopf algebra automorphisms of Uq (g) is derived from

Uq (g) be a coalgebra homomorphism. Then Lemma 15. Let .11) : Uq (9) there are elements y, G Q + and finite subsets 'il, 'i2, fi, fir2 of the set I from Theorem 14 such that 71b(Ki ) —

,

'CEO —

rti)(Fi ) = where

ar bS 7 Cr

E

E

arE,

E sE/i2

7

(55)

rEJii

ds E C. For r

E

I

(54)

Uf1 and

s

E /2,2 UJ2

we have

E r E Ug"ti (n + ) and F, G Uq- "li (n_ ).

(56)

Proof. By Theorem 14, the element 1/)(Ei ) can be written as a finite sum E ar -yr iE r K7 F r, with ar,fri E C. Since is a coalgebra homomorphism, we have A((E i )) = o A4(E). Using (17) and (18), we expand A(ti)(E i )) and (000) oA(Ei ) in terms of the basis elements Er .K.,Ye. Comparing coefficients we conclude that the term 0(Ei )07/)(Ki ) appearing in the expression op 0) o A(Ei ) must be of the form given by (54) with Er and F, as in (56). Formula (55) is proved analogously. 0 Let A = (a ii ) be the Cartan matrix of the Lie algebra g. A permutation fi of the set {1, 2,• • • , 1} of vertices of the Dynkin diagram of g such that = aii , j , j = 1,2, • • • , 1, is called a diagram automorphism of g. From the explicit form of Dynkin diagrams it is easily seen that the Lie algebras B1, 1 > 2, and C1, 1 > 2, have no nontrivial diagram automorphisms. The Lie algebras A1 and DI, 1 > 4, have precisely one nontrivial diagram automorphism p and p has order 2. The Lie algebra D4 has nontrivial diagram automorphisms of orders 2 and 3. Theorem 16. Let e l , c2, - • • be nonzero complex numbers and let II be a diagram automorphism of g. Then there exists a unique Hopf algebra automorphism V) of Uq (g) such that Y

i) = K12(i), 7P(Ei ) = ci .E12 (i),

CFO

ci7 1 Fii (i).

Every Hopf algebra automarphism of Uq (g) is of this form. Proof. It is easy to verify that each such rti) is a Hopf automorphism. In order to prove the last assertion we apply Lemma 15. Let be a Hopf algebra automorphism of Uq (0). Since 1/) is invertible, the element -yi satisfying = b(K) must be a simple root a p (i). Moreover, p defines a permutation of the roots in the Dynkin diagram of g. By (54), IP(Ei ) is of the form ai.Eii(i) bi.K.y,Fii( i). Since is also an algebra automorphism, we have 1/)(Ki)IP(Ei)C.Ki r 1 = egEi). This equation implies that bi = O. The assertion concerning 0(Fi ) is derived from (55) in a similar manner.

172

6. Drinfeld—Jimbo Algebras

6.1.7 Real Forms of Drinfeld—Jimbo Algebras As discussed in Example 1.10, real forms of the complex Lie algebra g correspond to Hopf *-structures on the Hopf algebra U(g). By a real form of a Drinfeld—Jimbo algebra Uq (g) we mean an involution on Uq (g) such that the Hopf algebra Ug (g) becomes a Hopf *-algebra. As for Uq (512 ), we distinguish three domains of parameter values. Case 1: q E R. In this case the real forms of Uq (g) are described by Proposition 17. Let ti be a diagram automorphism of g and let ci , i = 1, 2, • • , 1, be nonzero complex numbers such that tt2 = id and cp,( i) = There exists a real form of Uq (g) with involution determined by IC: = KA (i) 7 Ei

Fi*

Ci 1

(i)

p (10 .

Two real forms defined by the data ( 1a, c) and (fil, e) are equivalent if and only if there exists a diagram automorphism v such that vkl = jiv and um > 0 for all i satisfying p(i) = Proof. The proof is given by a direct verification. A closer look at the equivalence conditions shows that up to equivalence there are only the following standard real forms:

K: = K1., (i) , EiK = criK p (i)Fti ( i) ,

Fi* = cri E j(i) K ja(1.0 ,

(57)

where p is a diagram automorphism of g such that p 2 = id, ai = 1 for p(i) and ai = ±1 for tt(i) = i. The real form of Uq (g) defined by (57) with p = id and cr = 1, i = 1,2,• • • , 1, is called the compact real form. The compact real form of Uq (s1,,) is denoted by Uq (sun ). Putting p = id in (57) we obtain a real form of Ug (sin ) denoted by Uq (su(Gri, • • .) ai)). Case 2: q = 1. Proposition 18. Let 1u be a diagram automorphism of g and let e i 1,2, • • • , 1, be nonzero complex numbers such that p 2 = id and ci,( i) = 1. Then there is a real form of Uq (g) with involution determined by ,

K: = Kp (i),

E = ci Ei.,( i) 7 Fi* =

F

If q is not a root of unity, then two real forms defined by the data ( 1u,ci ) and (p', c) are equivalent if and only if there exists a diagram automorphism v such that up = jilt'. The equivalence conditions allow us to restrict ourselves to the following standard real forms: (i) 7

Ft:* = Fp

7

(58)

6.2 Poincaré-Birkhoff-Witt Theorem and Verma Modules

173

where p is as in Proposition 18. For Uq (slri ) and p = id this real form is denoted by Uq (s1,2 (R)). It is an analog of the universal enveloping algebra U(s1(n, R)).

Case 3: q E

q

Proposition 19. Then the Hopf algebra Uq (g) has real forms if and only if = sp(2n,C). In this case, the corresponding involutions are given by

1‘7 = Ki ,

=

Fi* =

where ci are nonzero complex numbers such that ci = (_1)d. Two real forms with data ci and c are equivalent if and only if eici-1 > 0 for all i. In this case we may restrict ourselves up to equivalence to the real forms K i* = Ki ,

E = (-1)d1 siKi Fi ,

of Uq (sp2n ), where si = 1 for even di and s i

Fi* = for odd di .

Using the form of Hopf algebra automorphisms of Uq (g) obtained in Subsect. 6.1.6 it can be shown that the above lists are exhausting. Theorem 20. If q is not a root of unity, then the Drinfeld-Jimbo algebra Uq (g) has real forms if and only if q E R orlqi = 1 or q E JTR g = sp(2n,C). All real forms of Uq (g) are then described in Propositions 17-19. ,

6.2 Poincaré-Birkhoff-Witt Theorem and Verma Modules In this section we construct explicitly a vector space basis for the DrinfeldJimbo algebra Uq (g) in terms of general root vectors. The main technical tool for doing that is an action of the braid group of the Lie algebra g as an automorphism group on the algebra Uq (g). 6.2.1 Braid Groups

If A =-- (aii ) is the Cartan matrix of g, then the numbers aij aii may be equal to 0, 1, 2 or 3. Let mii be equal to 2, 3, 4, 6 when aij aii is equal to 0,1, 2, 3, respectively. Recall (see [Bou2]) that the Weyl group W of g can be defined as the group generated by the reflections wi, w2, • • • , w (corresponding to the simple roots of g) satisfying the defining relations 14. = 1,

= 1, 2, • • • , /, .wiw i wi wi • • • =-- wiwiwiwi • - • , i

(59)

where in the latter equations there are mii w's on each side.

Example 1. The Weyl group W of sl(/ + 1,C) is the permutation group Pi+iIt is generated by elements wi, w2, - • - ,wi with defining relations

174

6. Drinfeld—Jimbo Algebras

7.4 = 1, wiwi+iwi =

and wi wi wi wi if — ji > 1.

A

Definition 3. The braid group associated with g is the group 9.3g generated by elements s i , • - • , si subject to the relations SiSiSiSi

• • • =

SjSiSi Si

(60)

• - -

s's on each side.

where there are

Example 2. The braid group 93g for g = sl(/ + 1, C) has 1 generators and the defining relations

s i si±i si = si±i si si±i and s i si = si s i if

ji > 1.

This group is usually called Artin's braid group on 1 strands.

A

Clearly, a braid group contains an infinite number of elements, because the elements s E Z, are pairwise distinct. The correspondence sri' n, n E Z, j= 1,2, - 1, defines a homomorphism from 939 to Z. If J denotes the normal subgroup of q3g generated by the elements 4, = 1, 2, • , 1, then the quotient group 93g IJ is isomorphic to the Weyl group W of g. Thus, we have a natural homomorphism cp from 93g onto W such that cp(si) = w i . The braid groups 93g have the following important property. ,

Proposition 21. Let w = wi 1 w i2 • - • wi,, k = 1(w), be a reduced decomposition of an element w E W. Then the element s t, = si1 si2 • • • si, of 93g depends only on w and not on the choice of reduced decomposition for w. Proof. The proof is given in [Bou2], Chap. 5, § 1.5, Proposition 5.

111

6.2.2 Action of Braid Groups on Drinfeld—Jimbo Algebras

The importance of the braid group Sg in the present context stems from the fact that it acts on the algebra Uq (9). Theorem 22. To every i, i = 1, 2, - • there corresponds an algebra automorphism Ti of Uq (g) which acts on the generators H E Fi as

—Fi K i ,

Ti (Ki )

T(Ej) =

i

a,3 E

Ti(Fi ) =

r=0

E

(-1raii q

r=0

11;(Fi ).=

r(Ei )(— aii—r)Ei ( Ei )( r),

(_iraii,r(Fi ) (r) Fi

—r) 7

(61) i

j, j1

(62) (63)

where

=

l[n] qi !,

(Fi ) (n)

The mapping si --+ 7; determines a homomorphism of the braid group 93 0 into the group of algebra automorphisms of Uq(g).

6.2 Poincaré-Birkhoff-Witt Theorem and Verma Modules

175

Proof. The proof is given by direct and lengthy calculations. First one shows T(F) given by (61)-(63) satisfy the definthat the elements Ti (Ki ), ing relations (12)-(16) of Uq (g). Thus, there exists an algebra automorphism

Ti as stated above. Then one verifies that the automorphisms 7; satisfy the defining relations (60) of the braid group B9. Hence the map s i -+ 1; defines 1=1 an action of 93 9 on Uq (g). The braid group 939 acts also as a group of algebra automorphisms on the h-adic Drinfeld-Jimbo algebra Uh(g). The corresponding algebra automorphisms Ti are given on the generating elements Hi , Ei and Fi , j = 1, 2, • - 1, by the formulas (62) and (63), with qi replaced by e dz h and by

Ti (Hi )= H -

Z(E) =

_FiedihHi Z(F)

e—dihHiEi.

(64)

Clearly, this braid group action induces the classical action of the Weyl group . W on the subalgebra Uh (1)) of Uh(g) which is generated by H1 , • 6.2.3 Root Vectors and Poincaré-Birkhoff-Witt Theorem

In the universal enveloping algebra U(g), there are root elements for every root of g. In Uq (g) we have so far only root elements Ei and Fi corresponding to the simple roots of g. Using the braid group action on Uq (g), we shall define root elements of Uq (g) for arbitrary positive and negative roots of g. Let wo be the longest element of the Weyl group W of g and let wo = , ai be the wi , wi, • - • w ir, be a fixed reduced decomposition of wo. Let a l , simple roots of g. Recall that by Proposition 4 the sequence =

Wii (Cti2), '

/372 = W il • • • W in-1

(ctiTi)

exhausts all positive roots of g. The corresponding root elements of Uq (g) are obtained by Definition 4. The elements

Efir

=

- • • Tir_ i (Eir )

and

Fpr

Ti1Z2 ' • • Tir- I (Fir)

( 65 )

from Uq (g) are called root vectors of Uq (g) corresponding to the roots Or and -A., respectively. Root vectors for the h-adic algebra Uh (0) are defined similarly. The root vectors of the universal enveloping algebra U(g) can be defined in the same way using the Weyl group W instead of the braid group. Then different reduced decompositions of wo give, up to signs, the same root vectors of U(g). This is no longer true for the root vectors (65) of Uq (g).

Example 3. For the Lie algebra g = sl(3,C) there are precisely two reduced decompositions wo = w1w2w1 and wo = w2wiw2 of the longest element wo in W. The corresponding sequences of positive root vectors of Uq (s13) are

176

6. Drinfeld-Jimbo Algebras

E1,

TI(E2)

E2 7

1-2(E1) = — E2E1

— E1E2 + Q-1 E2E11

T1T2(E1) = E27

E1E2,

T2 71(E2) =

Note that Ti(E2) is not proportional to T2(E1).

A

Proposition 23. (i) If w E W and ai is a simple root such that w(c) E L , then T(E) E Uq (n+) and Till (Fi ) E Uq (n_.). All root elements Ef3, and For ,

r =1,2, • • • n, from (65) belong to Uq (n+) and Uq (n_), respectively. (ii) If *xi ) = ai is a simple root, then T(E) = Ei and Tip(Fi ) = F.

(iii) For the root elements Er and For we have If),Eo r KT 1 = q( "r) Epr ,

If ),For lf = q-(43r) F3r .

Proof. The proof is given by direct calculation using the definition of Proposition 4 and the formulas (62), (63) and (65).

3f3 ;

Having root elements in Uq (g) corresponding to all roots of g we can state an analog of the Poincaré-Birkhoff-Witt theorem for Uq (g): Theorem 24. Let Epr , For , r = 1,2, • • • , n, be the root elements from Defi-

nition 4. Then the following set of elements is a vector space basis of Uq (e): • • - For:Kti l • • • Kit ' Ef3s : • • • Ef38117 ri ,si E No , ti E Z.

(66)

Proof. The proof of this theorem can be found in [Ros2] and [Yam] for Ug (sin) and in [Lus] for the general case. Corollary 25. The elements Eroli • • • E

r1, - • • , rn E No, form, a vector space basis of the subalgebra Uq (n+) of Uq (p). Likewise, the set of elements Fka • • • 1114, s,. 7 s i E No , is a vector space basis of Uq (n_). In particular we see that there is a natural one-to-one correspondence between basis elements of Ug (n+) and U(n± ). It can be shown by using Theorem 24 that the algebra Uq (g) has no zero divisors, that is, we have ab 0 in Uq (g) if a 0 and b 0 (see [DC1(]). An analog of the Poincaré-Birkhoff-Witt theorem holds also for the hadic algebra Uh(g) • Theorem 24'. Let E , For , r = 1, 2, • • - n, be the root elements of the

algebra Uh(g). Then the set of elements Fr1 [3].

on lit' • • • 1111 Est'.

• • • F rn

si Si'

ri ti , si E No ,

is a basis of the C{[h]j-vector space Uh(D)• From Corollary 25 we obtain additional information about the decomposition (52) of the subalgebra Uq (ni ). The dimension of the subspace UP (ni ) from (53) is equal to the number of basis elements in Uq (ni) of degree ± 3 . As in the classical case, the latter number coincides with the Kostant partition function K(3). If 3 0, then K (/3) is defined to be the number of partitions

6.2 Poincaré-Birkhoff-Witt Theorem and Verma Modules

177

of the linear form f3 into a sum of positive roots of g, where roots may enter into partitions with multiplicities. If = 0, then K(/3) := 0. Thus, we have dim U(n_) = dim U(n) K(/3).

(67)

6.2.4 Representations with Highest Weights Let T be a representation of the Drinfeld-Jimbo algebra Uq (g) on a vector space V. For any function A on the root lattice Q we set VA=IXEVI T(K,)x = A(a)x for all a E Q1,

• KT 1 for a = where K = vial. That is, each VA is a joint • eigenspace of the commuting operators T(Ki ), j = 1, 2, - • • , 1. Definition 5. If VA {0}, then we say that the function A is a weight, the number n-tA = dim VA is the multiplicity of the weight A and VA is a weight subspace of the representation T. The nonzero vectors in VA are called weight vectors. A representation T of Ug (0) is called a weight representation if its underlying space V decomposes into a direct sum of weight subspaces. Weight representations are the most important representations of Ul (g). If q is not a root of unity, then every finite-dimensional representation of Ug (9) is a weight representation (see Subsect. 7.1.1 below). Definition 6. A weight representation T of Uq (g) on a vector space V is called a representation with highest weight if there exists a weight vector E V such that T(K,)eA , = A'(a)eA , , a E Q,

T(E)en' = 0 for i = 1, 2,

, 1 and T(lq (g))eA , = V.

(68)

We then call the function A' on Q a highest weight and the vector eA , a highest weight vector of the representation T. Since Ug (n t ) contains the unit element 1, the first condition in (68) implies that T(Uq (n + ))e iv = Ceiv. The last formula in (68) means that the vector eA0 is cyclic for the representation T. From (68) we easily derive the relations V = T(Ug (n..4)T(UA))T(Uq (n + ))eA , = T(Uq (nAeiy,

(69)

V = E0EQ+ T(Elg- °(n_))eAl.

(70)

Remark 7. The definition of highest weight representations depends on the positive roots and so on the corresponding ordered sequence a l , a2 , - • - , ai of simple roots of g. As noted in Subsect. 6.1.1, we may also take the ordered sequence of simple roots -a i , -a2 , - , -al. In the latter case the elements Ki-1 , Fi play the role of the elements Ki ,Ei in Definition 6. That is, T is a highest weight representation with respect to a l , a2, • • , ai if and only if T o 0 is a highest weight representation with respect to -a1, -a2, • • • I -cti!

6. Drirdeld-Jimbo Algebras

178

where 0 is the algebra automorphism of Ug (9) from Proposition 9. Moreover, A'(a) is a highest weight of T if and only if A'(-c) is a highest weight of T o O. All results of Subsects. 6.2.4-6 and Chap. 7 remain valid (under appropriate reformulations) for highest weight representations with respect to -a l -a 2 , • • , -ai. We shall use this setting in Subsects. 8.4.1, 11.2.3, 11.5.3 and 11.6.4 below. A In the rest of this section we suppose that q is not a root of unity. Proposition 26. Let T be a representation of Uq (g) with highest weight A' on a vector space V. Then the sum (70) is a direct sum and gives the weight subspace decomposition of the space V. We have

T(t.1,7 (n_))e A l ,

where

,u(a)

A'(c).

(71)

All weights of the representation T are of the form pt(a) = ni E No and ai are the simple roots of g.

wher

Proof. Let s E U(n_). By (53), we have

T(K)T(x)e n ' = T(Ifa)T(x)T(K,V)T(If cr )eAr = q

(a)T(x)eAl ,

that is, T(x)en' is a weight vector with respect to the weight q-( ',43)A'(a). Because q is not a root of unity, the sum (70) is a direct sum. Since all forms E Q+ in (70) are sums of simple roots with nonnegative integral coefficients, all weights of T are of the form stated in the proposition. El

A representation of Uq (g) is called a representation of type I if it has

a highest weight of the form Ai(a) =-- q(), where A is a linear form on the Cartan subalgebra 13 of g. We shall prove in Subsect. 7.1.2 that any irreducible representation is a tensor product of a representation of type 1 and a one-dimensional representation. Therefore, we shall restrict ourselves in this chapter to representations of type 1. If T is a representation of type 1 with highest weight A'(a) = q (') , then (with a slight abuse of language) we call A also a highest weight of T and denote T by TA. The corresponding highest weight vector eA , will be denoted by e'A- . By Proposition 26, for any type 1 representation TA we have a direct sum decomposition V = vA_0 , where

_o71 (Uci-13 (11_))eA = {X E V j 71 (1Ca )x =

c, E Uq (4)}. (72) , Note that the functions q(11-0 ) and q (A— /31 ' c') on Q do not coincide if 13 because q is not a root of unity. VA

6.2 Poincaré - Birkhoff-Witt Theorem and Verma Modules

179

6.2.5 Verma Modules The formulas (67) and (72) imply that for a type 1 representation TA with highest weight A on a space V one has the inequality

(73)

dim VA_13 < K(3), where K is the Kostant partition function (see (67)).

Definition 7. Let TA be a type 1 representation of Uq (9) with highest weight A on a vector space VA. The corresponding Uq (g)-module VA is called a Verma module if (74) dim VA_o = K (fi ),

, where VA_o is the weight subspace (72). A Verma module for all fi E with highest weight A is denoted by MA and the corresponding representation of Uq (g) by TT . It follows from (67), (72) and (74) that the map Uq (n_) x —) (x)eA E MA is bijective. That is, a Verma module is freely generated by the action of the algebra Uq (n_) on the vector eA. Any Verma module MA can be realized directly on the vector space Uq (n_). In this case, the unit 1 of this algebra is a highest weight vector and the representation operators are given by the formulas

E (4()),

(75)

i = 1,2,.. •,

(76)

Ff3 E tV(n_),

(77)

TX (K,)1 = q (A 'a) 1, rrV

fr'\1 = Fi ,

TX (Ei )1 = 0,

TX (K)F0 = q (A) Fp,

TX(Fi )Fp = Fi Fp,

(Ei )Fp = Ei Fp,

F, , E Uq-13 (n_).

(78)

Let us explain why the right hand side of the last equality belongs to Uq (n_) and is well-defined. Since the element Ei Fs is contained in Uq (n_)Uq (11)Uq (n+ ), it is a sum of products x_hx +,x_ E Ug (n_), h E Ug (i)) , X + E Uq (n+ ). To every such summand there corresponds a vector of the underlying space Uq (n_) of the representation TX. Namely, if the degree of x + is nonzero, then this vector is TX(x_hx + )1 = O. If the degree of x + is 0 and x_hx + E Uq (n_)Uq (0), the corresponding vector is TX(x_h)1 = TX(x_)TX (h)1. This vector is uniquely determined by the formulas (75)-(78).

Definition 8. A linear form A on i) is called integral if the numbers (A, an, = = 1,2, • • • , 1, are integers. The set of integral linear forms is denoted by P. A linear form A on 4 is called dominant if (A, ai ) > 0 for i = 1,2, • • • ,l. The set of integral dominant linear forms is denoted by P.

Verma modules of Uq (g) have similar properties as in the classical case. Let us describe some of them. Let A E P be such that (A, an > 0 for fixed j E {1,2, • • , / } and put ni := (A, an. Let MA be a Verma module

180

6. Drinfeld—Jimbo Algebras

with highest weight vector eA . Using the relations (14) and repeating the reasoning of Subsect. 3.2.1 we find that

Ei F

l eA = 0,

EiFeA 0,

EiFin l +l en = 01

The vector eA, F+ l eA has the weight A' = A — (ni + 1)ai . We have

TX(Uq (9))eAt TX(Uq (n_))TX(Uq (1)))/I(Uq (n + ))eA, TX(Uq (n_))e x , that is, T (Llq (n_.))ex is an invariant subspace and the restriction of TT to TX(Uq (n...))eAt is a representation with highest weight A' = A — (n i + 1)a i . One can see that this submodule is isomorphic to the Verma module MA'. Thus, we have shown that if a Verma module MA has a highest weight > 0, then it has a Uq (g)-submodule isomorphic to E P such that (A, the Verma module MA'. Moreover, we have (A', an < 0. If A E P+ , then such a submodule MAI exists for every i = 1,2,- • • ,l. In the latter case we have the following stronger result. Proposition 27. Let MA be a Verma module with A E P. Then for every element w of the Weyl group W of g there exists the Verma submodule in MA with highest weight A y, = w(A + p) — p, (79)

where p is the half-sum of all positive roots of g. Every irreducible Uq (g)module contained in MA determines a type 1 representation with highest weight. These highest weights are of the form (79). Proof. The proof of this proposition is analogous to that of the corresponding III assertion in the classical theory [Dix]. 6.2.6 Irreducible Representations with Highest Weights

A Verma module MA has nontrivial Uq (9)-submodules in general. It can be proved that if MA is reducible, then there is always a unique maximal proper Uq (g)-submodule M of MA. That is, any other proper Uq (g)-submodule of MA is contained in M. We denote by LA the representation of Uq (g) determined by the quotient module MA IM. Proposition 28. (0 LA is an irreducible representation. (II) If TA is an irreducible type 1 representation of Uq (g) with highest weight A, then TA is equivalent to LA. In particular, for any linear form A on the Cartan subalgebra I) of g there exists, up to equivalence, a unique irreducible type 1 representation of Uq (g) with highest weight A. (iii) If A E P+ , then the representation LA is finite-dimensional.

Proof. (i) is clear by the mwdmality of M. (ii): Let VA be the underlying space of the representation TA and let eA be its highest weight vector. By Proposition 26, VA is spanned by the vectors TA (Fii. - • - FijeA, n E No. It is easy to see that the operators TA(X),

6.2 Poincaré-Birldioff-Witt Theorem and Verma Modules

181

X E Ug g), act on these vectors by the formulas (75)-(78). Hence the linear mapping T: Uq (n_) -0 VA given by T(Fil • • • Fin ) = TA(Fi i - - • Fir )eA intertwines the representations TT and TA. Since T(M) is an invariant subspace of VA and TA is the irreducible representation of Uq (g) with highest weight A, we have T(M) = 10, so that TA LA. WO: Let i be one of the numbers 1, 2, • • • , 1. As noted before Proposition 27, MA has the submodule MA', A' = A- (ni +1)ai , since A E P± by assump(

tion. The set of weights (counted with multiplicities) of the quotient module MA/MA , is invariant with respect to the element w E W corresponding to the simple root a i . Indeed, the multiplicity no, of a weight A in MA/MA, coincides with rn'Al rnl,;:, where rnI)1, (resp. mn is the multiplicity of A in the module MA (resp. in MA , ). The multiplicities of weights for Verma modules of Uq (g) coincide with the corresponding multiplicities for Verma modules of the Lie algebra g. Therefore, the multiplicities of weights in the Uq (g)-module MA/MA , coincide with those in the quotient MA/MA , of the corresponding Verma modules for g. In the classical case, the set of weights of the module MA/MA , is invariant with respect to the element wi of the Weyl group W. Therefore, this invariance holds in the quantum case as well. Since MA contains the submodule MA I, A' = A - (ni 1)a 1 , for any i = 1, • • • , 1, the set of weights of the representation LA is invariant under the whole Weyl group W. Since MA (and hence LA) has only a finite number of dominant weights, it follows that the set of all weights of LA is finite, that is, LA is finite-dimensional. Proposition 29. Let LA be the representation from Proposition 28 with = mi ai highest weight A E P± and highest weight vector e. Let

be an element from Q+ such that (A, c') > mi , i = 1,2,- - • ,l, and let VA MAIM be the carrier space of the representation LA. Then the linear mapping Uq-13 (n_) D VA(x)e is injective. Proof. By the definition of a Verma module, it is enough to show that 3 is not a weight of the maximal Uq (9)-subnaodule M of MA. This follows from Proposition 27, because no set of weights {Ai, niai ni E Nob w E W, contains 0. D 6.2.7 The Left Adjoint Action of Uq (g)

Recall that by Proposition 1.14 (i ) the left adjoint action adL(a)b =

E a(1)bS(a(2))

defines a representation of the algebra Ùq (g) on itself. In the classical case, this representation is the adjoint action ad(X) b = Xb - bX, X E g, b E U (g), and it decomposes into a direct sum of irreducible finite-dimensional representations over the center of U(g) (see [Dix], Subsects. 2.3.3 and 8.2.4). The last statement is not true for Drinfeld- Jimbo algebras.

182

6. Drinfeld—Jimbo Algebras

An element b E Ûq (g) is called locally finite if the space adL(Ig (g))b is finite-dimensional. Let F be the set of all locally finite elements of t/i g (g). It is obvious that F is a vector space which is invariant under the representation adL and that F decomposes into a sum of finite-dimensional adL-invariant subspaces. Since adL(b)cd = E adL (b( i ))c - adL(b( 2))d, the set F is a subalgebra of 6(g). Throughout this subsection we assume that q is not a root of unity. Our aim is to describe the algebra F. We give a brief exposition for the case Ûq (s12 ) and state the main results for the general case without proofs. Let Tq (r)) denote the multiplicative group of the algebra 69.(rj), that is, Tq (I)) is the group of elements KA, A E Q. Let us begin with the case g = s12 . From the formulas (3.12) and (3.13) for the comultiplication and the antipode of Oq (512 ) we obtain that

ad L (E)a

Ealf -1 — qlf -l aE, adL(F)a = Falf -1 — q -1 1f -l aF,

,

adL(K)a = KaK for any a E

adL(K -1 )a

If -l alf

(80) (81)

6q (512 ).

Proposition 30. The set .F C Ûq (512) is not trivial and

(7q (512).

Proof. Let b = EK. Since adL(E)b = 0 and adL (K)b = qb by (80) and (81), b is a highest weight vector of a weight subrepresentation of adL . A direct calculation shows that adia (F3 )1) = 0, so that b E F. Similarly, c= FK E F. Thus, F is not trivial. Using (80) we easily find that

fl adL(E n )E = (1 - q -2i+1)En±ilf-n. fl

i=1

Therefore, since q is not a root of unity, dim ada/q (s12 ))E = co and so E g _F. 1=1 Proposition 31. S := Tg ()n.F= {K2"

r C No}.

Proof. From the first formula of (80) we derive that ad L (En)Ifs=ll . -A-Lz=1

— qs -2"-2 )En

Therefore, if s 2N0, then we have 1 q8-2i+2 0 for all j E N and hence dim adL(Ûq (sl2 ))K = a), so that 1-(3 F. Suppose now that s = 2r E 2N0. Then the above formula implies that adL(Er+ 1)K2r = 0. Similarly, adL(F 1 + 1)K2r = 0. Further, by ( 81 ), (K-n)K2r = K2 for any n E Z. From the preceding facts and foradL mula (3.5) (more precisely, the corresponding formula for Ûq (s12 )) it follows that the vector space adaq (s12 ))1f2r coincides with the span of elements adL(EiF i ) 2--j < r 1. Thus, If 2r E 1- for r E NO. .1 2r ,

6.2 Poincaré - Birkhoff-Witt Theorem and Verma Modules

183

Let S-1 := fs -1 i s E SI. Clearly, 88 -1 is a subgroup of the group Tq (4). By Proposition 31, the coset space Tq (4)188-1 consists of two elements and we have Tg (4) = SS -I- + SS -I- K -I- . Since EK and FK belong to .F (see the proof of Proposition 30) and K -2 E 8 -1 , FK -1 and EK -1 lie in .FS-1 . Since .F is an algebra and s -i , { K -2r I r E No }, the set .FS -1 is also an algebra. Therefore, the polynomial algebras C[FK-1 ] and C[EK -1 ] are contained in .F8-1 . Hence C[Flf'] • 88 -1 • C[EK -1 ] C ,FS— 1 . Since C[FK -1 ] - T(13) • C[EK -1 ] = tTigf(s12) , we conclude that Ùq (512) = .F8-

1 ± .Fs -1 K -1 .

This equality shows that although the algebra .F does not coincide with Ûq (812) it is nevertheless rather large. Note that 5 -1 n.F contains only the unit element. An explicit description of the set .7. is given by Proposition 32. (i) For any r c No, the set ,F(r) := adL(Ûq (s12))K 2r is an adL-invariant vector space of dimension (r +1)2 with a basis formed by the elements (FK - 1 )mK 2r(EK -1 )n, 0 < m,n di , i = 1, 2, • • • , 1. As it is expected from the case of Uq (512) (see Proposition 3.15), there are many additional elements in the center 3q . Proposition 47. Let Ea and E„, c E 4 be the root elements from Definition 4. Then the elements Et: , Fg:, a E 4 and Kr, j = 1,2, • - • , 1, belong to hg.

6.3 The Quantum Killing Form and the Center of Uq (g)

195

Outline of proof. For Kr the assertion follows immediately from the relations qdiaii Ei and K i Fi K i-1 = q-diao Fi . Similarly, the assertion for and Ft is obtained at once from the defining relations. The proof for the D elements (Ea ) P and (Fa ), a ai , is more involved and is omitted.

Er

We denote the elements Eg, Pc:, and Iff by e a , fa and k , respectively. Clearly, ki-1- G 3 q . For the simple roots ai we write e a, and fa, as ei and fi , respectively. Let 30 be the subalgebra of 3 g generated by e a , fa , a C and k i , k i-1 , i =1, 2, • • • , 1. We also consider the subalgebras 32, 3 0- of 3o generated by the elements k i ,k,7 1 , 1 < i < 1, the elements ea, a E 2i± , and the elements fa , a E 2i+, respectively. The sets 32+ := 383 1-3F and 32- = 30- 38 are also subalgebras of 30 •

2+, 32-

are Hag subalgebras of Proposition 48. (j ) The algebras 3o, 3, 3 Uq (g). The comultiplication of Uq (g) acts on the elements e i , f , k i as

Lt(e) = e i k i + 1 ®e, 21(fi ) = f ®1 + k i-- i

L(k) = k o ki .

(ii) We have 3 = Ug (n±) n3q The multiplication map defines an algebra isomorphism of 30- 038 0 38- onto 30 • (iii) The action of the braid group 23 g on Uq (g) leaves 30 invariant. .

It is quite remarkable that the formulas for the comultiplications of the elements e = Er and fi = FiP are the same as the corresponding formulas (18) for the generators Ei and Fi , respectively. The following theorem determines the structure of Uq (g) in terms of the subalgebra 3 0 of the center 3q . Theorem 49. The algebra Uq (g) is a finite-dimensional linear space over 30 with basis Wrn 1,41 ri 8rL V81 1 < ri ,ti ,si < p. ,3n 11 ,31 '01 Let Spec (3 q ) and Spec (30 ) denote the sets of all characters on the commutative algebras 3 g and 30, respectively. That is, Spec (3 g ) and Spec (30 ) consist of all algebra homomorphisms of 3g and 30 , respectively, to C. These sets are of importance for the study of irreducible representations of Uq (g). For any irreducible representation of Uq (9), the elements of 3 g act as scalar operators. Hence they determine a unique algebra homomorphism of 3 g to C, called the central character of the representation. The following two important results allow us to describe the structure of Spec (30 and Spec (30). •

I-1

Theorem 50. The elements e a , fa, a c L ki , i =-- 1, 2, . • • ,1, are algebraically independent (that is, they do not satisfy a nontrivial algebraic equation with complex coefficients). The center 3g of Uq (g) is algebraic over 3o (that is, every element of 3g fulfills a nontrivial algebraic equation with coefficients in 30 ). ,

Example 4 (Uq (s12 )). The center of the algebra Uq (512) is generated by the elements EP, FP, KP, K -P and Cq , where Cq is the Casimir element (3.6).

196

6. Drinfeld-Jimbo Algebras

It follows from (3.6) that Cq - (q q -1 ) -2 (Ifq+ relation, one easily computes that

. f4= (cq P-1

( 17

--= FE. Using this

1 )2 (Kqi+1 1‹. 1 17 -±4 )) = EpFp

T0

that is,

+ 710: -1 + • • • + where y G C and X = (-1)P(q - q -1 ) -2P(KP

+ X = EPFP, K -P)

G 30-

(110) A

Since the elements e a , fa and ki are algebraically independent, the set Spec (so) is isomorphic to C2 ' x (C x ) 1 , where Cx C\{0}, 1 = rank g and n is the number of roots in A + . Thus, the set Spec (30) is characterized by 2m + 1 = dim g complex parameters. The restriction of characters on 3g to its subalgebra 30 defines a map v : Spec (3q ) -* Spec (30 ). The sets

v -1 (0 =-- {7/ e Spec (3 g ) I v(n) = for E Spec (30 ) are called the fibers of the map v. The second part of Theorem 50 implies that for every point G Spec (so) the fiber v -1 ( ) contains only a finite number of points. Proposition 51. The fibers of y have at most p i points and the generic fiber has precisely pi points, where 1 is the rank of g.

6.4 Notes The quantized universal enveloping algebras were discovered independently by V. G. Drinfeld [Dil] and M. Jimbo [Jim 11. The triangular decomposition of Uq (g) is proved in [Rosi]. The Hopf algebra automorphisms and the real forms of Drinfeld-Jimbo algebras were described in [Tw]. The Poincaré-Birkhoff-Witt theorem is from [Ros2] and [Yarn] for g = slN and from [Lusi in the general case. Braid group actions and general root elements were invented by G. Lusztig [Li] (see also [L2] and [Lus]) and S. Z. Levendorskii and Y. S. Soibelman [LS1]. Verma modules for Uq (g) and the corresponding descriptions of finite-dimensional representations of Uq (g) appeared in [Li]. The adjoint action of Drinfeld-Jimbo algebras was extensively studied in pLi , see also the book [Jos] and the references therein. The existence of a dual pairing of Uq (b+) and Uq (b_)°P was observed by Drinfeld [Dr21. In our exposition we followed the paper of T. Tanisaki ITan31. The description of the quantum analog of the Casimir element is also taken from [Tan3}. The results on the Harish-Chandra homomorphism in Subsect. 6.3.4 appeared in [Ros3]. The center of Uq (g) in the root of unity case was investigated in [L31, [Lus], and [DCK}.

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

Weight representations and Verma modules of Drinfeld—Jimbo algebras Uq (g) appeared in Subsects. 6.2.5-7. The present chapter is devoted to a detailed study of finite-dimensional representations of these algebras. In Sects. 7.1-4 we assume that q is not a root of unity. As we have seen in Sect. 3.3 in the case of Uq (s12 ), the corresponding representation theory of Uq (g) is similar in many aspects to the classical theory. In Sect. 7.5 we investigate representations of Drinfeld—Jimbo algebras Uq (g) in the root of unity case. Then the representation theory strongly differs from the case when q is not a root of unity.

7.1 General Properties of Finite-Dimensional Representations of Uq (g) 7.1.1 Weight Structure and Classification

The aim of this subsection is to prove the following theorem: Theorem 1. Any irreducible finite-dimensional representation of a DrinfeldJimbo algebra Ug (9) is a weight representation and a representation with highest weight. Such a representation is uniquely determined, up to equivalence, by its highest weight. Proof. The proof will be given by several steps stated as propositions. Proposition 2. Every irreducible finite-dimensional representation of a Drinfeld—Jimbo algebra Uq (g) is a weight representation.

Proof. Let T be a nontrivial irreducible representation of Ug (g ) on a finitedimensional vector space V. Since the operators T(Ki ), i --= 1,2, • • • , 1 = rank g, commute with each other, they possess a nonzero common eigenvector. Let V' be a maximal subspace of V on which all operators T(Ki ) are diagonalizable. Then we have dim V' > 1 and V' = V'A I where VA/ = {v E V/ I T(Ki)v = p = (p i , • • • , p i ) . Assume on the contrary that dim V' < dim V. Since T is irreducible, V' cannot be invariant under all operators T (Ei ), TWO, i = 1, 2, • - • , 1. (Indeed, if T(E)v' E V' and T(Fi )vi E V for all weight vectors Arf E V', then V' is an invariant subspace

198

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

of V.) Let y be an element of V' such that T(E i )v is not in V' for some j E {1, 2, • • • , /}. Without loss of generality we can assume that y is a weight vector. Then there is a p, such that y E V. Then we have T(Ei )v 0 and T(E)v, that is, y" T(Ei)v is a T(K i )T (Ei )v =T (Ei )T (If i )v = common eigenvector for all T(K1 ) and all operators T(Ki ) are diagonalizable on V" = V' ® Cy". Since dim V" > dim V', this is a contradiction. Lemma 3. If T is a finite-dimensional weight representation of a DrinfeldJimbo algebra Uq (9), then there exists N E N such that T(E 1 ) -T(E) = 0 -ip E {1, 2, • • • , /}, p N. for all Proof. Let V be the carrier space of T and let V =-- ED V12' be its weight It is enough to show decomposition, where Vo = fy E V 1 T(K i )v = that for every y c V ,i = (p i , • • • , we have T(Eii ) • • • T(Eip )v = 0 for sufficiently large p. Set y' = •T(Eip )v. Then y' c Vp1, where ,

= pti q nkaik . Here nk is the number of T(Ek) = (t4, • • • , p,;) and appearing in the product T(Eii ) • • • T(E). The representation T has a finite number of weights. We denote them by p, p( 1 ), • • - AM. We shall prove that there exists N G N such that for p > N, pt' does not occur in this list. Let /1■ . : s) = /49) 44, i = 1, 2, • • - ,I. We show that there exists a number / } such that i0 E {1, 2, - - • 7

gio

nka t o k

{1 /\% : 01) - •

Let us write the complex numbers q and

A i(0r)

(1)

}.

as q = exp 2i7rv and nkaik

exp (217rvy s) ). Since qi = V aio'i ) / 2 , the equality q i that there is an integer m such that (ai, ai) 2

_ (8) ,, nkaik —

= /1■ 8) implies

m V

-

•

By the definition of aik, the latter writes as Ek nk(ai,ak) = y's) + m/v. Since Ek nk (ai, ak) E Z, we conclude that y i(s) + E Z. Since y t Q7 there exists at most one integer m such that y s) + m/v c Z. Set zi 8) := yi(8) +miv. Assume that for every i E {1, 2, • • • , /} there exists s E {0, 1, • • , r} such that Ek nk (ai, ak) = 4s) with integral 49) . This is a system of linear equations with unknowns n 1 , n2 , • • , ni. Since the l x l matrix consisting of the entries (ai, k) is invertible, for a given 491) , • • • , 4 si) there exists at most one integral solution of this system. However, the number of possible sets fz i(81) - • • , 4 81) } is finite. Therefore, if the integers ni, n2 7 • • ni do not belong to a certain finite set 931, there exists an index io such that (1) is satisfied. Let N := sup(jni I + • • - + Intl) + 1, where the supremum is taken over 931. Then the assertion of Lemma 3 holds.

7.1 General Properties of Finite-Dimensional Representations of Uq (g)

199

Let T be a finite-dimensional representation of Uq (g) on a vector space V. Recall that y E V is a highest weight vector of T if y is a weight vector such that T(Ei )v = 0 for i = 1,2, • • • , 1 and T(Uq (g))v = V.

Proposition 4. Every irreducible finite-dimensional representation of ri g (9) is a representation with highest weight.

Proof. Let T be an irreducible representation of Uq (g) on a finite-dimensional vector space V and let Vo = ker T(Ei ). Let N be the smallest number in N for which the assertion of Lemma 3 holds. Then there exist a vector y E V , / } such that vo = T(Ei, • • - Ei„,_,)v and indices • • -5 iN-1 E {1, 2, and T(E)vo = 0 for i = 1, 2, • • • , 1. Thus, vo E Vo and so Vo {0}. From the formula (6.13) it is clear that Vo is invariant under all operators T(K i ). Since Vo {0}, there exists a common eigenvector e o E Vo for the operators T(Ki ), i = 1,2, • • • , 1. Because T is irreducible, the invariant subspace T(Uq (9))e o is equal to V. Since eo E Vo , we have T(Ei )eo = 0 for all i. The preceding LI shows that eo is a highest weight vector for T.

R

Let w = (wi , • • , we), wi = ±1. Then there is a one-dimensional representation T, of Uq (g) such that

T(E) = T,(Fi ) =0, T(K) =w,

i = 1,2, .., l.

Clearly, every one-dimensional representation of Uq (g) is of this form. If T is an irreducible representation of Uq (g) with highest weight A Z., is an irreducible represen(A i , • • - , Ai), then the tensor product T P i A i , • • • , wi AO. In particutation of Uq (g) with highest weight w - A lar, the irreducible finite-dimensional representation TA with highest weight A _ w (i n _ 7 wig ni is the tensor product of representations Z., and TA, with A' = q". Proposition 5. A = (A i , • - • , Ai) is a highest weight of an irreducible finitedimensional representation T of Uq (g) if and only if it is of the form A = j. q fl= (wiq fl1 ,..., wjq t2) with wi =- ±1 and ni G No .

Proof. Let U(s12 ) be the subalgebra of Uq (g) generated by the elements Ei , Fi , Ki , Ki-I- . It is isomorphic to Uqi (51 2 ). We restrict the representation T to Uqi (s12). Since finite-dimensional representations of Uqi (s12 ) are completely reducible (see Proposition 3.12), this restriction decomposes into a direct sum of irreducible representations of Uqi(s12 ). The highest weight vector e of T belongs to the carrier space of one of these irreducible representations and is its highest weight vector with weight A. By the results of Subsect. 3.2.1, Ai is of the form stated in the proposition. Conversely, let n = (n i , • - • ,ni), ni E No , and w = (w 1 , • • • , we.), w. = ±1. By Proposition 6.28, there is an irreducible representation L r, with highest weight qn. Then Z., L„ is an irreducible representation with highest weight A = w • qn. El Proposition 6. Every irreducible finite-dimensional representation T of Uq (g) is uniquely determined, up to equivalence, by its highest weight.

7. Finite-Dimensional Representations

200

of Drinfeld-Jimbo

Algebras

Proof. Let A = w • qn and A' = w' • qn i be highest weights of the representation T. From the form of eigenvalues of the operators T(K1 ) we see that w = w'.

Since w = w', the irreducible representation qn and q", so qn = qn by Proposition 6.28.

T has the highest weights

Remark 1. By Remark 6.7, we may take the roots -al -a2, • • • , --cti as a set of simple roots of g. In this case, the elements F , i = 1, 2, • • • , 1, correspond to simple roots. Then highest weight vectors are taken with respect to the operators T(Fi ) instead of T(E), and weights A = (A 1 , • • • , A1) and the corresponding weight vectors y are defined by T(Ki-l )v = Av. All results of this chapter are true (under appropriate reformulations if necessary) for this setting. This approach will be used in Sect. 8.4. 7.1.2 Properties of Representations Let T be an irreducible representation of Uq (g) on a finite-dimensional vector space V. Then T is a representation with highest weight A = (A 1 , - • - Ai) and corresponding highest weight vector e. We repeat the main properties of such representations from Subsects. 6.2.5-7: (i) V is spanned by the vectors e and T(F11 ) • • • T(Fip )e, {1, 2, • • - 1 } . Moreover, dim VA = 1.

(ii) The vector T(Fii ) • • • T(Fip )e in V is of weight t = (p i , • -

• • • , ip E

pi) with

aki3 where aki are the entries of the Cartan matrix of g. Every weight of the representation T is of this form. =

The counterpart to Proposition 5 for the algebras Ùq (g) is

Proposition 7. If (0.) = (w i ,. • • ,co), Wk E {1, -1, and n = (n 1 , • • • ,n1) is a dominant integral weight for the Lie algebra g, then A = ce-qn is the highest weight of some irreducible finite-dimensional representation of Oq (g). Moreover, every highest weight of an irreducible finite-dimensional representation of Ùq (g) is of this form. The classical H. Weyl theorem on 4Comp1ete reducibility has the following quantum analog.

Theorem 8. Each finite-dimensional representation of U(g) or of 6, (g ) is completely reducible. Proof. A proof of this theorem can be found in each of the papers [Ros11,

[APW1 and [JL].

Any dominant integral weight n = (n1 , n2 , - • • , ni) corresponds uniquely to a dominant integral form A on the Cartan subalgebra of g by ni = 2(A, ai)/(ai . We shall denote the irreducible finite-dimensional representation T of Uq (g) with highest weight q" by T. and also by TA with a TA are called of type, 1. slight abuse of notation. Such representations Ti,

7.1 General Properties of Finite-Dimensional Representations of U9(g)

201

Since any irreducible representation is a tensor product of a representation T„ and a one-dimensional representation Z.„ we study in the following mainly type 1 representations. The following properties of representations Ta-- TA are similar to those of irreducible finite-dimensional representations of 0. Proposition 9. (i) The Weyl group W of the Lie algebra 0 acts naturally on the set H A of weights of the type 1 irreducible finite-dimensional representation TA of Uq (g). This action of W leaves H A invariant and preserves the dimensions of weight spaces. (ii) If Q' is the element and ci.) is the operator from Subsect. 6.3.3, then q (A+p,A-i-p) we have TA W I )(.4) (Hi) For any type 1 irreducible representation TA of Uq (g) we have TA(Z) = , Z E 3 q , where 3 q is the center of Uq (g) and A A E P+, is the central character of Uq (g) from Subsect. 6.3.4. The central characters separate the finite-dimensional irreducible representations of Uq (g), that is, for any two representations TA' and TAI A' 74 A", there exists an element (Z) 74 Z c 3 q such that Proof. Since TA is equivalent to the representation LA from Subsect. 6.2.6, the proof of (0 is in fact given by the proof of Proposition 6.28(iii). The assertion of (ii) follows from Corollary 6.43. The first part of (iii) follows from the formula A = A oy o 411 (see Subsect. 6.3.4) and from the expression (6.105) for the elements of 3 q . The second part of (iii) is a consequence of Theorem 6.46. 1=1 Let us describe the highest weights A of type 1 irreducible representations of Drinfeld—Jimbo algebras Ug (0) corresponding to the simple Lie algebras sl(/ ± 1, C), so(21 + 1,C), sp(2/,C), and so(21, C). As in the classical case, it is convenient to characterize the highest weights A by the following numbers n = (n i ,n2, • • • ,ni), where ni = mi . Suppose that A Then the relations between the numbers n i and mi are

ni = mi mi+17 = 1, 2, • • -,l, for Uq(sli+1), ni mi — i = 1,2, • • • ,1 —1, n = 2mi for Uq(s 0 21+1), ni = mi — = 1,2, - - l —1, ni = mi for Uq(sP21), ni = mi — mi for Uq (s021)• = 1, 2, - 1, nI = —

Note that ni_i = ni for tlq (so2t) if and only if mj = 0. For Uq (sli+i ) and Uq (sp21 ) all numbers mi are integers, while for Uq (soN) they are all integers or all half-integers. They satisfy the dominantness con-

ditions

m1 m2

• • • > mi + i

m1 > m2 > • • • > mi > 0

for

for

Uq (slz+1),

Uq (80 21 +1 ), Uq(sp2i)

mi > m2 > • • • > mi_i > mi I for

Uq (sow).

(2) (3) (4)

202

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

For g = so2/ and g = sp2i there is a one-to-one correspondence between irreducible finite-dimensional representations of U(g) and collections of numbers (m1, m2, • - , mi) as described above. This is not true for the algebra describe •••, Uq (sli+i ). Two such sets (m i , m2 , • - - ,mi+i ) and the same irreducible finite-dimensional representation of Uq (sli+i ) if and only if there exists an integer m such that fhi m i + m, i = 1, 2, • • • , 1 +1. 7.1.3 Representations of h-Adic Drinfeld-Jimbo Algebras

By Proposition 6.8, the h-adic algebra Uh (9) is isomorphic to U (2)[[h]], where U(g) is the universal enveloping algebra of g. Therefore, as noted in Subsect. 1.3.1, if T is a finite-dimensional representation of the (complex) algebra U(g) on a (complex) vector space V, then formula (1.50) defines a representation U (9)[[h]] on the C[[h]]-vector space V[[h]]. Th of the h-adic algebra Uh(g) Conversely, let V be a finite-dimensional complex vector space and let T' be a representation of Uh (g) on Vh := VDT. Then it is clear that the equation

T(x + hUh(9))(v + hVh) := (x)v + hVh, x E U(g),v

E V,

V such defines a representation T of Uh(9)1hUh(g) _`2 U (g) on Vh/hVh that Th = . It is easily seen that T is irreducible if and only if Th is indecomposable. Thus, we have proved the following Proposition 10. The map T Th determines a one-to-one correspondence between finite-dimensional representations of the complex Lie algebra g and representations of the h-adic algebra Uh(g) on C[[h]]-vector spaces of the form V[[h]], where V is a finite-dimensional complex vector space. The carrier space V[{11 of an indecomposable representation (TA)h of

Uh(g) with highest weight A decomposes into weight subspaces V [[ h]] = e

[[ h]] , V [[h]] 1, = Vi,,

where VI, is the corresponding weight subspace of the representation TA of g on V. In particular, we get dimq[h]] = dimc V. A similar assertion is true for the representations TA of the Drinfeld-Jimbo algebra Uq (g). Proposition 11. A type 1 irreducible finite-dimensional representation TA of U7 (9) acts on a space of the same dimension as the corresponding irreducible representation TA of the Lie algebra g. Moreover, the dimensions of weight subspaces, corresponding to the same weight in these representations of and g, coincide.

Uq (g)

Sketch of proof. For g = sl(n,C) this will be shown in Subsect. 7.3.3. For a general Drinfeld-Jimbo algebra Uq (9) it can be proved by means of Verma MA are the Verma modules of g and Uq (g), respectively, modules. If M with highest weight A, then by (6.74) the dimensions of their weight subspaces, corresponding to the same weight, coincide. We can construct the

7.1 General Properties of Finite-Dimensional Representations of U9 (9)

203

irreducible finite-dimensional representations TA of Uq (g) and of g by means of their Verma modules MA and M. By Proposition 6.28, the representations TA are realized on MA/M and MA/M', respectively, where M and M' are the maximal proper submodules. By Proposition 6.27, the submodule (resp. M') coincides with the sum of all Verma submodules Mn w (resp. A/Au, ) with A w = w(A + p) p, w E W , w 1. This implies that the weight subspaces of M and M', belonging to the same weight, have the same dimensions. Therefore, the dimensions of the corresponding weight subspaces in MAN and MA/M' coincide. El —

Most of the above considerations and facts remain valid almost verbatim for representations of the quantum algebra N(g) over the field Q(q). In this case the representation spaces are vector spaces over Q(q). 7.1.4 Characters of Representations and Multiplicities of Weights

Proposition 11 allows us to define characters of finite-dimensional representations of Uq (g) which characterize these representations up to equivalence. If T is a type 1 finite-dimensional representation of Ug (0) on a vector space V with weight subspace decomposition V = (BoEp Vih , then the function x(T) =--

Et,Ep (dim 174 )ei-i

on the Cartan subalgebra I) of g is called the character of T. Here et is the function on I) defined by el.t(h) = eth(h) , h E b. Recall that the character of a finite-dimensional representation of the Lie algebra g is defined in the same way. Therefore, by Proposition 11, the characters of type 1 finite-dimensional representations of Uq (g) are in a one-to-one correspondence with characters of finite-dimensional representations of g. This leads to Proposition 12. (i ) Type 1 irreducible finite-dimensional representations of Uq (g) are determined uniquely, up to equivalence, by their characters. (ii) The character x(TA) of the irreducible finite-dimensional representation TA with highest weight A is given by the classical Weyl formula x(71/0

=

E (-1)/(w)ewo+P) wE w (_1)1(w) ew(p) E wEw

where W is the Weyl group of g, p is the half-sum of positive roots of g and 1(w) is the length of the element w E W.

Recall that the dimensions of weight subspaces 170 of the underlying space V of an irreducible finite-dimensional representation TA of Uq (g) are called multiplicities of weights p in TA and are denoted by mp,A. By Proposition 11, they coincide with the corresponding weight multiplicities in the irreducible representation of g with highest weight A. There exist several formulas for the calculation of weight multiplicities (see [Hum] or [The]);

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

204

7.1.5 Separation of Elements of Uq (g) The aim of this subsection is to prove the following theorem.

Theorem 13. Let a E Uq (g). If T (a) = 0 for all irreducible finite-dimensional type 1 representations of Uq (g), then a = O. Proof. Assume on the contrary that a nonzero element a E Uq (9) is annihilated by all such representations T. As shown in Subsect. 6.1.5, a can be represented as a=

E

ciiii fiK tL ei,fj E Uq- )31(i) (n_), ei E UP ) (n+ ), c

E CI

where only finitely many coefficients are nonvanishing. Let fio be the maximal O. element in the set of all OW for which there exists a coefficient c For A E P+ 1 let TA be the irreducible tybe 1 representation of Uq (g) on a space VA with highest weight A and highest weight vector e4 . If 0 is the algebra automorphism of Uq (e) from Proposition 6.9, then Tfi := TA 0 O is also an irreducible type 1 representation of Uq (g) on VA. Since 0(Ki ) K and 0(E1 ) = Fi , we have ri (IC ti )e A = qe4 and T(F2 )en = O. Let us denote the vector eji and the vector space VA considered for the representation 711 by e9A and Vi, respectively. Since 71 0 TAI is a direct sum of irreducible type 1 representations by Theorem 8, we have (7107'41)(a) = 0 for all A, E P. By the definition of the comultiplication of Uq (9), L(e) = ei g Kow + b, where b is a sum of terms s yE.i . Therefore, since TA(E i )eA 0, we have VI T Af)(b)(eeA 0 en/) = 0 and so

(T,1

(K)

Since (71

TA1)(ei)(4

eft')

q (i3(i )41) Ttei)e °A 0 eAl

= K 0 K 1,, we obtain

TA1)(K tiej)(e °A

0

= q (f3r4 ) '111)+( "' A+ /3(i)) 11(ei)e9A 0 en/

Further, we have A(fi) = Ki l(1) O fi + d, where d is a sum of terms sFi 0 y. Thus we get

VI

q

Tni)(fiK,ei)(e °A 0 eiti)

(p(i),4/)+(i.1,4I-4+0(i))-(01 (i),-A+00))/t ej ) e6A

O TAI(MeAl + g,

(5)

(recall that is a sum of terms from (11)-4_Fp 0 VA' with 3 < (11)_ 4 _Fp is the weight subspace of VI for the weight -A + 13). Since Po is the maximal weight in the set {OW}, the term (TIOTAi)(fiKpiej)(40eiv) lias a component in (V1)--4-00 0 only for OW = )30 and in this case this component coincides with the right hand side of (5) when g is omitted. Thus, the canonical projection of (Til 0 Tiv)(a)(e9A 0 en') onto (VD- A+0,, 0 VA' is where g

7.1 General Properties of Finite-Dimensional Representations of Uq (g)

A+0.), ( e5)eon ® TA , (f1)en'.

205 (6)

o(i)---00

Since WI TAf)(a)(e °A en,) = 0, this projection is zero. Let N be a positive integer such that fib < M ai +

/3'(i) < N (al + • • • + al)

+

for all j for which there exists a coefficient cito 0 in (6). Let PN be the set of highest weights A for Uq (g) such that 2(A, ai )/(ai , ai ) > N, j = 1, 2, • • • , l. By Proposition 6.29, the mapping e 71(e)e 6A of [430 (n+ ) to (V)-A+00 is bijective. Thus, the vectors 711 (ei )e°A with i3 (..7) = )30 are linearly independent. Therefore, the vanishing of the sum (6) implies that

E 2,12 ciwq (So, A/)+0,,,A , _A+00) _(0

, (, ),_A+00)T A

, (m eA ,

=

( 7)

for all A E PN and all j with OW = Oo• It is shown similarly that for all A' E PN the vectors TA , (f)e4' that occur in (7) with c i 0 are linearly independent. The factorse" /)- (13f (z)t - A-13°) in (7) are independent of If we cancel these factors, we get

E

Citi•q(j.1700

-A)

f q(12,A =

(8)

for all i, j with OW = 00 and all A, A' E PN. For fixed A, we consider the left hand side of (8) as a linear combination of the distinct characters A' q (12 '11') on the semigroup PN. Now we apply Artin's theorem on the linear independence of characters which is also true for semigroups. Thus, all coefficients cii,i g 00° - A) in (8) vanish. Hence c i = 0 for all i, j and j.t with This contradicts the choice of 00 and completes the proof. = The counterpart to Theorem 13 in the h-adic case is Theorem 13'. Let a E Uh(g). If T(a) = 0 for all indecomposable finitedimensional representations T of Uh(g), then a = 0.

Proof. The result follows from the discussion preceding Proposition 10 and from the corresponding result for representations of the universal enveloping algebra of the Lie algebra g (see [Dix], Theorem 2.5.7). D 7.1.6 The Quantum Trace of Finite-Dimensional Representations

If T is a finite-dimensional representation of a cocommutative Hopf algebra A, then we have Tr T(adL(a)b) = Tr

(2))TOS(a(1))) = Tr

E T(a(0)TOS(a(2)))

E TOS(a(1)))71(a(2)) = e(a) Tr T (b )

206

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

for a, b E A, where Tr denotes the "usual" trace of a linear mapping on a finite-dimensional vector space. In this subsection we give a generalization of this relation to the Drinfeld-Jimbo algebra Ûq (g). Let p denote the half-sum of positive roots of the Lie algebra g. Suppose that T is a finite - dimensional representation of the algebra I:41 (g). We define the left and right quantum traces of T by

Trq ,L T(a) = Tr T(aK2p ) and Trq ,R 71 (a)

Tr T(aK2-p1 ), a E Og (g).

The main properties of these quantum traces are given by Proposition 14. (i) For arbitrary elements a,b E Ùq (g) we have Tr q T (ad I, (a)b) = (a) Trq T (b ),

Trq ,LT(ab) = Tr q ,L Ma),

Trq ,RT (adR(a)b) = E(a) Trq ,R T , Tr q ,AT(ab) = Tr q ,RT(ba).

(II) If T is a direct sum of type 1 representations of (4(0, then

T1 q ,LT(1) = Trq,RT(1). Proof. (i ): We prove the first formula. Since adL is an algebra homomorphism by Proposition 1.14, it suffices to verify this equality for the generators a = Ki , Ei , Fi of Oq (9). First we let a = E. Since (p,ai )/(ct i ,ai ) = 1 (see, for example, [Hum]), we have (p, (xi ) = (a i , ai ). Thus, from (6.30) we obtain

K 1 K2E = q-( ai'ai ) /2 + ("i ) Ei Ki-1 K2p = Using the formulas (6.34)-(6.37) and the preceding relation we compute

Trq ,/, 71(adL(Ei )b) = Tr 71((adL(E i )b)K 2p )

= Tr T(Ei )T(bKi-1 1C2p ) qi Tr T(K i-1 )71(bEi K 2p ) Tr T(bK i-1 K 2p )T(Ei ) - qi Tr 71(bEi K2 p )T(K i-1 )

= Tr T(b)71(Ki-l- K2pEi - qiEiK2pKi-1 ) = 0 = E(Ei ) Trq,L T(b).

The proof for the generator a = Fi is timilar. In the case a = K we get Trq , L T(adL (Ki )b) = Tr T(K i )T(bK i-1 K2 p ) = Tr T(bK i-1- 1C2p )T(Ki) = E(K i)Tr q ,LT(b).

The proof of the second formula is similar. The third and the fourth formulas are obvious. (ii): It suffices to assume that T is an irreducible type 1 representation. Then T is a highest weight representation. Let ; be the corresponding rep resentation of the Lie algebra g with the same highest weight. If G is the simply connected connected complex linear group with Lie algebra g, then there is a representation T0 of G which is the exponential of Tg. From the structure of the representations T and Ta it follows that there is an element

7.2 Tensor Products of Representations

207

h G exp Ij C G such that T(K2 ) = TG(h). The Weyl group W of g can be considered as a subgroup of G. If wo is the longest element of W, then we have KV = K -2p = Kw 0 (2p) and hence h = w0h -l wc7 1 (see, for instance, [Me]). Thus we obtain Tr T(K2p) = Tr TG(h) = Tr Tc(wo)Tc(h -1 )TG(wo) 1

= Tr Tc(h) -1 = Tr T(K2p) -1 = Tr T(1C2p ), which gives the assertion of (ii). Clearly, we have Ti)(a) =

Ei Trq,r, Ti(a),

Tr q ,R (IED Ti) (a) =

Trq ,R

Ti (a).

Since 41(1(2p ) = K2p K2 p , it follows at once from the definition of the tensor product T1 0 T2 of two representations T, and T2 (see Subsect. 1.3.1) that Trq ,/, (T1 0 T2)(a) = Trq ,R (Ti 0 T2)(a) —

Trq, Ti(a(1)) Trq ,/, T2(a(2)) E Trq,R T1 (am) Trq,R T2 (a(2)) -

If T is a direct sum of type 1 representations of Og (g), then the number

dimq T := Tr q , L T(1) = Tr QR T(1) is called the quantum dimension of T. If T' and T" are two such representations, then the preceding formulas yield dimq (Ti 0 T2) = (dim q (dim q T2 ) •

Example 1. Let T1,1 be the type 1 irreducible (21 + 1)-dimensional representation of Ùq (512) from Theorem 3.13. Since K2 p = K for t'lq (512), we get 2/ q

dimqTi,/ =

i=0

(21+1)/2

-(21+10

-

q -1/2 _ q -112

= [2 1 + 141/2.

7.2 Tensor Products of Representations The investigation of tensor products and Clebsch-Gordan coefficients for representations of Drinfeld-Jimbo algebras Uq (g) of higher ranks is much more complicated than in the case of (4012 ). The reason is that multiple irreducible representations appear in the decompositions of tensor products. By Proposition 12, the multiplicities of irreducible components in tensor products are determined by the same formulas as in the classical case. We develop these results on multiplicities in Subsect. 7.2.1. Let us assume that all representations in this section are of type 1.

208

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

7.2.1 Multiplicities in Tensor Products of Representations

By Theorem 8, finite-dimensional representations of Uq (g) are completely reducible. Therefore, the tensor product TA 0 TA' of two irreducible finitedimensional representations of Uq (g) decomposes into a direct sum of irreducible components: TA 0 TA/ = (

A A' BO A MA' TA,

( 9 )

where mAA ' denotes the multiplicity of the irreducible representation TA in TA 0 TA'. This multiplicity mAA ' Ai can be expressed in terms of weight multiplicities of one of the representations TA and TA'. In order to give the corresponding formulas we introduce some notation. If p is a weight of some representation of g, let {p} be the dominant weight lying on the orbit Wp. The element w of W for which wp = {p} is denoted by w{4 }. If m the multiplicity of the weight p in the representation TA, then we have (see [K11]) ' A

AA'

77-1,À

E = M ,Ath (det w{ 4± ,v±p}),

(10)

where det w = (-1)/(w) and the summation is over all weights p of TA such that p + A' + p} = A + p. Since any such weight p can be represented in the form w(A + p) A' — p, w E W, formula (10) can be written as {

mA

= \--"‘ wEW

(-1) 1(14 M, Aw ( A±p )_

p

w Ew (-1) 1(w) 74+ 09 _ w (A' -1- P) .

From (10) one derives the following Proposition 15. The decomposition of the tensor product TA 0 TAI of two irreducible finite-dimensional representations TA and TA' of Uq (g) is given by TA 0

=

E

AZEITA

1-1

r 1 AN±P}-P1 14 ±A ±P r{±

(11)

where HA is the set of all weights of the representation TA and 0,, is the number defined as follows: 0, = O if there exists w E W, w 1, such that wv v and 0, det w{,} if no such ilement exists.

Note that the relation w(p + A' + p) = p + A' + p, w 1, means that the linear form { p + A' + p} p is not dominant, hence this linear form cannot be a highest weight and the symbol T{p± A. ±p }_ p has no meaning. But in this case the corresponding coefficient in (11) is equal to 0, so the summand in (11) does not occur. In (11), there are negative and nonnegative coefficients. Adding the coefficients of the same irreducible representation we obtain a nonnegative number which is the multiplicity of this representation. Formula (11) shows that the set of weights of one of the irreducible representations TA and TA' already determines the decomposition (9). Some special cases of the formula (11) are stated separately in Corollary 16. If the form

wA + A' + p is ,dominant for any w

E W, then

7.2 Tensor Products of Representations TA 0

TAI = El) rnA1.4 T

209

( 12)

+A l

where the summation is over all weights p E HA for which p+ A' is dominant. If the linear form w A + A' is dominant for any w E W, then the summation in (12) is over all weights p E HA. In particular, if all forms w A+ A' w E W, are dominant, then to every weight p E HA there corresponds a representation Tp+Ai on the right hand side of (9) and the multiplicity of Tp+Al in TA 0 Tx is equal to the multiplicity of the weight p in TA. If the weights A are given by the numbers ( ) i , - • , Ai) with A i = 2(A, ai )/(ai , ai ), then a direct calculation shows that for the classical complex Lie algebras A1, B1, C1, Di the forms wA+ A' are dominant for all w E W if and only if • + A l,

for A1: for B/ :

1

i

I,

•• + 2Al_i + , • • + 2 A 1- 1 + A i l • • + 2Ai - i + 2Ai,

for C/ :

1CiCI- 1,

• + At_1 + A1,

for D1 :

• • + 2A1-2 + A1-1+

1 < i < I.

If the decomposition (9) is of the form (12), then the set of highest weights A of irreducible representations which occur in (12) is contained in the set HA + A'. This statement is also true in the general case. Proposition 17. The highest weights of irreducible representations in the decomposition (9) belong to the set HA+ A'. Moreover, the multiplicity of an irreducible representation Tp+A, , p E HA, in TA 0 TA' does not exceed the multiplicity of the weight p in TA.

Using the embeddings of irreducible finite-dimensional representations of the Lie algebra g into its infinite-dimensional representations of the principal nonunitary series one proves the following result (see [ATK2], Sect. 4.2). Proposition 18. The multiplicity m/x1A1 of an irreducible representation TA in the decomposition (9) does not exceed the multiplicity of the weight A — A' in TA, where A' is the highest weight of the contragredient representation TA ,

of TN. Moreover,

74A1 ,

74T,

714 ,T■

Propositions 17 and 18 admit the following useful corollary. Corollary 19. (i) If all weight multiplicities of an irreducible representation TA are at most one, then Tx 'appears in the decomposition (9) of any tensor product TA 0 TA/ with multiplicity less than or equal to one. (ii) If the weight multiplicities of one of the representations TA and TA' do not exceed 1, then all multiplicities of irreducible representations in the decomposition (9) are at most one.

210

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

Next we develop formulas for the decomposition of the tensor product of an irreducible representation and the first fundamental representation (also called the vector representation) of a Drinfeld-Jimbo algebra. We characterize the highest weights by the numbers mi given in (2)-(4). Let Tm be an irreducible representation of Uq (g), g =--- A1_1, B1, Ci, Di, with highest weight m (m1, m2, • • - mi) and let T1 be the first fundamental representation of Uq (g) (that is, with highest weight (1,0, • • • , 0)). Then the tensor product Tm 0 T1 decomposes into irreducible components as Tm

=-

(13)

i=1

for Ug WO, where e = (0, • • , 0, 1, 0, • • , 0) (1 is in the i-th place), and as

Tm ® Ti. z.:__

ED

/

„„/

for Uq (8021 + 1) if mi = 0 and also for Uq (sp 21 ) and Uq (5021)• If mi for Uq(sov+i) we have Tm

= Tm ED Wr

(14)

Tm+ei ED giwir i= 1 Tm—ei

i=1

Trn+e, ED W

i=1

0, then

Trn—e •

(15)

•

If for some m ± ei in (13)-(15) the dominantness condition in (2)-(4) is not fulfilled, then the corresponding representation Tm±ei must be omitted. Let Vg) be the set of highest weights of irreducible representations of Uq (g) which are contained in some i--fold tensor product rr, r c No, of the vector representation. Using the decompositions (13)-(15), we derive the following

Proposition 20. If the highest weights are given by the numbers ni := (mil m2, • • -,mi) satisfying the corresponding conditions (2)-(4), then

13 (s1 1) = P, 443(sp2i) = P+1 Vso 21+1 ) = {m E RE I rn E Z}, s43(so21) = fm E P± rn E Z}. Note that in terms of A = (A1, A2 7 • 43 (50 N) are described as 443(so21+

= {A E

I

1 A1), Ai

= 2(A, ai )/(ai , ai ), the sets

Ai E 2Z}, i13(so21) =-- {A

c P+ J Aii + Ai E 2Z } .

If g is a classical simple Lie algebra, a closer look at the proof of Theorem 13 shows that it remains valid if we replace the set of all type 1 irreducible finite-dimensional representations of Uq (g) by the set of representations TA, A E T (g ). Therefore, we get

Proposition 21. Let g be one of the complex simple Lie algebras A1, B1, C1, Di and let a E Uq (g). If TA(a) = O for all irreducible representations TA with A E T (g ), then a = O.

7.2 Tensor Products of Representations

211

7.2.2 Clebsch-Gordan Coefficients

Let TA and TA' be type 1 irreducible representations of Uq (g) acting on finitedimensional vector spaces V and V', respectively. As already noted above, their tensor product decomposes into a direct sum of irreducible representations: MAs A' rr TA 0 TA' = (16) A. • '

A

The corresponding decomposition of the carrier space V 0 V' is V 0 V' = m m" ' Vv 3,8, Twhere m,V, := V8 ED • • • ED V3 (m, times). In order to distinguish different subspaces V, with the same index s we equip them with an additional index r and write V,,., r 1, 2, • - • , m181,A1 . The Clebsch-Gordan coefficients of the tensor product (16) will be defined as for the algebra Uq (512 ) (see Subsect. 3.4.2), but now multiple irreducible representations may appear in the decomposition (16). As in Subsect. 3.4.2, we consider two bases of the space V g V'. The first one consists of the vectors ei e'j , j = 1,2, • • • , dim V, j = 1, 2, • • • , dim V', where fed and {ei } are bases of V and V', respectively. The second is formed by bases {etr} of the subspaces V5r . We suppose that the bases {er}, r = 1,2, • • • , mA/, with fixed s are such that the representations Tit s are given by the same matrices with respect to these bases. Both bases fei and { eskr} are connected by an invertible matrix U with complex entries (e i , e eicr) such that

ekSr =

E

(ei , ej

eksr)ei ei •

(17)

The numbers (ei , e eZT) are called the Clebsch-Gordan coefficients (briefly, the CGC's) of the tensor product TA O TA'. Let q be real. Then we can assume that V and V' are Hilbert spaces and fed, {ei' } and { eZr} are orthonormal bases. Then the matrix U is unitary and

= U* transforms the basis fen into lei 0e'i l. The entries (ei, er) of the matrix U* will be denoted by (er f e i , e'i ). The CGC's (ei , ei' I er) are then equal to the scalar products of basis vectors, that is, we have (e i , ei' eZT) = (e i ei' , er),

(er e i ,e;) = (esr e• e'.) k 3

Since the matrix U of CGC's is unitary, one has the orthogonality relations

E

ij

e'• (eektr I ei , e'.)(e• 3 2' 3

ES,k,r (ei el3

:

eic t` ) =

er)(ei.r I e', e;)

ss'brribkki

(18)

=

(19)

As in the case of the quantum algebra Uq (812 ) we then have the following relations between matrix coefficients of the representations and CGCs:

044'. = 4131

4.—/A ryr k ki

ei i

A, I e ksr »kw (e ki

ar

•e

(20)

, )

212

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

(er I ei ,e1,)t,fli tfli i,(ei , iei, e;;';).

tz.„ =

(21)

For general complex g, the above formulas (20) and (21) remain valid if the numbers (er e, e1) therein are replaced by the corresponding entries of the matrix U -1 . Repeated application of formula (21) leads to the following Proposition 22. The matrix coefficients t of any irreducible representation TA with highest weight A from the set 113(g), described in Proposition 20, are polynomials of the matrix coefficients of the vector representation.

7.3 Representations of Ùq (g1) for 7.3.1 The Hopf Algebra

a Root of Unity

q not

6, (gin )

The aim of this subsection is to introduce the Hopf algebra Oq (gli,). The algebra ti/g (gln ) is generated by elements Ei , Fi , Ki , 1,2, • n - 1, j = 1,2, • • • , n, subject to the relations K 1 K' = KiEj.K.

i-

i

=1,

= ei.d2 q -siti+i /2 Ei7 KiFiK 1 = re-2 Te72

[Ei l Fri =6 2,r

Te.2

g g-1

7

[E1 E] = [Fi , Fi l = 07 ,

E?Ei±i (q + q -1 )E1E1±1E1 + Ei±iE? = F12 F11

27

07

+ F1 + 1 1? = 0.

(q+

The algebra Ûg (g1n ) is a Hopf algebra with structure maps given by

z(E1 ) = Et ® K1K

A(Fi ) = Fi A(K i ) =

1 + K 1 Ki+i E • , +

Ki+i Fil K i , e(E1 ) = E(Fi ) = 0, e(K 2 ) = 1.

8(E1 ) = -qEi ,

(Fi ) =

8(K 1 ) = K1 1 .

For ij/q (gln ) we also have triangular decompositions Oq (gln ) = Uq (n+ ) 0,1 (I) Uq (n...) = Uq (n_)

t/j" Uq (n+ ),

(22)

where Uq (n+ ) and Uq (it_) are the subalgebras of U4(g1) generated by Ei , i = 1, 2,• • • ,n 1, and by Fi , i = 1,2, • • • , n - 1, respectively, and Ùq (t) is generated by Ki , I = 1, 2, • •• , n. Root vectors of Ûq (g1) corresponding to positive and negative roots of the Lie algebra gl(n, C) can be introduced by means of the braid group action

213

7.3 Representations of i,e (gl,n ) for q not a Root of Unity

(see Subsect. 6.2.1). In the present case we prefer to define them explicitly. We set Ei ,i+ i := Ei and Ei+ i,i := Fi . Then the formulas

= Ej+i ,i =

Ei,j-Filq E 1j Ej,j+1 qEi Eiilq-i

t'

i < j, i j,

r'

determine recursively elements Eti and Eii , 1 < i < j < n, of Ûq (gln ) . A direct computation shows that

KiEjk.K.Ïl =q112Eik, KkEikK k-1 = q -112 Eik, that is, the elements Ejk, j L k, indeed have properties of root vectors. If i < k < 1 or i > k > 1, then we have

[Eik,Ek i ] =

K?K -2 - K7 2 K2 k _ 12 k7 g

Eik Ei2ci

(g g -1 )Ek/EikEki

g

Ei2c /Eik ==- REik

Ekdq- 1 O.

7.3.2 Finite-Dimensional Representations of tIs "q (g1n ) Since Ki K2 - - • Kn belongs to the center of Og (gl,i ), the results of Subsects. 7.1.1-4 can be extended to the algebra Ùq (g1) as follows. Theorem 23. Let T be a representation of N(g1„) on a finite-dimensional vector space V such that its restriction to the subalgebra OA) is completely reducible. Then we have:

(0 V is a direct sum of weight subspaces with respect to tilq (1)). (ii) The representation T is completely reducible. (iii) If T is irreducible, then it is a highest weight representation

and the

highest weight subspace is one-dimensional. (iv) If T is irreducible, then its highest weight is of the form ty • qm = (w i rl , • • ,w n eln), where wi E {1, -1, i, -0 and m = (m i , • • • ,m n ) is a highest weight of an irreducible finite-dimensional representation of the Lie algebra gl(n, C). Moreover, every weight of this form is the highest weight of an irreducible finite-dimensional representation of Oq (gln ). (y) If two irreducible finite-dimensional representations of N(g1n ) have the same highest weights, then they are equivalent.

The highest weights of irreducible finite-dimensional representations of gl(n,C) are given by n integers m = (m i , • - • , mri ) such that

mi > m2 > • • • > mn• As above, the irreducible representations Tm of Ùq (g1) with highest weights qm = - • • , qmn) are called representations of type 1. The characters of finite-dimensional representations of Ûq (gln ) can be defined as in Subsea. 7.1.4. They coincide with the characters of the corresponding representations of the Lie algebra gl(n, C) and the analog of Proposition 12 is also true for el'ir(glii ).

214

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

7.3.3 Gelland-Tsetlin Bases and Explicit Formulas for Representations The Gel'fand-Tsetlin bases of carrier spaces of irreducible representations of Og (glu ) are formed by successive restrictions of the representations to the subalgebras Og (g17,_ 1 ), Ûq (gln_2), - • • , iTg (gl i ) U(gl i ). From the character theory we know that the decomposition of the representation Tm of Uq (gli,) into irreducible representations of Og (g1,.,_ 1 ) is the same as for the corresponding representation Tm of gl(n, C). Hence the restriction of Tm , m= (m i ,. • • ,mn ), to Og (gln._ 1 ) decomposes into the irreducible representations Tm,_ 1 7 (mi,n-1/ • • • 7mn- Ln-i), such that 771 1

>

Mi,-1 >

m2

>

>••'>

Mn-1,n-1 > Mn

(23)

and each of these representations enters into the decomposition exactly once. Since the irreducible representations of U(g1 1 ) are one-dimensional, we obtain a basis of the carrier space Vm of the representation Tm of t/q (g1n ) labeled by the Gel'fand-Tsetlin tableaux • •

M =

Ml,n-1 M2,n

M2,n-1

• ••

•••

• •• • •

Mn- 1,n- 1 Mn,n •••

mil where mi,„

(24) mi . The entries in (24) are integers satisfying the betweenness

conditions

m,+1

mii

mi+Li+17

i = 1,2,• • •,j, j = 1,2,• • •,n.

(25)

The set of all tableaux (24), satisfying these conditions, labels the basis elements of the carrier space of Tm . The corresponding basis element will be denoted by 1M), where M is the tableau (24).

Theorem 24. Let q be a positive number and let Tm be the irreducible representation of Ûq (gln ) with highest weiglit m. Then the generators of 64(g1n ) act on the Gelfand-Tsetlin basis of this representation by

k-1

Tm (KOIM) = qak/2 IM), ak =

E mi,k - E i=i

1 < k < n, (26)

i=1

Tin (Ek)IM) = EA(M)IM), Tm(Folm) = E A3k(mk 1 0 > l', then its Gel'fand-Tsetlin tableaux are of the form 0

• • •

O

•• . •. •

• ••

• ••

711

0

0

0

0 0

1') ,

(30)

j'

• ••

where 1 > m > j > • • • > 0 > • • • > j/ > m/ > . The operators T11iv(K707 Tui(En-i) and TIV(Fn-1) act on the corresponding basis vectors IMP') by Tit

Tiv(E n_ i )

="

Kn mmim' = tio+r-m-m'oimwre),

(31)

mypn'

milm - + n 1][m - j + l][rn - + n - + n [m m' + n -

2. ]) m m+ 1 m'

frrl! - + 11 [j - mi + n - 3 1D' +n [m m' + n - 2][m - m' + n -

)

(32) T11 i(Fn-1)1Myr) ei-m+1][m-P+n-2][m-j][m-ji +n-31) i m-1 Tre [m - m' + n - 3][m m' + n -

+ (11

+ n 1J[nil - rip m' + n 2llf - m' + 11 ) 4 im3mi;mi_ i). [m-rnt+n-L2llm-m 1 +n - 1] (33)

For representations with highest weights (1, 0, • • • , 0), these formulas turn into To (K)lmrd 0) q (1--rn)/21 . iri 1 d 0) ,

(34)

218

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras Ti,o(En-Optcdo) =

([1 _ in][m _

71,o(Fn—i) i •M;3;° ) = ([1-

1/2 1mrin0+1,0 ), 11) m 1][m -ii)1/21K10-1,o).

(35) (36)

The formulas (31)-(33) show that IM71 ) is an eigenvector of the operator Tuf (K n ) and that the operators Tv (En-i) and Tip (Fn-i) change only the indices j and j' in the tableaux (30). Similarly, the matrix elements of the operators Tip (Kk+i), Tiv(Ek) and Tw(Fk) depend only on the (k+1)-th, k-th and (k -1)-th rows of the Gel'fand-Tsetlin basis elements (30). We see from the formulas (34)-(36) that the matrix elements of the operators Ti,o(Kn) Ti,a(En-i) and T1,0(Fn_1) do not depend explicitly on n. The next proposition shows that this is the case for all irreducible representations of ejg (gin ) with highest weights (mi, m2, • • •, mi, 0, • • • , 0). Proposition 26. If an irreducible representation Tm of t,q (g1n ) has a highest weight ni = (mi,m2,- • • ,mi3 O,• • • ,0) with more than two zeros, then the matrix elements of the operators T,-,,,(Kn ), Trn(E7,1) and Tm (Fn_i) with respect to the Gelfand-Tsetlin basis are independent of n. The corresponding formulas (27) for these operators contain only i summands.

Proof. The proof follows from the expressions for the coefficients A_ 1 (M) in (27) at mi + i = • • • = m n = O. 7.3.5 Tensor Products of Representations

In this subsection we consider tensor products of irreducible representations of E,q (g1n ) for which the multiplicities of irreducible components do not exceed 1. The most important such examples are the tensor products Tm where Tm is an irreducible representation and Ti is the vector representation of Oq (gl ii ). Recall that the decomposition of this tensor product was given in Subsect. 7.2.1.

Other tensor products with multiplicities not exceeding 1 are I'm Tp , where m is any highest weight and Tp is the symmetric irreducible representation with highest weight (p, 0, • • • , 0). From (11) one can derive that T

Tp =

IEDr Trn±r

(37)

where the summation is over all r = (r i , • • , rn ) such that ri E No and ri + - • + rn = p, mi + ri > mi > n12

+

r2 > rn2 > > mn + rn > mn .

(38) If the multiplicities of irreducible components in, the decomposition of a tensor product Tmn. 0 Tm in are at most one, then the CGC's with respect to the Gerfand-Tsetlin bases can be written in the form

42,

n

(M 1M") -=

(39)

219

7.3 Representations of 64,(gln ) for q not a Root of Unity

where mi are the rows of the corresponding Gerfand-Tsetlin tableaux. As in the classical case (see [VK2], Chap. 4), it can be proved that the CGC (39) is a product of the so called Oq (gln_i )-scalar factors (or reduced CGCs) -

Mn- 1 Mn - 2

In n tt ALAn, -1 M n-1 Mn

11/11 1M")=

t/

In n-1 t, M n-2 M n- 2 M n- 1

If

X•••X

( M2

111 2

M2 II)

rn1 m1 1111

(40)

7

which depend only on two rows of the CGC. 7.3.6 Tensor Operators and the Wigner-Eckart Theorem

In Subsects. 7.3.6-9 we suppose that q is a positive number. Let Tm be an irreducible finite-dimensional representation of Ùq (g1) with highest weight m on a space Vm with Gel'fand-Tsetlin basis {IM)}. Suppose that {Rr;vni } is a set of operators acting on a Hilbert space 55 and indexed by the Gerfand-Tsetlin tableaux 1M) of the representation Tm . Let T be a representation of Og (gln ) acting on b, which is a direct sum of irreducible finite-dimensional representations. We say that the set of operators {R741 } is a tensor operator transforming under the representation Tm of Ùq (g1) if for all generators Ei , F K i we have ,

-2)

T (Ei )lr,vni - q(ai-at+i)12R7417(

EA/1(m)R741,T(Ki-iKi+ ,),

(41)

k=1

T(F)R - q (ai - ai+ 1) /2 RZT(Fi ) = EA.(mi-k)RkT(Ki-lKi+1), (42) k=1 q a, /2 Rill T(K)RIAT41 T(Ki7 1) =

(43)

where A(M) and ai a•(M) are as in (26) and (27). Repeating the arguments of Subsect. 3.6.2 one proves the following Wigner-Eckart theorem for the matrix elements of the operators R. Theorem 27. If 55 = EBm, vni is a decomposition of 55 into irreducible subspaces for the representation T and flm' ,M')} are Gelfand-Tsetlin bases of the subspaces Vml , then the matrix elements of the operators Rik' with respect to these bases are expressed in terms of the CGC 1 's of Ùq (g1) by f

(rw, /4' I RR/ 1 m", M") = E(nif ornii m"), f

m'

M

If

M' M M"

•

(44)

)

If

Here Rfl m"), are the so-called reduced matrix elements of the tensor operator which do not depend on M', M, M". The summation index r in

220

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

the tensor product

(44) distinguishes multiple irreducible representations in Tm 0 Tm".

Note that if the tensor product Tm Tml, contains only irreducible components with multiplicities not exceeding 1, then the right hand side of (44) contains only one summand and the index r can be omitted.

7.3.7 Clebsch-Gordan Coefficients for the Tensor Product Tm 011 These CGC's are factorized into products of reduced CGC's of the form

(1,0)

ez mn-i +e3 )' Mn

(0,0)

where 0 = (0, • • , 0). The corresponding reduced CGC's are given by the formulas

ei )

(1,0) (0, 0)

q-1/2(i+1+E m

n- 1

Ei#i

jT

Me . 71)

3

Mn- 1

Fr J

. ,

-

[ in

,n

n-1 2

3±

)

1.

1/2

]

( 45)

njoi[mj,n mi,n —

+ei mn-1 + e3

(fl koi

[Mk,n mj,n-1

k+

[Mk,n mi,n

k + 4 (1di

= 190 -

TT [Mk,n-1 14i

f

M j,n-1

k+ k+

1

1,2 ]

(46) where [m]--=-- [m] q is the q-number (2.1), rri B ,n are the components of the highest weight mn and we have abbreviated V(j i) := 1 if j - i > 0 and .19(j - i) := - 1 if j i < 0. In the case of the tensor product 711 O Tm , the corresponding expressions for the reduced CGC's Ci

Inn-1

+ ei

are obtained from (45) and (46), respectively, if we replace q by q -1 . Both expressions (45) and (46) are special cases of CGC's considered in the next subsection.

7.3 Representations of C4(g1n ) for q not a Root of Unity

221

7.3.8 Clebsch—Gordan Coefficients for the Tensor Product Tm OTI, The corresponding CCC's factorize into products of reduced CGC's of the form

(p, 0) (r, 0)

(i) 0 ) (0,0)

(47)

The expression for the first reduced CGC is (p, 0) (0,0) [Pi,n - kj,n-

x

m.' mnai

=qa-vEll[iii,n i<j 1 ] 1 Fl i O. With such a tableau we associate the

Pn—i > •

RPn P m-

0

0•••0

0

Pn- 1

0

•••

n

EP= - Pi-1 H

111-1 [Pi - Pi - l P

H

i,n+

i=1

(52)

11.3 j=1

of the algebra Og (g172+1 ). Using the relations Ell,i+1 Ea = unmEr,1EZ i±i and E4_ 1 E171. 1,k = E

[nP[mPe - s)(772-8) si![in .5 Er_-1:10 Eigk E7i1; ]! 8 (n-m+1)

[s]![n sP[m - .51! where j

Erkr s

2 s K i2_; 1

± 1 < k, one proves that the elements (52) satisfy the relations

- i+i Ei, i± iRPn -1 -q -1 K-K7 1 RPn P.--i,"•,pi KK

([Pi+i - Pi][Pi Ei +1

+ 11) 1 / 2 RPn

(53)

Pn ([73i+1

-

Pi ± 1] [p

-pi_1 ] ) 1 / 2 RPn Pn-i 1" - 43= - 1,•")Pi

(54)

This means that for any finite-dimensional representation T of 61q(g1n+1), the operators ) form a tensor operator transforming under the symmetric representation Tyr, of tifq (gln ). Therefore, by Theorem 27 and formula (44) therein, for any irreducible representation Tinn+ , of ùq (g1) we have (M

RPn Pm- 11"*1P1 I M I) = BPI= :Pm -t (Inn+1)(M

P M').

7.3 Representations of ai (gin ) for q not a Root of Unity

223

Here M and M' are the Gel'fand-Tsetlin tableaux given by M =

and M'

Mn+1

mi

=

9 respectively, P is the Gel'fand-Tsetlin tableau (51),

(M I P M') is the Clebsch-Gordan coefficient and B

(mn+ 1) is the reduced matrix element depending on pn ,pn _ i , mn , mn' , where mn and mn' are the first rows in the tableaux M and M', respectively. As noted in Subsect. 7.2.2, the CGC (111 P j M') is equal to the scalar product (M P M'). Since q > 0, Tmn Tpn is a *-representation with respect to the involution of Oq (gln ) given by (Ei )* := Fi and (K)* := Ki , so that PI (Till n Tpn

(M

(En _ i )1141 = ((Tmn Tpn )(Fn _i)(M

P),

Inserting the formulas (26) and (27) we derive the recurrence relation n-1 -

E 3= •

n _1( 11 )(M P 1g

+

-= A (Pn71)q c (M P;-!11 M') An— (Ill n—i)q

Pn - Pn - 1

(Mn—i

P I AP),

where A3n. _ 1 and MZ-I 1 are as in (27) and c = m in + • • • + mnn mi,n-i •• - mn-i,n-i. Setting pn - 1 = pn_ 2 = • • = pi = 0, we obtain a recurrence relation connecting the reduced CGC's (p, 0) (0,0)

and

(p, 0) (0,0)

Inn Mn-1 )

where s takes one of the values 1, 2, • • • , n-1 and mils 1 is obtained from m n _ i if we replace ms,n—i by TY1 8 ,n _ 1 + 1. These recurrence relations determine the reduced CGC's (48) up to a normalization constant. This constant can be computed by using the orthogonality relations for the reduced CGCs. Using (53) one can easily verify that R P,..,0, ... ,0 coincides with ( [p - rPri [! [Pl!

1 r

E s=o

( _i)s q s(r-1—p/2)

[s]! Fr - sp

Er-s RP 7 1(n r ) n-1,n 0,•••,0Es n-1,n 1(7;-1

where p = pn and r -a- pn _ i • From the matrix form of this relation the second reduced CGC in (47) can be expressed in terms of the first reduced CGC's and the matrix elements of the operators Tri,(Enk_ i,n ). The latter matrix elements are given by the formula Mn-1

Tri ,n (EnP _ 1,n )

Mn-2 =

Tinn (EPn ,n_ 1 )

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

224

1 -1) nan-2) ri,ri- 1 (Inn Mn- 1) Sn-1,n-2( 111n (Sn- 1,n- 1 (Inn / _ir Mn- 1)) 2 Sn,n-1 (Inn

- 1) Sn-1,n-2(Inn-1 Mn-2) (55)

where the first rows mn in the Gel'fand-Tsetlin tableaux are omitted and

dn_ i (Mn- 1) Sn-1,n- 1 (Inn- 1 Mn- 1 ) • This leads to the expression (50) for the reduced CGCs. Formula (55) is proved by induction on p. For this, we use the recurrence relation (MI

n-1

Tmn (En in) M) =

AL 1 (M) ( 41 I Trnn (En) MZ 1) 2

x(Mnt i j Tmn (En-i,n) M), where Ani _ 1 (M) is as in (27). In this formula we substitute the expression (55) for the operator Tinr, (E1,) and use the equality

1- x1x2 ". Xn

=

.9=1

(1 —

H x8)

y, - xiyi yi iOs y s

(56)

7

which is well-known in combinatorics (see [Mac]). After some simplifications we obtain then formula (55) for Tir,„. (Er,'

7.3.9 The Tensor Product Tm T1 for e l —0 0 Let us consider Uq (glu ) as a one-parameter Hopf algebra deformation of the universal enveloping algebra U(g1) and assume that q > 0. The CGC's of Uq (gln ) are then functions of the parameter q and we can look at their limits for q 0 and q oo. If q 0, then a direct calculation shows that the reduced CGC's (45) and (46) vanish except for the following cases: =1

liM q -*0 Mn-1

lim

if

Mk,-1 = Mk+1,n,

mn

< k < n,

q-*0(mn-3.

Mn liM q -03 ma-i

=1

where in,t i := mn + ei. Similarly, when q (45) and (46) vanish except for liM q-, 00 •

fl-* 00

-

Mn Mn _

m±i

mn ( Mn

i < k < n,

1

M n- 1

(1, 0) Inn+i

(0, 0) mn-i = =1 if i

if

i = j,

oo, then the reduced CGC's

if

i=1

j +1 or i= j and mi n-1

Min •

7.4 Crystal Bases

225

Each CGC of the tensor product T., 0 Ti is a product of several reduced CGC's of the type (46) and one reduced CGC of the type (45). A closer analysis of these reduced CGC's shows that for fixed j and M' all CGC's ( m7, (1, 0) mrt i) vanish in the limits q 0 and q -* co except for one. ' Thus, by (17), for every basis vector Im M') of the representation space of T +. from (13), there exists a unique basis vector Ilan , M") 01(1, 0), M) oc of the carrier space of Tm„ such that in both limits q -* 0 and q we have M") 01(1,0),M). M') = Now we apply this procedure inductively starting from the tensor product T1 0 71 of two vector representations T1 of Ug (gin ). Let v 1 , v 2 , • - - , v the elements of the Gerfand-Tsetlin basis of T1. Since the irreducible representation Tmn with highest weight mi, = (min 7n2n • • • Innn), ninn 0 , is contained in the tensor product of m = min + • • • + m nn copies of the vector representation T1 , then at q = 0 and at q = oc every Gelfand-Tsetlin basis vector of the representation space of Tnr,„ takes the following simple form: ,

vi l

vi 2 0 • • •

v

,

= M in + • • • + Mnn

E {1, 2, • • • ,n}.

This observation is a starting point for the construction of crystal bases for spaces of finite-dimensional representations of Drinfeld-Jimbo algebras

N(g).

7.4 Crystal Bases The construction of convenient bases in representation spaces is one of the main problems of applied representation theory. For example, Gel'fandTsetlin bases for irreducible representations of the Lie algebras gl(N, C) and so(N, C) have been proved to be a useful tool for a number of applications of representations in theoretical physics. In the preceding section an analog of such bases has been constructed for representations of the algebras eldq (g1N ). However, it is not known how to get Gel'fand-Tsetlin bases for Uq (soN), because there is no Hopf algebra embedding Uq (solv_i) C Uq (soN). This section deals with the construction of Kashiwaras's crystal bases for finite-dimensional representations of the Drinfeld-Jimbo algebra U (0) over the field Q(q) defined in Subsect. 6.1.2. Roughly speaking, these are bases of vector spaces at "q 0" that behave well simultaneously with respect to the action of all generators E i and Fi of U4(9). (These statements are made precise in Definitions 1 and 2 below.) In applications to exactly solvable models in statistical physics the parameter q plays the role of the absolute temperature. Hence crystal bases desribe the behavior of the model at absolute temperature zero which explains the origin of their name. In this, section we give ra brief exposition, without proofs, of the main results on crystal bases. The complete proofs of the results are quite involved

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

226

and can be found in Kashiwara's original paper [Kas2I, see also [Kas1] and [Kas3]. For notational simplicity, we shall use the language of left modules rather than of representations, thus following the terminology used in [Kas2].

7.4.1 Crystal Bases of Finite-Dimensional Modules Recall that Q(q) is the field of rational functions in one variable q with coefficients in the rational field Q. Let A be the subring of 12(q) consisting of all rational functions without poles at q O. For a E A we identify the coset a + qA with the rational number a(0). Then we have A/qA = Q. Definition 1. A basis of a vector space V over Q(q) at q = 0 is a pair (L, B) such that L is a free left A-module satisfying V Q(q) OA L and B is a basis of the Q-vector space L IqL. Let V1 and V2 be vector spaces over Q(q) 'with bases (L 1 , B1) and (L2, B2) at q = 0, respectively. It is easily verified that the pair (L1, B1) ED (L2, B2) (L1 e L2 1 B1 U B2) is a basis of the vector space V1 ED V2 at q = 0 and that (L i , BO 0 (L2, B2) = (L1 OA L27 B1 0 B2) is a basis of 0 V2 at q = 0, where B1 B2 := -01 b2 I bi E Bi b2 E B21. The following example will play a crucial role in what follows.

-IN,

Example 2. For 1 E let V1 denote the (21 + 1)-dimensional simple left Uq' (s12 )-module defined by the formulas (3.22) and (3.23) with w = 1, and

let

h := e1

E V/ be its highest weight vector. Set F(i) := PIN! and fi(1) :--=

F (i) fo for i E No . The vectors 4(1) , j =-- 0, 1, • • • , 2 1 , form a basis of V1 such that

Kf 2(1) =

Efi(1)

[21 - i+l]fi(1) 17 Ffi(1)

(57)

If Li := iED Afi(I) and

B1 := Ifi(1) mod qL1 Ii= 0, 1, • • • , 211,

then the pair (L1, B1) is a basis of the vector space Vi at q = O. Next we define linear operators E j = 1, 2, - • • , rank g, on each finitedimensional type 1 left NW-module V. If g = $12 and V = V1 is as in Example 2, we set Éfi, = fi-1 and = fi+i for i = 0, 1, • • , 21, where f-i f214-1 := O. If V is an arbitrary finite-dimensional type 1 left U.'7 (s12 )module, then we decompose V as a direct sum of modules Vi and define È and Ê on V as direct sums of the corresponding operators for the modules Now let V be a finite-dimensional type 1 left NW-module. The subalgebra u,17 (9) i generated by the elements E, F, Ki, K1 -1 is obviously isomorphic to (812). Hence V is in particular a left U4, (sli)-module and the operators ti and Fi are defined by the preceding. Summarizing, if V = EDA VA is the weight space (k) F,i fk with decomposition, then every element f E VA is of the form f fk E 150-kail Eifk = 0 and we have defined ,

,

N,

Ek

7.4 Crystal Bases

227

tif Ek Fi(k-O fk, pif Ek Fi(k+i) fk. Finally, we can give the definition of crystal bases. Definition 2. A crystal basis of a finite-dimensional type 1 left Uq' (9)-module V is a basis (L, B) of V at q = 0 such that the following conditions hold: L = EBAE p LA and B = UAE p BA where LA = L n VA and BA =

B n (LxIqLx). (ii) t i (L) g L, F2 (L) Ç L, t i (B ) g B U 101, F(B) Ç B U {0 } for = 1,2, • • • ,n = rankly (iii) For any b 1 , b 2 E B and i G {1, 2, • • • , n} we have t ib2 = b 1 if and only if Fb1 = b2. Note that because of t i (L) C L and Pi (L) C L the mappings Et, and Pi pass to the quotient LiqL and define mappings, again denoted by ki and Pi , on LAL, so that t i (b) and Pi (b) are indeed well-defined for b G B. Conditions (i)--(iii) express the "nice" behavior of the basis (L, B) with respect to the actions of the generators Ki , Ei, Fi on V. For instance, if b E B, then either t i (b) = 0 or fi(b) G B by (ii). Further, if fi (b) G B, then b = FE(b) by (iii). The following assertion follows easily from Definition 2. Lemma 28. If (L i , Bi ) is a crystal basis of the N(9)-modules V3, j = 1, 2, • • r, so is the pair (L, B) := (ED i Li ,Lj i Bi ) for the direct sum EDi Vi .

7.4.2 Existence and Uniqueness of Crystal Bases Let us begin with the simplest case g = 512 . The pair (L 1 , B1) from Example 2 is obviously a crystal basis of the q(s12)-module Vi. Since any finitedimensional type 1 q(s12 )-module V is a direct sum of such modules V, it follows at once from Lemma 28 that V has a crystal basis. The existence of crystal bases in the general case is much more subtle. For A E P+ , let V(A) be the irreducible left N(g)-module with highest weight A and highest weight vector ex. We set CX)

L(A)=

E EA1-2

r=0 ,•••,i r =1

B(A) =

{

Pi,

• • • Pir

1 '••

Pi, (eA),

(e),) mod TWO I i , • • • , i r E 1 1 7 2 1' • ' nil.

Theorem 29. (L(A), B PO) is a crystal basis of V(A). Any finite-dimensional type 1 left Uq' (g)-module has 4 crystal basis.

Proof. By Lemma 28, the first assertion implies the second one. It is obvious that B(A) spans the Q-vector space L(A)/qL(A) and that Pi (B) Ç B U 01. The complete proof that (WO, B(A)) is a crystal basis is rather long. It is given in [Kas2) by an induction procedure called the grand loop.

228

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

The following proposition says that crystal bases are unique up to isomorphism. Proposition 30. Let (L, B) be a crystal basis of a finite-dimensional type 1 left (I:, (g)-module V. Then there is an isomorphism -y of V to a direct sum EDA V (A) of highest weight modules V(A) (possibly with multiplicities) which maps (L, B) to the crystal basis (EDA WO, UA B(A)) of EDA V(A).

Proof. A proof is given in [Kas21, p. 478, using Theorem 29. 7.4.3 Crystal Bases of Tensor Product Modules One advantage of crystal bases is their behavior under tensor products. Before we state the corresponding result, we consider an example. Example 3. From Subsect. 3.4.1 we know thai the tensor product left N(812)module 14 0171 12 (see Example 2) decomposes as a direct sum 17/±1/21EDV/-1/2. The vectors

w (1+1/2) =

(1) i

(1-1/2)

2 [

1

fo( 1 / 2)

1

21

(

[21]

_r(1) ® f1(1/2)

1/2) ( )

f

i= 0, 1, - • , 21+ 1,

q[i -I- 1 /1 (1) (1 [2/1 f' +1 ®f0

2)

7

= 0,1, • • , 2/ - 1,

respectively, on which K, E and F act by (57). are bases of 17/ ±1/2 and ii1/2, It follows from these formulas that =

of({1 11'2) ( mod q (L(1) ®L(1/2)

and

= f(i)

))

< 217 41++ 1 /2) =

f (1 /2) ( mod q (L(/) 1

®fP /2)

L(1 1 2))).

(58)

(59)

The pair (L, B), where L = L (1+ 112 ) e L (I -1 / 2) and B consists of the vectors (58) and (59), is a crystal basis of Vi 0 171/2. A

Let (L, B) be a crystal basis of a U(g)-module V. For b e B we set ei (b) = max{k

G NI

E(b)

G B},

O(b) = maxik G NI Pik (b) G Bl.

Theorem 31. (0 Let (L1, B1) and (L2, 132) be crystal bases of two N(9) modules V1 and 172 ) respectively. Then (Li, B1) (L2, B2) is a crystal basis of V1 ® V2. (ii) If bi G B1 and b2 G B2, then for any j e {1 2, • • , n} we have

, ki(b i b2)

b2 for b 1 2ib2 for

Pi (bi 0 b 2)

b2 b1 ®frb2

Oi(b1)

(bi )
Ei(b2),7 for MN) -15 4(b2).

7.4 Crystal Bases

229

Sketch of Proof. (i) follows easily from (ii). To prove (ii), it suffices to assume that g = 812 . In this case the assertion can be derived from the formulas for the Clebsch—Gordan coefficients. See [Kas2] for details. 1=1 To a crystal basis one can associate its crystal graph. A directed graph consists of a set of vertices and a set of ordered pairs of vertices, called arrows. A directed graph is called colored if it is equipped with a map from the set of arrows to another set, the so-called set of colors. Let (L, B) be a crystal basis of a N(g)-module V. We define the crystal graph of (L, B) as the colored graph with set of vertices B and set of colors {1, 2, • , n } , n == rank g, such that elements b, b' E B are connected by. an arrow, directed from b to b', with color i if and only if b' = F(b) (and hence b = Éi (b') by Definition 2(iii)). The crystal graph of a crystal basis (L(A), B(A)), A E P± , is a connected graph, because every vertex b E B(A) is obviously connected with the highest weight vector. Suppose that (L, B) is a crystal basis of a type 1 finite-dimensional U(g ) module V which is isomorphic to a direct sum Ep i V(Ai ) of irreducible modules VP4), Ai E P. It can be shown that then the crystal graph of (L, B) is (isomorphic to) the disjoint union of the crystal graphs of all (L(Ai ), B(A)). Thus, the decomposition of V into irreducible components is equivalent to the decomposition of the crystal graph of (L, B) into connected components. From Theorem 31 we can read off the crystal graph of the tensor product V1 0 V2 of two N(g)-rnodules V1 and V2 . Without going further into technical details we only state that it is obtained as the union of tensor products of each component of the crystal graph of (L 1 , B 1 ) with each component of the crystal graph of (L2 , B2 ). Combined with the result of the preceding paragraph, one obtains in this manner an interesting graphic algorithm for the irreducible decomposition of the tensor product V1 0 V2 (see [KM and [Nap. We illustrate this by a simple -

Example 4. The crystal graph of the crystal basis of the N(s12 )-modu1e is fi: 1)

f (1)

f 1) . Hence the crystal graph of V1 0 Vi is hçi l )

0 41)

f(l)

f

fi:(:1 1 )

p) fio)

fil) A l) Therefore, as discussed above, V1 0 Vi decomposes as V2

e Vi ® Vo .

7.4.4 Globalization of Crystal Bases

Let (WO, B(A)) be the crystal basis of the N(g)-module ITN, À E P±. Then -B(A) is a basis of the Q-vector space WNW). In this subsection we

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

230

shall show how to obtain a basis of the Q(q)-vector space V(A) from B(A). Let us fix an irreducible finite-dimensional N(g)-module V(A), A E P±, with highest weight vector e),. There exists a Q-algebra automorphism rl of defined by

N(9)

ri(Ei ) = Ei , rI(Fi ) = Fi ,

?IWO =

, r(q) =

From Proposition 6.28 it follows easily that there is a Q-linear mapping ri : V(A) -4 V(A) such that ri(ae),) = ri(a)e)„ a E N(g), Clearly, A := ri(A) is the subring of functions in Q(q) without poles at q = Do and L(A) = 71(LN) is a free A-module such that V(A) --= Q(q) OA MA). Further, we set VQ(A) =

E (Kg, q -1 1F4n1) • Fi(n r ) eA.

The crucial step for a globalization of crystal bases is Proposition 32. The natural map ( : L(A) an isomorphism of Q vector spaces.

n L(A) n vQ (A)

L(A)IqL(A) is

-

Let B(A) be the set ( -1 (13(A)). Then we have Theorem 33. MA) is a basis of the Q(q) - vector space V(A).

The basis B(A) of V(A) is called a global crystal basis of V(A). One can also prove that FiniV(A) =

Q(q)b, bEB(A),e i

where

Ei

(

ETV(A) =

IED

Q(q)b,

b)]rri

and Oi are as in Subsect. 7.4.3.

7.4.5 Crystal Bases of Uq1 (n_) In Subsects. 7.4.1-4 crystal bases of type 1 finite-dimensional N(g)-modules have been investigated. A similar theory can be developed for the subalgebra U(n) of which is generated by the elements Fi , i = 1,2, • • • , n = rank g. This will be sketched in this subsection. In order to imitate Definition 2 we first need to define the counterpart to the operators ki and P for Uqi(n_). Putting

N( 9)

E(x) :=

- qi7 1 )1Ci (Eix xEi), x E

Uq'

one easily verifies that E(Fi x) = q -( ""al ) FiE(x)+ bijx,

(60)

Ei'(xy) = gi (x)y + cri(x)E:(y)

(61)

where ai (r) := KixiCr i and x, y E Uqi (n....): Equation (61) means that g is a skew derivation of the algebra r.74(n....) with respect to the automorphism ai. -

7.4 Crystal Bases

231

By (60) and (61), we have E(F) =1 and g(Firn') = g2l _m[rn]1 im -1 . Further, the linear mapping g leaves N(n_) invariant and for any x E U(n....) there exists m E N such that (E)m(x) = 0. Using the preceding facts it can be shown (see [Kas2], Subsect. 3.2.1) that any x E U(n_) is of the form

x

nym

) = 0, k E N, Finy m ), m=o with uniquely determined elements ym E N(n_). Then we define E' (x)

= E

k

Ek

(m _ 1) .F14

(Yin) Pi (X) = Fi(In+1) (Yin) m=0 Now Definition 2 can be adapted to the present situation as follows. A crystal basis of Uq' (n_) is a basis of the Q(q)-vector space Uqi(n_) at q = 0 (see Definition 1) such that the conditions 00 and (iii) of Definition 2 are fulfilled. Note that condition (0 of Definition 2 is not needed here. Since obviously EF = I, it follows from (ii) that Pi (b) E B for any b E B. One can prove that Piki is the projection of U(n_) onto Fi • N(n_) with respect to the direct sum decomposition q(n_) = Fi • (.41 (n_) ker.g. Let goo) be the A-linear span of elements Pi , • • • Pir 1 for 7:1) • • , ir E {1,2, • , n } , and let B(oo) be the set of images of these elements in L(oo)/qL(oo). Further, by Proposition 6.28, every irreducible N(g)-module ITN, A E P+, is isomorphic to a quotient of the Verma module MA, which can be realized as a N(g)-module N(n_) by Subsect. 6.2.5. Therefore, there exists a homomorphism 7r), of the U:2 (g)-module U:i (n_) to V(A).

=

1.1(711,=-1

Proposition 34. (0 (goo), BOO) is a crystal basis of Uq' (n_). (ii) The crystal basis (L(A),B(A)) of the Uq' (g)-module V(A) (see Subsect. 7.4. 2) is obtained by L(A) = 71),(goo)) and B(A) = fx E BOO Tr),(b) 01, L(A)1qL(A) is the induced map of 7rA : L(oo) —> where Tr), : L(oo)/qL(oo)

L(A). Also the globalization of the crystal basis (L(oo), B(oo)) of q(n_) proceeds in a similar manner as in the case of N(g)-modules. If U4(n.4 denotes the Q[q, q -1 1-subalgebr a of Uqi(n_) generated by the elements Fi(m) , then it can be proved that the canonical map

: U4(n_) n goo)

nL(°°) —› goo)/qL(oo)

is an isomorphism of Q-vector spaces. Theorem 35. (j ) The set B(oo) := ( - '(B(oo)) is a basis of the Q(q)-vector space N(L.). (ii) For each A E P±, the homomorphism 7r), maps the set of all b E 13(00) such that 'WA(b) 0 to a basis of the Q(q)-vector space ITN. The set B(oo) is called Kashiwara's global crystal basis of Uq' (n_). It can be shown (see [L5] and [GL]) that it coincides with Lusztig's canonical basis which was constructed in [L4] and [L5] using a different approach. A detailed exposition of the latter can be found in the book [Lusl.

232

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

7.5 Representations of Uq (g) for q a Root of Unity In this section we study representations of Drinfeld—Jimbo algebras Uq (g) when q is a primitive p-th root of unity, where p is odd and p > di , i = 1, 2, • • • , I (l is a rank of g). We denote the parameter q in this case by E.

7.5.1 General Results In what follows, we shall use the notation and some facts from Subsect. 6.3.5. Recall that 3, is the center of the algebra U,(g) and Spec (3,) is the set of C. Let Rep (U (g)) denote the set of all all algebra homomorphisms x : 3, equivalence classes of irreducible representations of U,(0). If T is an irreducible representation of U(g) on V, then any T(Z), Z E 3,, is a scalar operator on V, that is, there is a çomplex number XT (Z) such that T(Z) = XT(Z)I. Obviously, XT : 3, C is an algebra homomorphism, called the central character of T. Clearly, equivalent representations have the same central character. Therefore, the assignment T XT defines a map

W: Rep (U ,(g))

Spec (3 6 ).

Proposition 36. Every irreducible representation finite- dimensional.

(62) of the algebra U6 (g) is

Proof. Let IXT be the two-sided ideal of (4(g) generated by the set ker XT {Z — XT (Z)1 I Z c 3, } . Since ker XT C ker T, the irreducible representation T of U,(9) induces an irreducible representation T of the quotient algebra Ur := Ue (9) /IXT . By Theorem 6.49, the algebra Ur is finite-dimensional.

Hence its irreducible representation T is finite-dimensional (see [CR]).

LI

Proposition 37. The mapping W is surjective, that is, for any x E Spec (3,) there exists an irreducible representation T of U,(g) such that xg-, = x. Idea of proof. As in the preceding proof, U := U,(9)/J7 is a finitedimensional algebra, where ix is the two-sided ideal generated by ker x. It can be shown that r./c {0}. Any subrepresentation of the regular representation of U,x gives an irreducible representation, say T, of

U,(g). Since T annihilates ker x, we get xg-, = x. The mapping W is not injective in general. The main properties of this mapping are contained in the following Theorem 38. There exists a nonempty closed proper subset D -of Spec (3,) such that: (i) If x E Spec (3,) and x D, then W -1 (x) consists of one irreducible representation. This representation is of dimension pn , where n is the number of positive roots' of g. (ii) If x E D, then W -1 (x) consists of a finite number of irreducible representations and their dimensions are strictly less than pn

7.5 Representations of Uq (g) for q a Root of Unity

233

Proof. A proof is given in the paper [DCK]. During this proof it is shown that for any X E Spec (3,), x D, the quotient algebra E/c = U,(g)//x is isomorphic to the matrix algebra Mpn (C). This algebra has, up to equivalence, a unique irreducible representation which is of dimension pn

Recall from Subsect. 6.3.5 that the center 3, is algebraic over 30 (that is, each element of 3, satisfies a nontrivial algebraic equation with coefficients in 30) and that the set Spec (30) is isomorphic to C 2n x (Cx ) 1 , where Cx = C\{0}, n is the number of positive roots of the Lie algebra g and 1 is the rank of g. From these facts we conclude that Theorem 38 gives a parametrization of the irreducible representations T of U, (g) such that XT D by 2n +1 = dim g complex parameters. If XT E D, then additional parameters are needed. Note that the set Spec (3,) is a complex algebraic variety of dimension p2n+ 1 . It can be shown (see [DCK]) that D is a proper subvariety. This means that almost all irreducible representations T of UE (g) are of dimension pn • Let us illustrate Theorem 38 by the simplest example. Example 5 (U,(s12 )). By Proposition 3.15, the center 3, of the algebra U,(s12 ) is generated by the elements EP, FP, KP, K - P and C,. (Note that q has been replaced by E.) All irreducible representations of U,(s12) have been classified in Sect. 3.3. Recall that these are the cyclic representations TabA l the semicyclic representations Too, and 714", and the irreducible representations T1 1, 21 < p (see Propositions 3.17-19). All representations TabA, 2 -1(j bA and T„, (p_ i)/2 are p-dimensional. The only irreducible representations of dimensions less than p are T1, 1 < (p-1)/2. For these representations we have T1(EP) = T1(F) = 0, T1 (K') = (ALT and T„i(C,) = wci := w ( 6 21-1-1 + 6 -21-1)( 6 _ 6 -1)-2 1 . The set Spec (3 E ) is determined by the points (s, y, z, c) E C 4 , where T(EP) = T(FP) = yI, T(KP) = zI, T(C) = cI and T is an arbitrary irreducible representation of U,(s12). The 3-tuple (x, y, z) runs over the set C2 x Cx and for each such 3-tuple (x,y,z) the corresponding values c are related to x, y, z by formula (6.110). Thus the subset D c Spec (3,) consists of the points

(0, 0,co,wci),

=

= 0,

1, • • •

5 - 1.

A

Proposition 39. For any nonzero a E U,(0) there exists an irreducible representation T of U(g) such that T(a) 0. Proof. By Theorem 6.49, the algebra U,(0) is a finite-dimensional vector space over 30 with a certain basis, say xi, • • • , xd. Hence we can write a as a = Ei xizi with zi E 30 . Since a 0, there is an index k E {1, • • • , d} such that zk 0. By the remarks after Theorem 6.50, Spec (30) separates the elements of 30. Hence there exists X I E Spec (3 0 ) such that x'(zk) 0. We choose a character X E Spec (3,) such that x' = x on 30. Since D is a proper subvariety of Spec (3,), we can choose x' such that x fl D. Now we show that a has a nonzero image in U,(0)11-x. Assume on the contrary that a E jX. Since Tx is the two-sided ideal of U(g) generated by ker x, it follows then that a can be expressed as a = Ei biyi , where

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras

234

bi E (.4 g), yi E 3, and x(Yi) = O. If we represent the elements bi also as bi = Ei xizii with zii E ho, we obtain a --= Ei xz = Eiii sizii yi. Comparing the coefficients of the basis element xi, we get zk = Ei zk i yi . Since x(zk) = xi(zk) 0 and x(yj ) = 0 for all j by construction, this is a contradiction. Hence the canonical image of a in U,(9)//x is indeed nonzero. As noted in the proof of Theorem 38, U6(g)/IX is isomorphic to the matrix algebra Mpn (C). Hence the composition of the quotient map (Mg) U6 (g)//x with the isomorphism U6 (9)/27 Mp n (C) defines a pa-dimensional irreducible representation T of U(g) such that T (a) L O. 1=1 (

Let T be an irreducible representation of U(g) on V. As in the case when q is not a root of unity, one can prove that T is a weight representation, that

is, V = EDA V,,„ where V), {v E V I T(Ki )v = e (A, az ) v, = 1,2, • • • , /1. However, T is not necessarily a highest or lowest weight representation. The problems of classification and explicit descriptions of all irreducible representations of U(g) are only partially solved. For instance, the dimensions of the representations T with )0-, E D are not even known (see [DCKP]). 7.5.2 Cyclic Representations An important class of irreducible representations of the Drinfeld—Jimbo algebras 14(g) are the so-called cyclic representations. Definition 3. An irreducible representation T of (Mg) is called cyclic if T(E) and T(Fi ) j = 1,2, • • • , 1, are nonzero scalar operators.

For Proposition 40 and Corollary 41 we assume that p and the determinant of the symmetrized Cartan matrix (da) have no common prime factor. Proposition 40. Let T be a cyclic representation of (1,40 on a vector space V. Then all weight subspaces of V are of the same dimension and the dimension of V is divisible by p 1 , where 1 is the rank of g. Moreover, T is neither a highest nor a lowest weight representation. Sketch of proof. If VA and VI, are two weight subspaces of V, then A — = kiai, where ki E Z and • • • , cei are the simple roots of g. Set X:ci if ki > 0 and 'Vic' = Ft' for k i < 0. By the definition of a cyclic representation, T(4 1 ./TC 2 • • • 41 ) is an invertible operator from VI, to VA.

E.

Therefore, dim VA = dim V. Since ker T(E) = ker T(F i ) =- {O} by Definition 3, T is not a highest and not a lowest weight representation. _ Finally, we prove that dim V is divisible by 1p. Because V = VA as noted above and dim VA = dim VI, as just shown, it suffices to prove that VA = VI if and only if A — E pQ, where Q is the set of all Ei niai, ni E Z. By (6.13), VA = ym means that Ei diaij kj 0 (mod p), i = 1, 2, • • • , /. Since p is coprime to det (d iao) by assumption, this is true if and only if all ki are divisible by p. Hence dim V is divisible by ,

7.5 Representations of Uq (g) for q a Root of Unity

235

Corollary 41. If T is a cyclic representation of U(g) on a vector space V, then pi < dim V < pn, where 1 is the rank of g and n is the number of positive roots of g.

In order to construct explicitly cyclic representations of t4(g), one uses compositions of certain algebra homomorphisms of U,(g) to an auxiliary algebra 233 chm defined below with irreducible representations of 233q,,• Let 213, denote the algebra with generators x,x -1 ,z,z -1 and defining relations -1 ZX = EXZ, XX = X 1 X = ZZ -1 = Z-1Z =.• 1. In the literature the algebra 213, is occasionally refered to as the q- Weyl algebra.

Let X and Z be the operators acting on a p-dimensional vector space V with basis e l) , e l , • • • , ep _ i by the matrices 0 0

X

1 0

0 1

.••

• .•.

.. •

•

••• •

• • •

1

000 (

1

0

0

• ••

0

0 )

0

Z

0

0 ••

0

•

0

..

E

2

•••

• • •

0

0

( .0 00

••

0

0 0 • • .

•••

ep— 1

that is, Zei = fi e,

Xe i= ei+i ,

where

ep e0 .

(63)

Since EP = 1 by assumption, we have ZX = EXZ and XP = ZP = I. Thus there is an irreduciblerepresentation 7r of 213, on the vector space V such that r(x) = X and 7r(z) Z. For m E N, let ar,,, be the m-fold tensor product of m (commuting) copies of the algebra al,. The corresponding generators of 2/3,, m are denoted by xi and zi , i = 1, 2, • • • , M. Let g = (g, g, • • • gm) and h = (hi, h2, • • • , hm ) be m-tuples of nonzero complex numbers gi , hi . We define an irreducible representation rgh of the algebra 213,, m on the vector space V®m by irg h(xi)=10•••010giX010.•.01, rg h(zi)=10-••01ghiZolo•••01,

where X and Z both act on the i-th tensor factor. 7.5.3 Cyclic Representations of the Algebra Uf(slt+1) Set m := 1(1 + 1)/2. We shall label the generators of 211,,m and the entries of the m-tuples g and h from Subsect. 7.5.2 as xi,, zji and gii , hii , respectively, with i < j, i,j = 1,2, - • 7 1. The main step for the construction of cyclic representations of Wsli+ i) is the algebra homomorphism pri, from

236

7. Finite-Dimensional Representations of Drinfeld—Jimbo Algebras

Proposition 42. Let r = (ri7r27• • ,r i ) and s = (s i , s2, • • • ,si) be two 1tuples of nonzero complex numbers. Then there exists a unique algebra homomorphism pr . : Uf (sli +i ) 90, 5,„ such that prs (Ki) = ri si-1 4zillo zT+11,1 and —1

prs(Ei)=E 1k =i f Prs(Fi) =

faJikZi,k+1 • • • Xi/ 7

E k=1 , -1

-1

X X 2+1-0-1-1—k X 71 2-1-2—k,i+2—k - - • .L, i 1

1

where {z} denotes the expression {z} = (z - z -1 )(f Proof. A proof is given in PDJMM2].

Now we define a representation a -a- o-rsg h of UE (sli+i) on Vom by composing the homomorphism Prs : Uf(s 1/4-1) ME,,, and the representation G(V°m). The representation o-rsgh depends then on 21 + 2m 71-gb : 23 ,m complex parameters ri , si , gii , hii . However, not all of these parameters are independent. It can be shown that the same set of representations arei gh is obtained if we put si = 1, i = 1,2, • • • ,l. Then a o-rgh depends on dim sli±i = 1(1 + 2) complex parameters. Theorem 43. For a generic choice of the parameters r, g, h (that is, except for a set of Lebesgue measure zero in C1(1± 2) ) the representations a = ar gh of UE (sli+ i) defined above are irreducible and cyclic. Proof. The proof can be found in [DJMM2].

LI

Recall that the dimension of the carrier space V (3'4 of any representation argh is pm, where m = 1(1 + 1)/2 is the number of positive roots of the Lie algebra sl(/ +1, C). That is, all irreducible representations o-rgh obtained from Theorem 43 have the maximal dimension (see Corollary 41). For special values of the parameters gii , hii , ri , the representations a are reducible (but not completely reducible in general) and we get invariant subspaces of the carrier space V®/ (1±' ) /2 . Let us treat such an example. Let i and j be integers such that O' < i < p 1 and 1 < j < I - 1. In (63) we used the basis ek, k = 0,1,2,• - • ,p 1, of the space V. Let ei ,i be the vector from the tensor product of j(j + 1)/2 copies of the space V of the form e, = ei ei 0 • • • 0 ei , where the vectors ei belong to the tensor factors V labeled by all pairs (k, n), 1 < k < n < j. We consider the linear • • V of Vol(i+ 1) 12 , where the tensor factors subspace Vi d Ceij 0 V V correspond to labels (k, n) such that 1 < k < n, j 4 and (1 + q 2 )(1 q-N )(1 q -N-2 ) Uq (spN ) admits the spectral decomposition

0, then the matrix ik for

qp+ _ (r i p _ q _N—i po, where

+

( 17 -1

+ q-N- 1 )I

(79)

q - 1)(q q - N - 1) ((if

UNI

CN - 1)

q _N_i ) 7

C1) k

PO =

q - N- 1)(q - 1

(80)

I UN - 1)

Vi) and the one-dimensional are the projections onto S(Vi 0 VI), (VI invariant subspace, respectively. The matrix R for the algebra Uq (sp N ) also satisfies the relations (77) with K

E.1,3 =1 .

Ei,N+1_,)•

(81)

As above, K is a multiple of Po , but now we have K2 = ( 17

1 + q -N- 1

q N-1-1)( q,

q --1)-1K .

8.5 L-Operators and L-Functionals 8.5.1 L-Operators and L-Functionals

Throughout this subsection, let Tv be a representation of the h-adic DrinfeldJimbo algebra Uh(g) on a finite-dimensional vector space V. The elements

= (id 0 Tv)(7?,), L -v = (Tv 0 id)(7Z -1 )

(82)

of Uh(g) 0 .C(V) and L(T) Uh(g), respectively, are called the L-operators of the algebra Uh(g) with respect to the representation T. Proposition 27. The L-operators LiFf and L v satisfy the relations

(4)1(4)2Rvy = Rvv(4)2(4)17

(83)

(L)1(4)2Rvy = Rvv(4)2(4)1,

(84)

276

8. Quasitriangularity and Universal R-Matrices

(2A id)Li4; — W413(4)23, (E

(id 0 llgv= (Lv)12(Lv)137 (id 0 E)./;12 = I.

= /,

(85)

(86)

Proof. Applying the operator id 0 Tv 0 Tv to both sides of the QYBE Ri2R13/Z23 = TZ23/Zi3R.12 we obtain the relation (83) with sign +. The QYBE is equivalent to the identity Re1-31R. 2-31- TZ —12 = Rel2R2-317-C131. Applying Tv 0 Tv 0 id to the latter relation we get formula (83) with sign -. Equation (84) is obtained by applying the operator Tv 0 id 0 Tv to the identity R.1-21 /Z.23 -R.13 = 1Z13 R.231 121 . The formulas (85) and (86) are immediate consequences of (2) and (7), respectively. For example, using the second relation of (2) and the fact that Tv is a representation, we derive that -

-

(id 0 .6)L v- = (Tv 0 id)((id = (Tv 0 id) (R.1-21 R1-31 ) =.(Lv - )13. - )12(Lv

CI

Let us choose a basis fed of V. Then there exist linear functionals tki on Uh(g), called the matrix coefficients of the representation Tv, such that (87)

Tv( . )ei = Ek tki( - )ek-

Further, by (82), there are uniquely determined elements of Uh (g) such that (1 ei ) =E ® e and L v- (ei ®1) = ei These elements it. z3 E Uh(g) are called L-functionals associated with the representation T. This terminology stems from the fact that the elements it are commonly treated as functionals on the corresponding quantum matrix groups (see Subsect. 10.1.3). Let (4) and ( (R -1 )) be the matrices of the transformations Rvv and R v i-T , respectively, with respect to the basis {e i ek} of V 0 V. Since (Tv 0 id)/4 Rvv and (id 0 Tv)L -v = R-v iv by (82), we have tki(r) = Mc" and tia(ii-j ) = (88) In terms of L-functionals the formulas (85) and (86) read as .6(1 i±i ) Ek /Pic t it and

en) = Uij .

(89)

In the remainder of this subsection, Tv, will be vector representation T 1 of Uh(g) on V1 . We fix the basis of V1 used in Subsect. 8.4.1. Let R be the corresponding matrix from Subsect. 8.4.2 and let L+ = (F) and L - = (1,7 ) denote the N x N matrices of L-functionals in the vector representation T1. Then the relations (83) and (84) yield the following matrix equations L-14- L 2-1- R =

L iTLR =

RL4r.

(90)

Properties of L-functionals in the vector representation are contained in

8.5 L-Operators and L-Functionals

277

= 0 for i > j and ll = 177i riFi =-- 1 for

Proposition 28. (j ) l =

i= 1,2,•••,N. 1- N = 1 for g = (ii) /iFi / F2 • • • / N (iii) L±ct (L±)t(c-i)t _ ct (L±)t (c-i)tL± =

I for g = soN, sp N , where C

is the matrix from (62). Proof. (i ): The relations It = Ç = 0, j > j, follow from the particular form (44) of the matrix R, and its inverse R, -1 combined with the fact that T1 (Ei ) and T1 (Fi ) are lowering and raising operators, respectively (see Subsect. 8.4.1). For the same reason, only the Cartan parts of R, and R,-1 contribute to the diagonal L-functionals. From this we derive that ll = ll = 1. (ii) follows from the explicit formulas in Subsect. 8.4.1 and (iii) is obtained from Proposition 20. ,

8.5.2 L-Functionals for Vector Representations The first aim of this subsection is to derive explicit expressions of the Lifunctionals /in terms of the generators of Uh(g). For this we 1, +1 , a+1,1 use the formula (44) for the universal R-matrix R. of Uh(g). In order to treat the first factor in (44), we change the standard basis {111, H2, - • H1} of the Cartan subalgebra I) which appears in the defining relations of Uh(g) to an orthonormal basis {HL 1 1, • • • , Hi} with respect to the symmetric bilinear form (., •) on x 1j (see Subsect. 6.1.1 and Remark 6 in Subsect. 8.3.2). Such a basis will be called self-dual. If {I-I; - • - Hi} is a self-dual basis of 4, then we have EBH 0 H = Ei 0 H. The L-functionals can be calculated by inserting this expression into (44) and using the formulas for the vector representations from Subsect. 8.4.1. We collect the results of these computations and the corresponding self-dual bases in the following list: -

,

Uh (g 1 N):

Hi , tif:i+

1± 71,

1,2,•••, N,

i

(47

— — I i+10, —

(

q q -1 )F2 qH

(q

= (q - q -1)(_1)i-3-1-1 q -H:E 3i31 1- r

1 )F023 q11;.

k

where j > i 1 and the positive roots are as in Subsect. 8.4.1.

Uh(s1N): N-1 k 4--ak=1

=--- 1, 2, - • ---‘1'7-1 Hs,

Hi = .11"i

=

= 1,2, •••,N

lti+ = (q -

j =1,• • - ,N - 2, -

1,

lN =

-(q -

kHkIN

7

8. Quasitriangularity and Universal R-Matrices

278

l. = (q — q -1 )(-1) i-3 + 1 1iti Epii ,

,

= — (q

where j > i + 1 and the positive roots are as in Subsect. 8.4.1.

Uh/2(s02n+1):

g - H: 7

1

= gH:

1, - - n,

+

+ Hi+i + • • • +

=

- 1 7 2 7' ' • 7 n7

i1 = In-4-1,n+1

1-60-1 = (q q -1 )q -14 Ekl

-

1 1c-+1,k = - (q q -1 )Ficek

(q

1 2-n-k+2,2n-k+1

1,

q -1 )q 1141 Ek q -1 ) Ficq -11141

1 < k < n — 1, ,(q ii 2

1+ n ,n+1

q -1/2‘)q -H'n E n

c 1/2 = _ (q

q -1/2)Fn e:,

[2] 1/2

where c

h

q 1/2 q =7" e

1/2 (q 1/2

1 n+1,n+2 + =

,

I n-± 2,n+ 1 =

eq 1/2( 17 1/2

-1/2)En, q -1/2) .Fn

= 2n + 2

and

h (Sp 2n ):

= 1, 2, • • • , 72,

+ Hi+i + • • • +

=

= q-

1 4k+1 = (q

1 ) (1 - 14

1+ 2n k,2n k } 1 = - (q q -1 )q 1141 E1c7 -1‘ -11'1, icq 1- n-k+1,2n- k = (q q -

-

1 ) Fidlik

i k+1,k = - (q -

- -

< k < n 1, Intn+1 = (q 2 Q -2 )q -lini En)

(q2

1 17L+1,n =

2 ) Fnql-Lin

where i' =2n+1 - i.

Uh(so2n): =H +H 1+••-

lin-2 (Ha-1 Hn)/ 2 1

Hn )/2

=

11 : 41;q 11:

44; =

=

(q

1 k-+1,k =

-g (q

-1) g -H;,Ek

(q,

i + 2n-k,2n-k-1-1

g -1 ) Fkg irk

q -1) gHi: +1 Ek

= (q q -1 )Fkq -111:+1 ,

1 < k < n — 1,

1+ n-1,n+1

t,

1 :+1 ,n- 1 =

17 -1) 47 -1-1:i _ i En,

-(

q

i+ n,n+2 -

17-1)(r11:1. En,

(4 - 1 )Fnq H:., ;:+2,n = (q 1

Expressions for the other L-functionals/ti , Ç, i < j, can be derived from the previous formulas combined with the relations (90).

8.5 L Operators and L Functionals -

279

-

Proposition 29. The L-functionals satisfy the recurrence relations

= (/;Ei /i+r - /Pri;Ei )477.,

r< j,

(91)

47,67,),

j j, in (102). Proposition 32. U(R) is a Hopf algebra with comultiplication ZA, counit e and antipode S determined by

'6 (1 ) =

it,

E(itj) = ij

S(L) = (L± ) -1 •

(104)

Proof That Li(R) becomes a bialgebra is easily verified. (The proof of Proposition. 9.1 below contains the details of this reasoning.) The matrices L+ and

286

8. Quasitriangularity and Universal R-Matrices

G — are upper resp. lower triangular and the entries of their main diagonals are invertible in U (R ). Hence both matrices are invertible. The entries it of the inverses (C±) -1 obviously satisfy the opposite commutation relations to (102) and (103). Hence there is an algebra anti-homomorphism S :11(R) 11(R) such that S(tt) We then have the relations S(.&).& --= I which ij mean that the antipode axiom is fulfilled on the generators. Therefore, by Proposition 1.8, U(R) is a Hopf algebra. LI

Now we set R = R, where R is one of the matrices from Subsect. 8.4.2. Let 1(9) be the two-sided ideal in 11(R) generated by the relations

- • • tk N =

fOr

= ct(c±)t(c-i)t

L±ct(L±)t(c-i)t

= MN, = r fror

g =soulsPN-

Since 1(g) is a Hopf ideal of U(R) as easily checked, the quotient 11(R) 11(9) is a Hopf algebra which will be denoted by UqL(g). The following important result says that this Hopf algebra is isomorphic to the extended Drinfeld—Jimbo algebra Urt (g) defined in Subsect. 8.5.3.

q(g)

Theorem 33. Let g be one of the Lie algebras glu , 51N, SON or sp N and set Uqext (g1N ) Ug (g1 N ) and U (s02 11+1) := Uq l/ 2 (S0211,+ 1 ). Then the Hopf algebras (9) and Uqext (g) are isomorphic with a Hopf algebra isomorphism Ut( g) kg) such that 0(t) 0: uql,(0 ) i < j, where lt l and 0(t.;:i ) =

and

are the elements of Ur(9) defined in Subsect. 8.5.4.

Outline of proof. As noted at the end of Subsect. 8.5.4, the elements E Urt (g) satisfy the defining relations of the algebra UL ( g ) . Hence Ut 9 \) such that 0(tti ) = there is an algebra homomorphism 0 :UqL (g) and 0(fi ) = i < j. Comparing the formulas (89) and (104) for the comultiplications and the counits, it is clear that 0 is a bialgebra and hence a Hopf algebra homomorphism. From the formulas listed in Subsects. 8.5.2 and 8.5.4 we check easily that the image of 0 contains all generators of U t ( g) . Therefore, 0 is surjective. It remains to show that 0 is injective. We carry out this proof in the case g =- slN . The proof in the other cases ig similar. First we solve the formulas of / 33 t. 1 t in Subsects. 8.5.2 and 8.5.4 for the generators of U'(s1N) i and set

(q q ' )t1, F: :=

.73

If'71

:= tt. 337

(q

q ' ) ' Ç...1

j = 1,2, • • • , N.

ii

(105) (106)

Using the relations of q(s1N), we next show that the elements of Uql-a(s1N) satisfy the defining relations of the extended Drinfeld—Jimbo algebra Ur t (s1N). In order to do this, we shall write down the relations (102) for Uql-a(s1N) explicitly. The equation rl-L-21- R = RLLt for Ugla(51N) is equivalent to the following relations:

8.5 L-Operators and L-Functionals

(q 'T1)-62+84+31 j < r, j < s, ij ri

tPjt;Es = £rst ij is

,

ii

= -6-3Y

t ij trj

ftcy Li- can be written i > r, j > s,

qt r

> r7 j < .5 1

(113)

> s,

(114)

t n tiz' R = R4LT is equivalent

where i > j and r > s. The matrix equation to the relations

s, Le = q

ri tr+j

q

ltFr

i > r, tZit7t., =

.6 27i ttj =

,

> r,

tTrEs ti::i + (q - q-1 )(tTiEs t; tr±i tL),

tZi tits

(111) (112)

tiitis - qtistiil t it t33

as

>

,

Çjtir-7.9 =

r, j

(110)

,t2:

= t77,-eTi + (q

= tits ti7j , i

(109)

j < s,

s. The relation £jL R =

where i < j and r

(108)

i r, j> s,

= ii

(107)

i < r,

=

287

j < s, < s,

(115)

Ç, i > r, j > s,

-e 33 7-t± sz = -etvi -e7. 33

where i > j and r s. Relation (107) for (i, j) = (i, i 1) and (7-, s) = + 1, + 2) gives

t t-1-1,i+2 tt,i+1 = 4+1,142 and The latter equation expresses tti+2 in terms of t. (by (110)) Inserting this expression into the relation -e-P,i+2-et,i+1 = we obtain 1 tt+1,i+2

t t,i+1 tiF+1,i-F2 t1i-,i+1) tz+1,i-E1 7

t t+1,i-F2 t-P,i+1

=

If we multiply this equation by (47 - q -1 ) -3 q(1 i- i) 2 and substitute the elements E and E4 1 from (105), then a slight simplification yields the Serre relation

E' i2+1 E:

-

(q q - 1 )E:±1 E:E:+1

E:g i2+1 = 0.

The second Serre relation for E follows from (107) and (109). The Serre relations for Fi can be derived similarly from the formulas (111)-(114).

8. Quasitriangularity and Universal R-Matrices

288

1 Equation (108) for i = j 1 and r = j and equation (110) for j =--and s = i lead to the first two defining relations (94). The third relation in (94) follows from (109). The corresponding defining relations for the elements can be obtained from (112)-(114). Relation (95) follows easily from (115). Thus, there exists an algebra homomorphism t9 : Urt (siN) q(s1N)

V(ki ) = K; for i 1, 2, • • • , N - 1, j = such that z9(Ei ) 19(Fi) 1,2, • • - , N. The formulas (105) and (106) imply that 9(E ) ---= E, 0(F21 ) =8WD =-- kJ . Since the sets of elements E F k and g, K.; generate the algebras Urt (s1N) and q(s1N), respectively, 9 is the inverse of 61. ,

,

8.6 An Analog of the Brauer-Schur-Weyl Duality In this section we develop an analog of the Brauer-Schur-Weyl duality for the Drinfeld-Jimbo algebras Uq (g), g =-- slN, SON, sp N • For this we need to extend the Hopf algebras Uq (soN ) by the group algebra CZ2 . Throughout this section we assume that q is not a root of unity. 8.6.1 The Algebras (// (soN) Let CZ2 be the group algebra of the group Z2 consisting of the two elements 1 and x. We denote by Uq (5o2+1 ) the tensor product CZ2 0 Ug li 2 (so2,i+ i) of the Hopf algebras CZ2 and U0/ 2(s02n+1 ). The group Z2 acts as a group of Hopf algebra automorphisms of Uq (50 2n ) such that 10 = id and x ( ) is the Hopf algebra automorphism defined by

x(E) = E(0, x(Fi) = Fx (%), x(Ki) =K,(0, x(i) = i for i = 1,2, • • • ,n - 2, x(n - 1) = n, x(n) = n 1. Since obviously x x = id, this defines indeed an action of the group Z2Let Oq (502n ) = CZ 2 y Uq (s02n ) be the corresponding right crossed product algebra (see Subsect. 10.2.1). That is, aq (so2n ) is the algebra generated by the algebra Uq (so2n) and an additional element x such that x2 1 with the commutation rule x(a) = xax, a E (.1q \so2n). ( One easily checks that Oq (so2r,) is a Hopf algebra with comultiplication defined by the comultiplication of Uq (so2,,) and the relation 21(x) = x x. The finite-dimensional irreducible type 1 representations of the algebra f/q(so2n ) are described by the following proposition. Proposition 34. (1) Let A --= (A 1 , A2 7 Ar,), A i 2(A, ai)/(a i , ai ), be the highest weight of a type 1 irreducible finite-dimensional representation of the algebra Ug

An_1 = A.

There exist exactly two nonequivalent irreducible representations 1;), and i3 of Clq (502,,) such that their restrictions to Uq (502n ) are

Case 1:

8.6 An Analog of the Brauer-Schur-Weyl Duality

289

equivalent to the irreducible representation TA. If v and y0 are highest weight (x)v° = y 0 . vectors for 11), and 7=3; - , respectively, then ii),(x)v = y and -

Case 2: An _i A. Then there exists a unique irreducible representation i"), of Oq (s02n ) such that its restriction to Uq (s02n ) is equivalent to the direct sum representation TA e Two, where x(A) = (Al, - • - 1An--2, An, An-i) if A = (A 17 - - , An _2 ) An -1, An ). (ii) Every type 1 irreducible finite-dimensional representation of fig (so2n) is equivalent to one of the representations from Case 1 or Case 2. Every finite-dimensional representation of Oq (s02n ) is completely reducible.

Proof. The proof of this proposition is similar to the classical case and we 111 omit it. The vector representation Ti of the algebra Oq (soN) is given by the same formulas of ill ' (Ei), ti (Fi) and Ti (Ki ) as for the vector representation of Uq (soN) and by (x) = I if N = 2n + 1, -

ti (X) - En,n+1 En+1,n

— Enn E+1,+1 if N = 2n.

8.6.2 Tensor Products of Vector Representations In this subsection we decompose the r-fold tensor product of the vector representations of Oq (soN) and Uq (g), g = slN,sp N , into irreducible components. This result will be used in the proofs of Theorems 38 and 11.22 below. In order to treat all needed cases at once, we write flq CO instead of Uq (g) for g = slN spN • In what follows, g denotes one of the Lie algebras 51N, SON spN . A set of integers n = (n 1 , n2 , • • • , nk) such that n i > n2 > • • • > nk > 0 is called a partition. Every partition is representable by a Young diagram 0) with ni boxes in the i-th row. consisting of boxes placed in k rows (if nk If n is a partition, then its transpose is the partition n' = (rel ,n'2 , • • • ,nt), where n'i is the number of boxes in the i-th column of the Young diagram of the partition n. In particular, WI is equal to the number of rows in the Young diagram corresponding to n. The number ni := Ei ni is called the length of the partition n. Let P be the set of all partitions (including the partition 0 = (0)). For A n Sin+1 1 B n = 802n+11 Cn sp2n and Dn 802n we define the following sets of partitions (see, for instance, {Wey1): P(A n ) = {n E P

P(B) = In

E P

n+ 1 } , P(C) =

E P 1 nÇ + n'2 < 2n + 1},

P (D n )

= In

I

n},

E P I nÇ + n'2 2n1

and

Pr (An ) = fn E P(An ) I Ln = r} Pr (Xn ) = {n

E

P(X)

I ni

r, In'

r (mod

2 , )}

xn

=

Bn ,CnI Dn.

290

8. Quasitriangularity and Universal R-Matrices

For Ay, and Cn , the sets P(An ) and P(C) coincide with the sets of highest weights (mi, m2, • • .) described in Subsect. 7.1.2 (see (7.2)-(7.4)). Note that different partitions of P(An ) may correspond to equivalent representations of Uq (s1,41 ). If n E P(B72 ) or n E P(D), we set n° := (N n'3 • • •)', where N = 2m + 1 for B7, and N = 2m for D. In fact, n° coincides with n when the first column is appropriately changed. If WI > n in n, then nr < n. Clearly, n" = n. Further, we have n° = n if and only if n E P(D) and nÇ = n. With every partition n E 7"(9) we associate a type 1 irreducible finitedimensional representation f7(n) of tIg (g). It is defined as follows. For g =s1 n+1/ sP2n/ it is just the irreducible representation of Uq (g) with highest weight n. For g = 502 n+1 , the restriction of t(n) to Uq 1/2 (S0271+1) coincides with the irreducible representation with highest weight n if WI < n and with highest weight re if WI > n, and the operator t(n)(x) is given by t(n)(x) =--- (-1) 1 *. In the case g --= s02n we set T(n) := t( n) if < n and (n) i')°,(n) if > n, where i'A and are as in Proposition 34 and A(n)

(ni - n2, • • • , an -1 nn ,nn _i + n r„) A(n) := A(n°)

if i4 < n,

if r4 > n.

Except for the case where g = 502n and nÇ = n, the restriction of the representation T(n) to Uq (0), g = spN , 5027„ resp. Uq l/ 2 (S02n+1) is irreducible. By Proposition 34, if g = s02, and WI = n, then the restriction of i(n) is the direct sum of two nonequivalent irreducible representations of U q (so2n)•

Proposition 35. If g = so2n+1/sP2n/so2n and if n, m E 2(0), n m, then the representations t(n) and T(m) of OW are not equivalent. Proof. Since t-J-q (span ) = Ug (span ) and P(sp 2n ) is the set of all highest weights of type 1 irreducible representations of Uq(sP2n)/ the assertion for 0 =----- sP2n follows. If g = s02n+1, then î'(n)() Pre)(x) and hence i; (n) Pn°). It follows from the definition of t(n) for fig (s02n+1 ) that t(n) and t(m) are not equivalent if < n and m< n or if WI > n and > n. This implies the assertion for g = s02n+1. For g -=-1' 502n, the assertion follows from the results of Subsect. 7.1.2 and Proposition 34. For partitions m, n E P(g) let us write m n if m = n + ei for some i in case g = slN and m = n ei for some i in cases g = soN , spN . Here ei is the vector with 1 in the i-th component and 13 otherwise.

Proposition 36. The tensor product of an irreducible representation i(n), n E 2(g), and the vector representation I-11 of CI q (g) decomposes into irreducible representations as (116) rn,,,n,mEP(g)

8.6 An Analog of the Brauer—Schur—Weyl Duality

291

Proof. For g = SON, one easily verifies that both sides of (116) coincide for the element x. Thus it suffices to prove (116) for elements of the algebras = SiN 7 spN 7 SO2n 7 resp. U47 1/ 2 (SO2n-1- 1 ) • If g = slNI SO2n ) spN or if .9 = SO2n+1 and n'1 n, n + 1, then the decomposition (116) follows from the formulas (7.13)-(7.15) and Propositions 34 and 35. If g -= 50 2n+ 1 and n'i = n (resp. n'i = n+1), then the representation corresponding to the first summand on the right hand side of (7.15) appears when a box in the (n + 1)-th row is El added to (resp. subtracted from) the Young diagram of n. Corollary 37. (i ) The r-fold tensor product ir of the vector representation 1-1. of 0q (9) decomposes into a direct sum of irreducible representations as

ir

(117)

neP,(9)

where m n , m n > 0, is the multiplicity oft(n) in the decomposition. (ii) If g = slN , then ir contains the trivial irreducible representation (that is, the representation with highest weight (0, 0, • • • , 0)) in the decomposition (117) if and only if r = kN , k E No. If g = SON or g --r-- sp N , then Tr contains the trivial representation if and only if r E 2N 0 . 8.6.3 The Brauer-Schur-Weyl Duality for Drinfeld-Jimbo Algebras First we briefly describe the corresponding classical results. Let G be one of the groups SL(N,C), 0(N, C) or Sp(N, C) and let T1 be the vector (first fundamental) representation of G on V1 = CN. It is well-known that the problem of decomposing the tensor product representation Tr of G into irreducible components is closely related to the structure of its centralizer algebra rr (G)l = {A E L (Vi®r ) I AT' (g) -= rr(g)A, g E GI. For G = SL(N, C), a classical result of I. Schur says that the algebra rr(G)' is generated by the flip operators ri,i+i , i =-- 1,2, • • • ,r - 1, of the i-th and (i + 1)-th tensor factors in Vi®r. In the cases G = 0(N, C), SAN, C) the Brauer-Weyl duality theorem asserts that Tr(G)F is generated by ri,i+i and Ki,i+i , i = 1, 2, • - • , r - 1. Here K is the projection of V1 0 Vi onto its one-dimensional r 2-invariant subspace and Ki, i+i denotes the operator K acting in the i-th and (i + 1)-th factors of Vi.°T. We now turn to the quantum algebras. Let Ti be the vector representation of Oq (g) on the vector space 171 , where g = slN , soN , sp N . The image f? r (0 9 (9)) of fig (g) under the r-fold tensor product representation it is a subalgebra of the algebra L (II') of linear operators on K®r. Let R and K be the matrices for Uq (g), resp. Uq 1/3(802n+1), from the formulas (60), (61), (78) and (81). As usual, ki,i+i and Ki, i+i are the operators on 171®r acting as k=T0R and K, respectively, in the i-th and (i ± 1)-th tensor factors and as the identity elsewhere. We denote by B(r) the subalgebra of r(V?r) generated by the operators fli,i+17i = 1,2, - - - , r 1, for g = slN and by ki,i+1 and .

-

292

8. Quasitriangularity

=

and Universal

R-Matrices

—1, for g =7 sopr, spN . Let f?r(C/q (g)) 1 and B(r)' be the sets of operators in £(1/r) commuting with all operators from ir (0q (g)) and B(r), respectively. Recall that a complex number is called transcendental K•, i+i , j

1,2,• • • ,r

if it is not a root of a nontrivial polynomial with integral coefficients. The q—analog of the classical Brauer—Schur—Weyl duality is stated as Theorem 38. Let r E N, r > 2, and let q be transcendental. Then we have

ilr(0q (9)Y = B(r)

and B(r)' =

Moreover, the algebra f?r(tIq (13)) decomposes as a direct sum of algebras

t?r (Oq(0)) = IEB nepr(g) L(V(n)))

(118)

where V(n) is the space of the irreducible representation t(n) of tIg (g). Proof. We carry out the proof for g ----- so 2#7,41 . The other cases are treated in a similar manner. Since the irreducible representations t(n), n C Pr (9)) are mutually inequivalent by Proposition 35, the decomposition (117) of the representation fer r implies (118). Further, since the multiplicities mn in (117) are independent of q, so is m := dim i 1® r(0-q (g))F E nEPr (g) 1 "n2 . On the other hand, let ti be the vector representation of the classical group G = 0(2n+1,C). The r-fold tensor product ti"r of this representation decomposes also into a direct sum of irreducible components according to the formula (117). Therefore, dim inGY =

E

nEpr m2n

= m.

Let us express the dependence of R and K on q by writing R(q) and K(q), respectively. By the Brauer—Weyl duality theorem for the group G =-0(2n 4- 1, C) (see [Bra] [Wey]), there exists a basis vk, k = 1,2, • • • , m, of the vector space TT(G)1 consisting of monomials of the operators Ti , i+1 and j --= 1,2, • • • , r —1. Replacing Ti,i+i by f?(q) i ,i+i and Ki,i+i by K(q)i,i+i in these monomials vk we obtain vectors denoted by vk(q). Consider the matrix of coefficients of the set {vk(q)1 k = 1,2, • • - , m} with respect to the standard basis Eid, 0 • • • 0 Eirir of L(Vi@r). By (61) and (78), the entries of this matrix are Laurent polynomials in q 1/ 2 with integral coefficients. For q = 1, the vectors vk = vk (1), k = 1,2, • • - m, are linearly independent. Hence there is a regular m x m submatrix of the coefficient matrix. For general q, the determinant of this submatrix is a Laurent polynomial, say f, in q 1/2 with integral coefficients. Since f(1) 0, f is nontrivial. Because q O. Therefore, the vectors is transcendental, so is q112 and hence f (q112) {14(4) I k = 1 7 2 7• • .7m} are linearly independent. By Proposition 19, the matrix k(q) and so the polynomial K(q) of k(q) intertwine the representation T1 0 1-'1 . This implies that B(r) Ç T (Uq(g))'.

8.6 An Analog of the Brauer—Schur—Weyl Duality

293

Since the vectors fvk(q) I k = 1, 2, • • • , ml of B(r) are linearly independent and m = dim inffq (9)Y, as noted above, we conclude that B(r) = ( q (g)) 1 . Hence B(r)' = ( q (g))" = (Cfq (9)), where the second equality follows immediately from (118). 8.6.4 Hecke and Birman-Wenzl-Murakami Algebras

In this subsection we introduce Hecke and Birman-Wenzl-Murakami algebras and show that the algebras B(r) appearing in Theorem 38 are images of representations of these algebras. Definition 6. Let q E C, q 0, and r c N, r > 2. The Hecke algebra 11,(q) is the complex unital algebra with generators gi, g, 1 and defining relations (119) gigi+igi = gi+igigi Fil -

gigi =gi g , for Ii -jI>2, = (q 1 ) gi ± 1.

(120) (121)

In the case q = 1 the equations (119)-(121) are the defining relations for the permutation group Pr . Hence Hr (1) is just the group algebra CPI-

0, q2 1, and r c N, r > 2. The Definition 7. Let p,q E CVO}, p,q Birman-Wenzl-Murakami algebra BWMr (p,q) is the complex unital algebra with invertible generators by]. g2, - - • ,gr---i subject to the relations (11 13), (120) and (122) e,g, = p- l e,, —1 (123) eigi_lei = pei, eig i _ l ei = p -1 ei , where ei := 1 - (q - q -1 ) 1 (gi

—1\

gi 1•

(124)

From the preceding relations it follows in particular that ei is a complex multiple of a projection and that g, satisfies a cubic equation. More precisely, we have e=

+ (p

p -1 )(q

q -1 ) -1 )e,,

(g, - q)(g i + q-1 )(g, p -1 ) = 0.

With another set of relations the algebra BWM,(p, q) can also be defined in the cases q = ±1 which have been excluded above. In order to give vector space bases of both algebras, we consider the following sets of monomials: Mn

gn, gngn-11 • " gngn-1 " • gib

M-n = fe1,n+11e2,n+11" •

for n = 1, 2, - r - 1, where

en,n-Fil

294

8. Quasitriangularity and Universal R-Matrices eii := gi_igi__2• • • gi±leigi+1 " •

+ 1 < j, and e,+1 := ei .

Further, we set Mo = {1 } . Proposition 39. (i) The set Br := {s1x2 • • • xr-i j xi G Mil is a basis of the vector space Hr (q). (ii) The set Br := {xni • • • x rir l-r +1 < ni < n2 < • • • < nr < r 1, n i + ni 0 if i, j, sn, C Mn, } is a vector space basis of BWMr(P, q). Proof. See [Bou2j, pp. 54-56, [Wen]] or [HI<W1 for fir (q) and [BW1 or [Wen21 for BWMr (p, q).

The above description of bases is recursive, that is, we have BH-1 = Br 'Mr for the Hecke algebras and Br +i = M_ r •Br Li Br Mr for the Birman-WenzlMurakami algebras. Since the sets Mr and M, consist of r+1 and r elements, respectively, it follows that dim Hr (q) --= r!

and

dim BW1144,q) = (2r - 1)(2r - 3) • - • 3.1.

Both algebras play an important role in knot theory (see, for instance, [HMV], [BW], [Wen2}). The interest in these algebras in the present context stems from the following Proposition 40. Let I-='? and ki ±i be as in Subsects. 8.4.3 and 8.6.3. Set e := 1 for SON and f ::= - 1 for spN . There exists a unique representation 7rr of the Hecke algebra Hr (q) for g = slN and of the Birman-Wenzl-Murakami algebra BWM, (q, EqN - €) for g = soN,spN on the vector space (C N )®r such that Irr (gi ) = j = 1, 2, • , r 1. That is, the algebra B(r) defined in Subsect. 8.6.3 is the image of Hr (q) resp. BWM r (q,eq N- c) under this representation Trr • Proof. We have to show that the matrices --- 1, 2, - • • , r -1, satisfy the defining relations of the algebras _1%40 and BWMr (q, cqN- '), respectively. Since R satisfies the QYBE by Proposition 4, ik fulfills the braid relation (15). This in turn implies (119). Condition (120) is obviously fulfilled. Equation (121) for 13 slN follows from (69). The relations (122) and (124) of the algebra BWMr (q, EqN- ') for g = soN, spN follow from Propositions 25 and 26 and the second equality of (77). Relation (123) can be easily derived (for instance) from (64). D

8.7 Applications The universal R-matrices are the main bridge connecting quantum groups with integrable systems. However, in the theory of integrable systems one needs R-matrices that depend on a spectral parameter. There is a procedure, called Baxterization, for obtaining solutions R(A) of the quantum YangBaxter equation from the R-matrices considered in previous sections. In this section we give a brief review of this method and the relations between Rmatrices and integrable systems.

8.7 Applications

295

8.7.1 Baxterization The parametrized quantum Yang-Baxter equation gives an approach to solvable (integrable) systems of statistical mechanics. A state of such a system is defined by a configuration at the vertices of a square lattice in the plane (for example, by spins). Each edge of the lattice is to be thought of as an interaction and contributes an energy ga, a') to the total energy, where a and a' are the spins at the ends of the edge. For instance, in the Ising model the total energy of a state a is E i (k i crjj cr j+ij + k2crij a,,i+ i). One of the most important characteristics of thé system is given by the partition function

Z

= Ea

exp(-47)1kT)

depending on parameters /e l , k2 , k, T. Usually one collects this dependence of the function Z by a single parameter called the spectral parameter A. Z(A) does not make sense for an infinite sysThe partition function Z tern. For that reason, one considers rectangular approximations of the system with N vertices. If ZN is the corresponding partition function, then the free energy of a site is FPO = ZN VN, where the approximating rectangles have to tend to the whole lattice. One needs to calculate F(A) as a function of A which means the model is solved. The usual technique of finding F(A) is the method of transfer matrices. A transfer matrix is a matrix T(A), depending on the horizontal dimension m of the approximating rectangle, such that Tr (T(A)) is the partition function for the m x n rectangle. The limit of F(A) when n —) oc is given by the largest eigenvalue of T(A). But the size of T(A) grows exponentially with m and its diagonalization is not straightforward. R. Baxter proposed considering systems for which the transfer matrices commute among themselves for different values of A. Then the matrices T(A) have a common eigenvector, say T(A)e = f(A)e. It turns out that one can then determine all possible functions f(A) including the largest one, that is, one can solve the model. In order to find systems for which the transfer matrices T(A) commute it is natural to factorize the matrix T(A) as R,i (A)R,_ 1 (A) • • • R i (A), where Ri corresponds to the contribution obtained by adding energy at the i-th position. Thus, it is necessary to look for local conditions on the matrices R i , which guarantee that the matrices T(A) mutually commute. It can be shown (see [Bax]) that if for each A and A' there exists A" such that

Ri(A)Ri+i (A')Ri (All ) =14+1(A ll )Ri(À i)Ri+i(A), Ri(A)Ri(Y) = Ri(Y)Ri(A),

2,

(125) (126)

then under certain boundary conditions we have T(A)T(A') = T(AW(A). If the matrices Ri are independent of A, then (125) is just the braid relation (15) which is known to be equivalent to the QYBE (6). The method of obtaining solutions Ri (A) of (125) from solutions of the constant(!) QYBE

296

8. Quasitriangularity and Universal R-Matrices

is called Baxterization. Recall from Subsects. 8.1.2 and 8.3.3 that representations of Drinfeld-Jimbo algebras give rise to solutions of the latter equation. In this subsection we carry out the Baxterization procedure for the Rmatrices of the vector representations of the Drinfeld-Jimbo algebras Uq (g), g

= SIN, SON, SPIV

First let R be the R-matrix for the vector representation of Uq (s1N) from 0 R (recall that, as usual, r is the Proposition 24. By Proposition 19, k flip operator) satisfies the braid relation

(127)

k12É‘?23k 12= k23R 12k 231

which is equivalent to the QYBE (6). We shall consider the Baxterization of this R-matrix when the parameters A, A' and A" are related by the condition A" = A'A'. Then the Baxterized R-matrix is a solution of the equation R 12 (x)k23(xY)k12(y) — k25(y) k12 (x023

(x).

(128)

We look for solutions of this equation of the form

k(x) = g (x) (I + f (x)k) , where f(s) and g (x) are functions of the parameter s. Inserting this expression into (128), and using the braid relation (127) and the Hecke condition 12, - 1) És? 2 (q I for the matrix k (see (69)), (128) turns out to be equivalent to the equation + (q - q -1 ) f (x) f (y) = f (xy)

f(s) + f

(129)

for the function f. The function g (x) is arbitrary. Substituting f(s) = (q - q -1 ) -1 (f (x) - 1), we find that the general solution of equation (129) is given by f (s) = (q - q -1 ) -1 (x.)' - 1), where 1/ is an arbitrary parameter. An important particular solution is obtained for -y = -2 and g (x) = (q q -1 )x In this case we get

I(x) = Now we turn to the Drinfeld-Jinito algebras Uq (soN) and Uq (sp N ). In this case we represent the Baxterized matrix k(x) in the form

I(x) = h(s) (I

f (x)k g(x)K)

where k and K are given by (61), (78) and (81), respectively. As in the previous case, inserting this expression into (128) we find that k(X) = h(X)

—

_1

a±x ,

K)

where h(s) is an arbitrary function, -y is a parameter and a* = q -1 )x and "t = -2 we have In the special case when h(s) =

297

8.7 Applications

x -1A -

k(s)

v + -ci-1)(ce±1+ 1) ±x -1 + (q

xr?-1 cf±x - lf

q -1 )(a± + 1))

x - x -1

(x -1 q xq -1 )P + (xq - (xq) -1 )P _

a(x)P o ,

(q, r eg eN ct± ) x2) where a(s) = (x a±x-1 ) -1 (a± x-1 )(eq' N ex± and e, P + , P_, Po are as in Subsect. 8.4.3. The last equation determines the spectral decomposition of kx). Another relation between the values A, A' and A" is obtained under the following change of parameters: x = e (q-1-q)(0-0')/2

y = e (q-1-0072

(130)

Then instead of (128) we have the equation

112(0 - 023012(91 ) = k23(012(6)R23(0 - 0').

(131)

The substitution (130) transforms a matrix k(s) satisfying (128) to a matrix Ék (0) that fulfills (131). 8.7.2 Elliptic Solutions of the Quantum Yang-Baxter Equation The previous solutions of the QYBE are called rational (dependence on x) and trigonometric (dependence on 0). Elliptic solutions are expressed in terms of elliptic functions of the spectral parameter. In this subsection we shall give elliptic solutions which are symmetric with respect to the commutative group Zp Zp p G N, where Zp := ZipZ. Let X and Z be the matrices from Subsect. 7.5.2, where e = exp(27ri/p). We set

71 (a)

71 (al , a2 ) := zaixa2 ,

Cle2 — 0 1 1,2 7 • • •

- 1.

Clearly, this equation defines a projective representation of the group Zp Zp that is, we have T(a)T(13) = ea2 01 T (ct 4_ 0) . Since the algebra generated by X and Z coincides with the matrix algebra Mp (C), any solution of the QYBE on CP CP can be written as R(0) = woMT(a)07(/(3) with wao G C. We are looking for solutions of the following form R(0) = wa (0)7-1 (a) T(a) -1 , w a (0) E C. (132) One immediately checks that such solutions are invariant under the projective representation T of the group zp 0 Zp in the sense that they satisfy the condition

R(0)(T

T)(a) = (71 T)(a)R(0)

for

a E Zp Zp.

298

8. Quasitriangularity

and Universal R-Matrices

In order to obtain elliptic solutions we impose the additional conditions

R(0 + 1) = Z1-1 R(0)Zi = Z2R(0)ZV, R(e + T) = e -i"T

R(0) =

XT 1 R(0)Xi =

0 T',

e -2"i° X2R(0) X 2-1 ,

(133) (134)

where 7- is a complex parameter. These equations are in fact consistent with the quantum Yang-Baxter equation. If we substitute (132) into (133) and (134), we obtain the equations WΠ(9

+ 1) = E Ct2 W a

2i0 7r E-

0) 7 Wa

wa (0 N)

WΠ(0)

= 1.

These equations are satisfied by the elliptic function /D a

m=

&a(11)

where n is an arbitrary parameter and

ea (o)

E

ec:(o)

ce2 ) 2 exp (i7rT (m + —

is the following 0-function:

27ri (m

711=-

c ) (0 + — al )) N N

8.7.3 R-Matrices and Integrable Systems In order to construct integrable systems with periodic boundary conditions corresponding to a given Baxterized R-matrix one uses relation (131) written in the equivalent form Ri 2 (0 0 1 ).R13 (0)R23 (0 1 ) = R23 (01 )R13 (0)R12 (0 — 01 where

R ii := ro î

.

) 7

(135)

We construct a transfer matrix (T(\)) by setting

(0) i 1k22 T1m.km 8 11 - 1 8 m-1

R(0) isik 'l R(0) 81 ,k2 • • • R(0) is1-,7,7' km 82 1 2

Here rn denotes the number of rows in the rectangle. The matrix entries T(0) are linear transformations of L(Vr).Ule partition function Z R(9) is defined by

ZR (9) = Trviem(Tr TOW , where n is the number of columns of the rectangle. If we represent T(0) as a product of R-matrices Ri (0) and use the quantum Yang-Baxter equation, we easily derive the relation

E

s,r

R(0 - 0')T(0) i8 T(0 1 ) 1.

E

8

r

To/yrcT(0):R(0_0/) .;1..

The latter can be expressed by the matrix equation R(0 - 0 1)12T1(0)T2(0 1 ) =-- T2(0')T1(0)R(0 - 0')12

(136)

8.7 Applications

299

in the vector space 171 0 V2 0 Virl, where T1 := R13R14 • • • Ri, m +2 and T2 := R23 R24 • • ' R2,771+2. Equation (136) is called the fundamental relation of the quantum inverse scattering method. Now we multiply both sides of (136) on the right by R(0 — 0') 21- and take the trace Tr in L(V1 0 V1). Then we obtain Trvi ovi T2091 )7'1(9) = Trvi ovi (R(61 — 9')12T1 (0)T2(0')R(0 — 0%7 21 ) =

Tr vi ov,T1(9)T2( 9'),

that is, we have Tr Ti (0)Tr T2 (01 ) = Tr T2(0')Tr Ti (0).

This means that the system is solvable. To every two-dimensional lattice system of statistical mechanics there corresponds an associated one-dimensional quantum chain. This is a quantum system for a particle in the space consisting of a chain of n sites. In the continuum limit n —> oo we obtain a (1 + 1)-dimensional quantum integrable system. This quantum system is the quantization of an integrable system of the equations of motion in 1+1 dimensions. In such a way we also obtain the conformal field theory. As an example, let us consider the elliptic R-matrix

R(0) =

sn (07 + sn T 0

T)

0

0

0

sn (07) sn T 1

sn Or) sn T

ksn (07- )sn (OT + 7)

0

0

ksn (07-)sn(07- +7) \

1

0 ,

0

5n (Or + T) sn T

l

where T is a complex parameter and sn T is the Jacobi elliptic function depending on the parameter k. This R-matrix describes the so-called eight vertex or XYZ model. The term XYZ originates from the existence of anisotropies in the associated quantum model in all three space dimensions. In the limit k —> 0 we have sn T -) sin T and then R(0) describes the trigonometric six vertex or XXZ model. In the limit T -> 0 we obtain the R-matrix for the rational six vertex or XXX model. Equation (135) can be rewritten as 523(0 — 0' )513 (0)Si2 (0') = S12(0')S13 (0)S23(0 — 0'). Under the conditions of unitarity and crossing symmetry

Si2(0)S21(-0)' = I- ,

S12

'---

(S21 Or

— i1\ \t 1

this equation determines factorized S-matrices describing the scattering of particle-like excitations in (1 + 1)-dimensional integrable relativistic models. The term oSvii i2 (0) IS interpreted as the S-matrix for the scattering of two

300

8. Quasitriangularity and Universal R-Matrices

particles with isotopic spins j 1 and i 2 to two particles with isotopic spins j i and 3 2 , and the spectral parameter 0 is the difference of the rapidities of these particles. The many-particle S-matrices in such models factorize into products of two-particle S-matrices.

8.8 Notes The fundamental concepts of a quasitriangular Hopf algebra and of the quantum double and their basic properties were established by V. G. Drinfeld [Dill, Pr2]. Various general approaches to the double have been given by S. Majid (see [Mail] and [Maj], Sect. 10.2). The importance of the quantum double for the construction of universal R-matrices was also observed by Drinfeld who obtained the universal R-matrix for Uh (s12 ) and also Theorem 9. The universal R-matrix for Uh (s1N) was computed in [Ros2]. The formula (44) for the universal R-matrices of arbitrary h-adic Drinfeld-Jimbo algebras was determined in [KR21 and [LS1j. The R-matrices and L-functionals for the vector representations of the Drinfeld-Jimbo algebras and their properties have been derived and studied in [RTF]. The construction of R-matrices for fixed q given in Subsect. 8.3.3 is due to T. Tanisaki [Tan3]. The corresponding approach to L-functionals for fixed q developed in Sect. 8.5 appears here for the first time. The results on the Brauer-Schur-Weyl duality in Subsect. 8.6.3 are taken from {113]. The Birman-Wenzl-Murakami algebra was invented independently in [Mur] and [BW]. The Yang-Baxter equation arose in the work of C. N. Yang [Yam] and R. J. Baxter [Bax] on exactly solvable models in statistical mechanics. Finding solutions of this equation was one of the main reasons for the development of quantum groups. There is now an extensive literature on solutions of the QYBE and on its algebraic aspects (see, for instance, [DV], [Gul ] , [Hi], [Isl.], [Jim5], {LR], [Majl], [Rad3], [Res1], [S1]). The braid relation first came up in E. Artin's theory of braids [Al ].

Part III

Quantized Algebras of Functions

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

With this chapter we take up the detailed study of quantized algebras of functions. The rest of this book will be mainly devoted to this topic, although many results are obtained for general Hopf algebras. The aim of this chapter is to develop basic definitions and facts on the standard quantized simple Lie groups and their quantum vector spaces. Following common terminology in the physics literature, let us adopt the convention from now on to sum over repeated indices belonging to different terms. However, if confusion is possible or if a formula looks too unfamiliar, we shall write the sum sign.

9.1 The Approach of Faddeev—Reshetikhin—Takhtajan 9.1.1 The FRT Bialgebra .4.(R) Let V be a finite-dimensional complex vector space and let R be a linear mapping of V 0 V to itself. Let /4 := T o R, where r is the flip of V 0 V. We fix a basis of V. Then there are complex numbers Ridm kim such that R(x i 0 xi) = kirxn,

xm and .k(xi 0

xm .

=

Let quij ) denote the free algebra with N2 generators uii , j , j = 1, 2, • • N and let 3(R) be the two-sided ideal of quii ) generated by the N4 elements := RVi um k

ult4R 14

j, rn, n

1,2,., N.

Let A(R) denote the quotient algebra C(u.i )/J(R), that is, A(R) is the algebra generated by elements uii , i j = 1, 2, • • • , N, subject to the relations ,

p ji k 1 _ Rik -"klU m Un UlcUl mn)

m, n = 2, • - • , N.

(1)

The N x N matrix (uii ) of the generators uji is called the fundamental matrix of A(R) and is denoted by u. In matrix notation the defining relations (1) of A(R) can be reformulated as

304

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces R111112 = 112111R

(2)

itu 1 u2 = u 1 u2 I

( 3)

or equivalently as ,

I and where u i and u2 are the N2 x N2 matrices defined by u1 = u = I u (Kronecker product of matrices). Then all products in (2) and (3) are just matrix products. The reader should notice that u1u2 u2u1 unless A(R) is commutative. In this matrix notation we shall write the algebra A(R) as (4) A(R) = C(u) I (Juiu2 u1u21). Similar notation as in (2), (3) and (4) will often be used in the following. From the defining relations it is clear that A (r o R o T) is the opposite algebra A(R)°P and that A(ToR-1 o T) = A(R) if R is invertible. Example 1 (R = id and R = T). If R is the identity map, then (1) reads as i44 = uu-1.7, and A(R) is the commutative polynomial algebra in N2 that is, A(R) is the algebra of coordinate functions of theindetrmas, matrix algebra MAC). At the other extreme, if R is the flip T 7 then A(R) is the free algebra C(u) with generators uji . ,

Proposition 1. There is a unique bialgebra structure on the algebra A(R) such that

= uik

IL .I; and e(uii )

6ia ,

j = 1,2, • • , N.

( 5)

In matrix form the equations (5) are commonly written as ,A(u) = uOu and E(u) = I . Proof Since C(u) is the free algebra with generators uji , there are unique algebra homomorphisms A : COLD C such C(u) 0 C(u 32 ) and e : C that (5) holds. Then C(u) becomes a bialgebra with comultiplication A and counit e Indeed, by Proposition 1.8, it suffices to check the axioms (1.15) and (1.16) for the generators uji which is easily done. Since A(I 2 ) = uni k u nt + and e(a) = 0, we get A(j(R)) Ç ,7 (R) C(u) + C(u) J(R) and e(J(R)) = {0 } . That is, ,7(7Z) is a biideal, so the quotient algebra A(R) =-- C(i4)/J(R) is also a I=1 bialgebra. .

Definition 1. The bialgebra A(13) from Proposition .1 is called the coordinate algebra of the quantum matrix space associated with the linear transformation R or, briefly, the FRT bialgebra. Of course, the preceding definition of the bialgebra A(R) depends on the matrix coefficients /Alm of R and so on the basis {xi } of V. The next proposition gives a basis-free characterization of the bialgebra A(R). In particular, this implies that the bialgebra A(R) is uniquely determined up to isomorphism by the linear transformation R which justifies the notation A(R).

9.1 The Approach of Faddeev-Reshetilchin-Talchtajan

Proposition 2. There is a linear map (pv : V

305

V 0 A(R) such that V is a right comodule of A(R) with coaction (pi/ and 14 G Mot (Soy (pv). If A is another bialgebra and "Iv : V -W is a right coaction of A".' on V such that f? E Mor (1,bv 000, then there exists a unique bialgebra homomorphism 0 : A(R) .A such that (id 0 0) 0 (ICI 17 = 11) v -■

Proof. By Proposition 1.13 and (5), the linear map cpv : V

V 0 A(R)

defined by i = 1, 2, - • • , N, (Pv(xi) xi 0 is a right coaction of A(R) on V. Using (1), we obtain (P i/ O

v v )k( xm ® x n ) = = =

(

So y o vv)(14

z 0 xi 0

=

(6)

x k zi)

4î4R

x

=

0

Xj

O Rikii

urn k un /

(14 id)(xk 0 xiu rn k uln ) o

O (pv)(xm 0 xn),

so that 1 E Mot ((py O Soy ). This proves the first part of the proposition. To prove the second assertion, let A and Ov be as above. There are elements t G A , j, j = 1, 2, • • • , N, such that Ov (xi ) = z 0 Pi:. Since Oy is a by Proposition 1.13. right coaction, we have 2k(t) = t O tl and e(t) = From the computation carried out in the first part of this proof it follows that ian for k tni tikqiim E Mor (7Py O 7Pv ) implies that kkiii tni Therefore, the map O: extends to an algebra homomorphism 0 of A(R) to A which obviously satisfies the condition (id 0 0)0 Soy= ftp v . On the other hand, the latter relation implies that 0(u3i-) = t hence the uniqueness assertion follows. ,

By (6), the elements lei are the matrix coefficients of the corepresentation Soy with respect to the basis tx i l. We call the coaction (pv and likewise the matrix u = (14) the fundamental corepresentation of the FRT bialgebra A(R). A similar terminology will be used for the quantum groups GL q (N),S.L q (N),0 q (N),S0 q (N), and Spq (N) which are derived below from FRT bialgebras A(R).

Remark 1. The FRT bialgebra A(R) is a special case of the following more general construction. Let C be an arbitrary coalgebra and let r be a linear functional on C OC. Let M(C, r) be the quotient algebra of the tensor algebra T(C) by the two-sided ideal J generated by the elements /(x,y)

E (r(x( 1 )0y(0)x( 2)0y( 2) - y( l )0x( i )r(x( 2)0y(2))) , x, y E C.

By the universal property of 'T(C), the comultiplication and the counit of C extend uniquely to algebra homomorphisms zl : T(C) T(C) o T (C) and : (C) — ■ C. Then, T(C) becomes a bialgebra. Since e(i(x, y)) = 0 and .6(/(x,y)) = E (/(x ( 1 ) 1y(i ) )

x(2)

Y(2) + Y(1)

1 (1)

0 /(x(2) 7 Y ( 2 ) )) 7

306

9. Coordinate Algebras of Quantum Groups

and Quantum Vector Spaces

j is a biideal of T(C). Hence .A4(C, r) is a bialgebra with structures induced from T(C). Let us consider the special case C =-- MN (C)', see Example 1.4. Let u c C be given by ulic(g) := gici for g = (gm) c AIN(C). As above, let R be a linear transformation of V 0 V with matrix coefficients RiL i and define r E (C 0 C)' by r(uini g := Riditn . Then the bialgebra .A4(C, r) is nothing but the FRT bialgebra A(R). We close this subsection by studying *-structures on FRT bialgebras A(R) and on Hopf algebra quotients of it. Recall that a *-bialgebra (resp. a Hopf *-algebra) is a bialgebra (resp. a Hopf algebra) A which is a *-algebra such that 2i(a*) --= 2(a)* and E(a*) = e(a) for a c A. Suppose that the vector space V is equipped with an involution, that is, with an anti-linear mapping y -+ v* such that (v*)* = y for y E V. Define an involution on V 0 V by (v w)* := v* w* and assume that the basis elements x i of V satisfy xl = x i . Then' the transformation R of V g is called real if R((v g w)*) =-- (R(v w))* y, w E V, and inverse real if R((y Ø w)*) = (R -1 (v w))*, v, w C V. Clearly, R is real if and only if frir = RI 11 for all i, j, m, n (that is, It = R) and R is inverse real if and only if Rrir = (R- l) ir for all i, j, m,n (that is, _II = R-1 ). Since C(uii ) is the free algebra with generators uii , there is a unique antilinear multiplicative mapping 0 of C(u) (that is, 9(Aa +,ub) = A9(a) + FA(b), O(ab) = 0 (a)0 (b) for A, p E C, a, b C C(u)) such that 6+ ( 4) = uji . The next proposition will be used for the treatment of real forms in Subsects. 9.2.4 and 9.3.5 below. Proposition 3. (i) S'appose that R is real. Let I be a subset of C(u) such

that the two-sided ideal J :=(3(R), I) of C(u) is a biideal and the bialgebra A = quii ))1,7 is a Hopf algebra. Assume that 0(I) C j. Let 1,1) be an automorphism of the Hopf algebra A such that 02 = id. Then A is a Hopf *-algebra with involution determined by (uii )* = SOP(u3i:)), i , j = 1,2, • N. (ii) If R is inverse real, then A(R) is a *-bialgebra with involution defined by (uii )* = uii , = 1,2, • • •, N. Proof. (j ): We first prove the assertion in the case V) = id. Define an antilinear anti-multiplicative mapping 19 : C(u) -> A by 19(4) = S ( LI). Let : C( u) * A --= C(t4)/J be the canonical map. Using the assumption that 3 3 R is real and the properties of 5, 0 and /9 we get V(V,.n = 0, .)= so V(J(R)) = {0 } , and 19(x) = S(/(9(x))) for x C quii ). Since 0(I) ç J. by assumption, it follows that 19(J) = 01. Hence 79 passes to an anti-linear antimultiplicative map a -* a* := V(a) of A = quii )1,7 to A. By definition we then have (uii )* = S(u). For A being a Hopf *-algebra we still have to show that a** = a for a E A and that 2i and € are *-preserving. It suffices to do this on the generators uji . The relations 46(( u )*) = .6(u)* and €((uij )* ) = are easily verified. Applying the *-operation to the identity E,n (4)*u.7 = -

9.1 The Approach of Faddeev-Reshetikhin-Takhtajan

Et, s(uni )u.73.

= bij we get

En (uriL)*(u7)**

307

Er, s(u4) (wit)** = 6. This proof in the case 0 = id. =

implies that (4)** = tirii. and completes the Let us turn to the general case. Put a*' := i,b(a)*, a E A. Since 7,b is a Hopf algebra automorphism, V) 0 51 = So 7,/) and hence 0(a*) = 1/0) * , a E A. Using the latter and the assumption 0 2 ,---- id one verifies that (A, *1 ) is also a Hopf *-algebra. (ii): The bialgebra quii) is a *-bialgebra with involution defined by ------ uji . By It z= R -1 , we obtain (Iim )* = - E,,, t ,,, s (R - 1)ik i Irk.19 (R - 1),nrsm , so the involution of quii ) passes to the quotient A(R) --= C(u)/J(R). 0 9.1.2 The Quantum Vector Spaces XL (f; R) and XR(f;R)

We retain the notation from the preceding subsection. Let f = { fi , • - • 7 fn} be a set of polynomials f i in one variable with complex coefficients. We first give a basis-free definition of the algebra XL(f; R). Namely, XL (f; R) is the quotient of the tensor algebra T(V') over the dual V' of the vector space V by the two-sided ideal ,A(f; R) of 11 (V') which is generated by the ranges of the mappings fm (R) t , m = 1, 2, • • • , n, of (V 0 V)'. Here fni (R) t denotes the transpose of the linear transformation fr,-,(i?‘) of V 0 V. Now we describe the algebra el'af; R) in terms of coordinates. Let us identify V and V' and so V 0 V and (V 0 V)' by identifying y E V with the linear functional '6 on V given by i3(w) = E, cii0i, where y = E, a i s i and w = E, 13 ix, E V. Then the tensor algebra T(V') is (isomorphic to) the free associative algebra C(x i ) with generators xi, • • • , xN and Ji,.(f;R) is the two-sided ideal of C(xi) generated by the nN2 elements fni (.1b13/ xkxi, where m = 1, 2, • • •, n and i, j = 1, 2, • • • , N. That is, XL (f; R) is the algebra with generators xi, • - - , xN and defining relations

frn (lbxkx/ = 0,

m = 1,2, • • • , ri; i, j = 1,2, • • • , N.

( 7)

Let x denote the column vector (x1, • • - , xN) t of generators. In matrix form the relations (7) can be written as frn (ft)xi x2

( 8)

Indeed, if x 1 denotes the matrix x0/ and x2 is the matrix /0x, then the expression f„,(ii)xix2 in (8) is just the product of the three matrices fm (Jet), xi and x2. Similarly, the algebra XR(f, R) is defined as the quotient of the tensor algebra T(V) over V by the two-sided ideal generated by the images of the transformations fm (11), m = 1, 2, • • • ,n, of V 0 V. Clearly, XR(f;R) is the algebra with generators xi, - • • , xN subject to the relations

fm (10 121 X k X1

-= 0 1

m=1,2,•-•,n; i,j=1,2,•••,N.

(9)

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

308

Proposition 4. Let A be a bialgebra which is a quotient of the bialge bra A(R). Then there are algebra homomorphisms p : XL(f ; R) —> A0 XL(f R) and (p R XR(f R) —> XR(f; R) 0 A such that

(PL(xi) = î4 0 x i and (pR(x i ) = xi 0 u3i .

(10)

These mappings (p L, and (pR turn the algebras XL (f;R) and XR (f;R) into left and right A-quantum spaces, respectively.

Proof. We carry out the proof for XL,(f ; R). By the universal property of the free algebra C(x i ), there is a unique algebra homomorphism çoL : C(x i ) A® C(xi ) such that çoi,(xi) = u 0 x i . Since PL(frn(MSkXj)

(

frn (ft)Ukr U is SrSs

ki _x r X_ s Tn rs Un k i f (ft) i

by (3), 40L(Si(f; R)) ç A 0 ,71,(f; R). Hence (pi, passes to an algebra homomorphism, still denoted by çoii , of XL(f;R) to A 0 XL(f; R). To prove that ciaL is a left coaction, it suffices to check the conditions (1.52) on the generators si. We omit this simple verification. Definition 2. The algebras XL (f; R) and X R (f; R) are called the coordinate algebras of the left and right quantum vector spaces, respectively, associated with the set of polynomials f and the transformation R. Suppose that the matrix 1 is symmetric, that is, 1 = kj. The matrices

.h for the quantum groups Glig (N), SL q (N), 0 q (N) and Spq (N) considered in Sects. 9.2 and 9.3 will have this property (see formulas (13) and (30) below). Then the defining relations (7) and (9) are the same, so the algebras XL( f; R) and X R( f ; R) coincide and are denoted by X( f ; R) . By Proposition 4, X(f; R) is a left and right quantum space for the bialgebra A.

Example 2 (R = id). Let R be the identity map and f(t) = 1 — t. Then X(f; R) is the commutative polynomial algebra in N indeterminates and the left coaction (pi, is given by the matrix multiplication of u and x. A Next we consider some quantum spaces which are obtained by modifying the preceding construction. For sim - plicity we restrict ourselves to the right handed coordinate versions and to the case of single polynomial relations. For any triple of polynomials f, g h in one variable and any complex number 'y, let X(f,g,h, R) denote the unital algebra with 2N generators X1, ,XN,Y1, n 7 YN and defining relations

f(MVxkxi = g(f)gYkY1 = h(MYfrol Yisi +

Pybki = 0, i j = 1, • • • , N. ,

(11) Proposition 5. Let A be a Hopf algebra which is a quotient of the FRT bialgebra A(R). Then the algebra X E.'. X(f,g,h,ry;R) is a right quantum space for A with respect to the coaction (pR determined by the equations (PR(xi) = xi 0 u-1: and çoR(yi ) = yi S(u).

(12)

9.2 The Quantum Groups GL q (N) and SL(N)

309

Proof. The proof is similar to the proof of Proposition 4. One first defines C (xi, y i ) g A by (12). As usual, an algebra homomorphism çoR : C(xi,y i ) j =1,- • - ,N. Using C(xi, yi) denotes the free algebra with generators (3) one then derives that çoR passes to the quotient algebra X of C(xi , yi ) to CI give a right coaction there. The defining equations (11) may look strange at first glance. However, they become very natural in terms of corepresentation theory (see Sect. 11.1). 1, = g(Nki and Clci = WO. Then the matrices A, B Set At! = f(f?) 4 and C belong to the intertwining spaces Mor (u u), Mor (ue g uc) and Mor (tie u, u uc), respectively, and it is only these properties which are used during the proof of Proposition 5. The term yisi in (11) may also be replaced by Tikil ykx/ with T E Mor (ucou). The second equation of (12) means that çoR acts as the contragredient corepresentation uc on the generators yi . In the following Sects. 9.2 and 9.3 we realize the constructions A(R) and X(f ; R) by taking as R the matrices R which have been derived in Subsect. 8.4.2 for the vector representations of the Drinfeld-Jimbo algebras Uq (g), g -= giN , sip/ 1 502,, spN , and Uq i/ 2 (S027-14-1). Our aim is to develop the coordinate Hopf algebras of the quantum groups Glig (N), SL q (N), °q (N), 80q (N) and SPq (N). Since A(R) is only a bialgebra, this requires additional constructions.

9.2 The Quantum Groups aL q (N) and SL q (N) In this section R is the matrix q l/N R1,1 obtained in Subsect. 8.4.2 for the Drinfeld-Jimbo algebra Uq (s1N). By (8.60), Ê = r o R has the matrix entries

Ran = q6i2 bin 6i, + (q q-1 )Sim,6in 0(j - i),

(13)

where 0 is the Heaviside symbol, that is, 0(k) = 1 if k > 0 and 0(k) = 0 if

k < O. We suppose that A + := q + q-1 0. By (13), the matrix k is symmetric. From Proposition 8.24 we know that k satisfies the quadratic equation

qi)(k + q -11) = 0

(14)

and that

P + := A 1 (k + q -1 /)

and

P._ :=

+ TT)

(15)

are projections (that is, Pi = P±) such that P + P_ = P_ P + -.----- 0 and

= qp+ - q --1 p _ .

(16)

310

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces

9.2.1 The Quantum Matrix Space Mq (N) and the Quantum Vector Space CqN We begin by describing the FRT bialgebra A(R) explicitly. Definition 3. The bialgebra A(R), with R = (Kil n ) given by (13), is called the coordinate algebra of the quantum matrix space ltici (N) and is denoted by 0(Mq (N)).

Inserting the matrix entries R;l:Lim from (13) into the relations (1), we see that the algebra 0(Mq (N)) is generated by the elements uii , i, j = 1, 2, • • • , N, with defining relations:

uik tric -z--- quik uik , -';

uik uik = qujk uik , i < j,

(17a)

i < j, k < 1,

(17b)

u-kul,

nik uji - ui ut = (q - q-1 )uik ul,

i < j, k < I.

(17c)

These relations are equivalent to the requirement that for any k, 1, m, n with k 7 b = unk t c = Ulm 1 d = uniof the corresponding k r(v(i) W (1)*(0) 0 V(0), V E VI E W.

10.1 Coquasitriangular Hopf Algebras

333

If V = W is the right A-comodule A with respect to the comultiplication, then we obviously have (e e)rvw = r, that is, the universal r-form r can be recovered from the map rvw in this case. Proposition 1. If U, V and W are right A-comodules, then we have:

(1) rvw is an isomorphism of the right A-comodules V OW and W 0 V. (ii) ruo v,w = (ruw Oid)(id0rvw) and ru y ow = (id 0 ruw )(ruv 0 id). (iii) (rvw 0 id)(id ruw)(ruv Old) (id 0 ruv)(ruw id)(id® rvw)• Proof. (i): Define a linear mapping fwv : WO V —> 170 W by wv(w 0 y) = v E V, w e W. Using (1), we obtain

(fw v 0 rvw )( y

w)

=

wv

(E r(V(1) 7 W (i)*(0) 0 V (0))

Er(v (2) , w( 2))*( 1 ), w( i ))y( 0) O w( 0) EE(v( oe(w ( l) )v (0) w (0) = v w and similarly rvw 0 i'wv(w O y) = w v. Hence rvw is invertible. Let cpyow and cpwov denote the coactions of A on the right comodules V W and W 0 V, respectively. By (6), (1.61) and (5), we get

(Pwo v o rv, w ( Y

111)

= (Pwcov (E r(v(i), w(i)*(o) v(0)) Ew (0)

O V (o) O w( i)v( i)r(v ( 2), W(2) )

Ew (o )

0 v(0) 0 v(2)w(2)r(v(i) w(i))

= (rvw 0 id) 0 wv o w (v w). That is, rvw E Mor (wvovv,(Pwov)• Using (3) and (4) the assertions of (ii) are proved in a similar manner.

(iii) follows easily from (II).

111

Of course, the preceding results hold for left A-comodules as well. By Proposition 1(iii), for each right A-comodule V the linear mapping rvv defined by (6) satisfies braid relation, hence T O rVV is a solution of the QYBE. That is, coquasitriangular bialgebras induce solutions of the QYBE on their comodules just as quasitriangular bialgebras do on their modules. A large part of the theory of coquasitriangular bialgebras can be developed by dualizing the corresponding theory for quasitriangular bialgebras. Some basic facts on coquasitriangular bialgebras are collected in Proposition 2. For any coquasitriangular bialgebra A with universal r-form r we have:

(i) r12r13r23 = r23r13r12 on A0 A 0 A. (ii) r(1, a) = r(a, = e(a) for a E A. (iii) i is a universal r-form for the bialgebras A°P and A"P (see Subsect. 1.2.3 for the definitions of these bialgebras).

334

10. Coquasitriangularity and Crossed Product Constructions

:= f(b 0 a). (iv) P21 is also a universal r-form for A, where f21(a (y) If A is a coquasitriangular Hopf algebra, then for a,b E A we have r(S (a), b) = f(a, b), t(a, S(b)) = r(a, b), r(S(a), S(b)) = r(a, b). Proof. All proofs are straightforward verifications. As a sample, we show the first relation of (v). Set ri(a b) = r(S(a)0b). From (3) and (ii) we obtain

E r(a( i ),b(0)r/ (a( 2) 7 b(2))

Er(a(i) ,b(1) )r(S(a(2) )7 b(2)) = = Er(a(i) S(a(2) ),b) = e(a)r(1,b) =

--=

so that re = E 0 E. Similarly, r'r = E 0 E. Hence r' = F.

1=1

The next result describes the square and the inverse of the antipode of a coquasitriangular Hopf algebra. It is dual to the corresponding assertion for quasitriangular Hopf algebras (see Proposition 8.5). Proposition 3. Let A be a coquasitriangular Hopf algebra with universal r-form r. Define linear functionals f and f on A by Pa) =

E r(a( i),S(a( 2)))

and J(a) =

EF(S(a (i) ),a(2) ),

a E

A.

Then we have ff =Jf =e in A'. The antipode S of A is bijective and satisfies S2 = * id* f and S-1 = f * S * 1. That is, for a E A we have

82 (a) =

E f(a(o)a( 2)f(a( 3)),

1. (a) =

Proof. Applying (5) with a 0 b replaced by S(a( 3))a( i)f(a( 2)). This yields

E

E 512 (a(2))f(a(1))

=

E f(a(i))S(a(2))f(a(3)).

E a( 1) S(a( 2))

we get

f (a)1 =

E 8 2 (a(4))S(a(3))a(i)f(a(2)) E S(a(3)S(a( 4)))a( i)f(a( 2)) = E amf(a( 2)).

(7)

Using the first and the third equality of Proposition 2(v), formulas (7) and (4) and Proposition 2(ii) we compute'

f f(a)

E f(S(a(i)), a(2))f(a(3)) = E r(82 (a(1)), a(2)f(a(3))) E r(82 (a(0),8 2 (a(3))f (a(2))) = E r(a(i),a(3))f (a(2)) E r(a( i ), a(4) )r(a( 2), S(a( 3))) = E r(a( i), S(a(2))a( 3)) r(a, 1) =

Similarly, fj=-- e. From the relations ff =

E and

(7) we obtain

E f(a(i))a(2)f (a(3)) = Ef(am)f(a(2))82(a(3)) Since 82 = 1* id*

f as just shown, we

= 82 (a).

have 82 *1= j*id, that is,

10.1 Coquasitriangular Hopf Algebras

E s2 (a(1))f(a(2)) = E f(a(1))a(2)•

335

(8)

Put T := f * S * J. Using (7) and the relation fj-= E. , we get

E T(a(2))a(i.)

E f (a(2))51 (a(3))f (a(4))a(i) = E S(a(3))f (a(4)) 82 (a(2))f(a(1)) = E(a)1.

Similarly, (8) and the equation fl = E imply that E ct( 2)T(a(1)) = This shows that T is an antipode for the bialgebra A°P. By Proposition 1.6, T= Another interesting feature of a coquasitriangular bialgebra A is that its comodules carry (left and right) module structures. Proposition 4. The equation y 1± (a) are bialgebra homomorphisms of A° 13 to U. For a,b,c E A, we have 1 ± (ab)(c) = E 1 ± (b)(c(l))1 ± (a) (c(2)) (23)

A(1 ± (a)) = E

1±(a (1) )

/ ± (a(2))•

(24)

(ii) Suppose that A is a Hopf algebra. Then II is a Hopf subalgebra of A° with antipode determined by the relation S(1±(a)) = 1±(S: 4 1 (a)), a E A. Moreover, for all a,b E A and g E A° we have the identities

A(1(a)) =

/(a(2)) 0 Sr(a(1))) 1± (a(3)))

(25)

46(4a)) =

/±(a (i) )sula(3)))

(26)

4a(2))7

adR (g)(1(a)) = E 1 (a(2) ) g(a(1)S(a( 3))),

(27)

E t(a(2)) g(a(1)S(a(3))))

(28)

adL(g)(i(a)) =

1(a)1(b) = /(amb(2) *(b( i), S(a( 3)))*( 3), a( 2)),

(29)

T(a)l(b) = i(a(2)b(3))*(1), b(2))r(a(3), S(b(1))).

(30)

10.1 Coquasitriangular Hopf Algebras

343

Proof. (i): Since 1 - (a) (.) = f21(-1 a) and 121 is also a universal r-form on A by Proposition 2(iv), it suffices to prove the assertions for 1+ . By (3), we have 1+ (a)(bc) = r(bc, a) =

Er(b, am)r(c,a( 2)) = E 1+(a( i ))(b)1 + (a(2 ))(c).

By the definition of A° (see Subsect. 1.2.8), this implies that 1+ (a) E A° and hence 1,1 C A° . The preceding equation rewrites as (24) using the definition of the comultiplication of A° . By (24), 11 is a subbialgebra of A° . Formula (23) follows from (4). The equations (23) and (24) combined with the relations 1+(1) = e mean that the linear maps r-± are bialgebra homomorphisms of A°13 to U. (ii): The first assertion of (ii) follows at once from (i). Recall that the Hopf algebra A °P has the antipode Si'. Formula (25) is obtained from (22), (24) and the fact that S(1 - (a)) = 1(574 1 (a)). Next we prove (27). Using condition (5) twice, we compute

(1(a)g)(b) =

E q(b(i), a)g(b(2)) E r(a( i ),b(0)r(b( 2 ), a(2))9(1)(b(3)a(3))9(2)(S(a(4))) E r(a( i ), b(1))r(b(3) , a(3))9(1) (a(2)b(2) )9(2) (S(a(4))) Er(a (2) , b(2))r(b(3), a(3))9(1)(bwa(1))9(2)(S(a(4))) E /(a( 2 ) )(b(2))g( 1 ) Om )9(2) (a(i) S(a(3) ))

That is, we have

1(a)g =

E g( 1 )1(a( 2 )) g(2)(a(1)S(a(3)))

(31)

and hence

adR(g)(1(a)) =

E s(g (1) )/(a)g (2) = E S(91))9(2)/(a(2)) g( 3)(a( 1 )S(a (3) )) E1(a(2) ) g(a( 1 )S(a( 3))).

Now we verify equation (29). First we apply the QYBE r12r13r23 = r23r13r12 to x • y for ac, y E A. We then obtain the identity

E S(1 - (s(i)))r(s(2), y ( 1 ) )/+(Y(2) ) = E CE (Y(Or(x(i), Y(2))

(x(2))). (32)

Using formulas (22), (23), (5), (32) and (4) we compute

E

(amb(2))r(b( i ), S(a(3)))r(b(3) a(2)) = Er(b ( l ) , S(a(4)))8(1 - (a(1)))5V - (b(2)))[r(b(4), a(3)))1 + (a(2)b(3))] =

ow 7 5(a(4)))57(1- (a(1)))45(1 (b(2)))r(b(3), a(2)

a(3))

344

10. Coquasitriangularity and Crossed Product Constructions

= E x.0( 1 ), S(a(4)))Sr(a(1)))[S(1 - (b(2)))r(b(3),a(2))/ + (a(3))1/ + (b(4)) = E r(b( i), S(a( 4)))Sr(a( 1 )))/ + (a( 2))r(b( 2 ), a(3))Sr(b(3)))/ +(b(4)) E r(b( i ), a( 2)S(a (3)))/(a( i))/(b( 2)) = 1(a)1(b). The formulas for the functionals ï(a) are derived in a similar manner. Instead of (31) we use the equation gt(a) = E 4a(2))9(2) (S(a(o)a(3))• Equation (30) gives the motivation for the definition of the covariantized product and the transmutation theory of coquasitriangular Hopf algebras, see Proposition 34(ii) below. An immediate and perhaps surprising consequence of formulas (29) and (30) is Corollary 12. If A is a coquasitriangular Hopf algebra, then the sets 1(A) =

S(T(A)) and T(A) = 8 (1(A)) are subalgebras of the Hopf dual A°. In general the algebras 1(A) and i(A) 'are not Hopf subalgebras of A°. Let us recall the notation a = a - E(a)1. From the formulas (25) and (27) we easily obtain the following Corollary 13. Let V be a linear subspace of a coquasitriangular Hopf algebra A such that ,A (2) (V) C AOV A. Then, T(V) MO a E VI is an adR-

invariant linear subspace of A' satisfying X(1) = O and d(X) - e 0 X E T(V) 0 A° for X G T(V). We shall see in Subsect. 14.2.3 that (under some technical assumptions) the subspaces of A° with the above properties are just the quantum Lie algebras of bicovariant first order differential calculi on A. Next we consider *-structures. Recall from Corollary 1.11 that the algebra A° for a Hopf *-algebra A becomes a *-algebra with respect to the involution f f* (a) := f(S(a)*), a E A. Proposition 14. Suppose that A is a Hopf *-algebra.

(i) If the universal r-form r is real, then = /T(S -2 (a*))

and

1(a)* =-1(8(a)*).

(33)

(ii) If the universal r-form r is inverse real, then 1± (a)* = 1± (S -2 (a*)) and 1(a)* =

r(a*( 1 ),S (a* (3)))!(a*(a*(4))1(a* ( 2)) , (34)

where J is the linear functional from Proposition 3. Proof. As samples, we prove the formulas for 1+(a)* in (33) and for 1(a)* in (34). Using the assumption that r is real and Proposition 2(v), we get l+(a)* (b) = 1+(a) (S

= r(8 (b)* , a) = r(at, S(b))

= r(8-2 (a*), S-1 (b)) = 1(S-2 (a*), b) =

10.1 Coquasitriangular Hopf Algebras

345

The second proof will be only sketched. Let b, C E A. Using the first formula of (34) and the relation * o S -1 = S o* (see (1.39)), we compute

S(1 - (b))* = (S -1 (b))* = (8 -2 (8 -1 (b)*)) = 118 -2 (8(b*))) =

(b*))

and hence

1(b)* = (E 8(1 - 0(1)))1 + 0(2))) *

/±(s-2(b* (2) ))sq--(b* (1) )).

(35)

Farther, by Propositions 3 and 2(v), we have

b = E Ab (i) )s2(b(2))f(b(3)) =

r(b(i) 51-2 0(2) ) 15-2 (b(3) )f ( b(4) ) • (36)

Using the formulas (22), (36), (32), (35) and (7) in this order, we get

1(c)

=

= E

(c(i) ))*(2) , 8 - 2(c(3) ))/+(s - 2(c (4) ))f(c (5))

E 1±(s-2(c (3) ))r(c (1) , 8-2(c(4) ))sr (c(2)))f(c(5))

= = E r(, ) , coof(c(3))1((c(2))*)*• Inserting the latter formula into the expression E r (c(1), S(c(3)))/(c(4))/(c(2) ), it simplifies to 1(e*)*. Setting e = a*, this gives the assertion. Now let u=(ziii )i,j-1,...4 be a fixed matrix corepresentation of A. Setting = r(uii ,unm )

(37)

i ) given by we obtain a matrix R = (R) with inverse R -1 = ((R -1 ) A = t(uji , uT,'„ ).

(R -1

Comparing (37) and (6) we recognize .k = TOR as the matrix of the map rvv, when V is the right A-comodule Cd with coaction determined by the matrix corepresentation u = (uji ). Therefore, by Proposition 1(iii), it satisfies the braid relation and so R fulfills the QYBE. From Proposition 1(i) we obtain ut " klu'n rn

We define dxd matrices LI = A by = l(u), 1 = The linear functionals

15i on A

pplk 4 -nm 7 - 1

i,j=1,•••,d.

[= (4)

of functionals on

ti = 1(u)), i. =

(39)

L = (11i ) and

1- 4), (

(38)

are called L-functionals associated with the matrix corepresentation u. Since A(uii ) =utOull and 6(14) = 6ij , (24) and

Proposition 2(ü) imply that

346

10. Coquasitriangularity and Crossed Product Constructions

Li (1 ii ) = IV 0 1 ±k and

(40)

c(iti ) = 6ii .

The latter relations mean that the mappings LI(.) : A > Md(C) are algebra homomorphisms. Further, if A is a Hopf algebra, then the matrices L and E are just the matrix products L = S(L)L+ and E = L -Es(L - ). From the corresponding definitions we see that the evaluation of the functionals /ti 3 at the matrix entries unm, are given by —

ii.(un ) = (142)rm. iti (li n ) = Rni . / -:i ( Un ) =-(R -1 ) in / i(m nti ) = (fer i im) 3 my 3‘m ‘ 3m m3 7 3 m .

In matrix notation these formulas can be expressed as L(u2 ) = R21,

1-1- (I-12) = Ri-21 1 Li(u2) ----- (M)21, E1(u2) =

(j )12. (41) 42

Proposition 15. The matrices LI satisfy the relations LiF 14R =- RL -21- LiF, LT L2- R = R1_ 2- LT, L i- L-2F R = RL-2F L 1- .

(42)

Proof. Using the formulas (21), (4) and (38) we conclude that

(/ +j/+ kl k I i )( a)R nm

=E =

nkini

E r(a( i ), ujk )r(a( 2),W)Rnkim

= r(a,Wujk )Rtim = kit *, unk ulm ) . Rikil l:.il le (a), that is, LiEL -2FR = RL -2F Lt. Replacing r by the universal r-form 1 21, the same reasoning shows that LT L2- 17 = RL2- LT . The QYBE for r (by Proposition 2(i)) implies that f 12 r23 r 13 = r13 r23f- 12 • Applying this equation to uni 0a0u-In and using formula (37) we obtain

E 17 (4, a( 1 ))r(a( 2),uji )Rnklm = E Rr(a( i ),uim )i(u nk ,a( 2)). This means that rkii+iiRnkim = R i i+mi rnk . Thus, L i- L -2F R = RL-2I- L i- .

0

Proposition 16. Suppose that A is a coquasitriangular Hopf algebra and is a matrix corepresentation of A. (0 The matrices L = S(L)L+ and I = L+ S(L- ) defined above satisfy the reflection equations (43) L2R 1 21_1R21 = R12 1-1 R2 1 1-2)

[ 2R21 1--1R12 = R21r-1R12 1 2-

(44)

(ii) The elements Tr Lk D-1 , k E N, belong to the center of A°, where D -1 is the matrix given by (19) and L = S(L1L+. Proof. (1): By the third equality of (42), we have LiER1 2,5(Ln .= S(Ln.R1214. Using this relation and the first equality of (42) we get

10.1 Coquasitriangular Hopf Algebras

347

L2142L 1 = S(LnI4R12S(LT)LiE = S(L 2-- )S(LT).R12L -21- Lii= S(Li- L 2- )1414R 1 2.

(45)

Interchanging the lower indices yields L 1 R21 L2 =

)L -2E 1-jR21•

(46)

Multiplying (46) by R12 on the left and using the first two relations of (42) we obtain R12 Ll R21 I-2 := S( LT L2- Lt 14Ri2R21. By (45), the right hand side is equal to L2R12L1 R21. Thus we obtain formula (43). The proof of the second reflection equation (44) is similar. (ii): From formula (31) we derive by induction on k the identity 1(a 1 ) • • • 1(ak )g = E g(1) 1(a (' 2) ) • . • /(alt2) )g( 2) (ati) S(at3) ) • • • a/4 ) S(4) ))

for arbitrary elements al, • • , ak E A and g E A°. Using this equation and -1 )ii = (D -1 )r-n "g-1 (uji ) by (20), we obtain the relation S(urri)(D 3 i (L kp -iyig

1 Zi2 l i 32 ..

- 1) 3i

E g( 07,12 4,-23 • • . l g(2) (uin S(u in:)uS(u in33 ) .. = E g(1) l i 2 iz . • • l g(2)(uS(uT)(D -1 )3i. ) Eg(i)(L k )rn , g(2)( ,_, (u3i s(u)))(D-1) g

j (LIC

1 )i3

El

We now assume that A is a Hopf *-algebra. We shall express the conditions in Proposition 14 in terms of the functionals / ±ii and l. First suppose that r is real and (7.L i )* = S uii ) for i,j = 1,2, • • • d. Since then S-2 ((4.)*) = S -1 (trii ) and S(74)* =74, the equations (33) yield (

(1±11)* = S(i i')

and

(13i.)*

(47)

Next we assume that r is inverse real and (uii )* = uji for i, j = 1,2, • • • , d. By un m (D-1 7 and (D -1 ) 3i. (20), we have S-2 04) Du(D1 )T , where the numbers are given by (19). Inserting this into (34) and setting fliknin := r(4, Sei4.0), we get (48 ) (111) * = D inl± „?(D7 1 )T and = fek7n(D-1)jmink It might be necessary to emphasize that the matrices L±, L and of functionals depend on the choices of the universal r-form r and of the matrix corepresentation u of A. If we take the universal r-form t 21 (by Proposition

348

10. Coquasitriangularity and Crossed Product Constructions

2(iv)) instead of r, then 1+1 and l j interchange. Further, if A is a coquasitriangular Hopf algebra and u is replaced by y = (vi) with v := then the representations LI of A go into the contragredient representations L, where (LIfl i := S(/-1 ). Let us specialize the preceding to coquasitriangular FRT bialgebras. Suppose that R is an invertible solution of the QYBE. Then, by Theorem 7, the FRT bialgebra A(R) is coquasitriangular and its canonical r-form r of A(R) ) denotes the fundamental corepresentasatisfies equation (37), when u = tion of A(R). Thus, all the above facts are valid in this case. Let U(R) denote the bialgebra U defined above for A = A(R). That is, U(R) is the subalgebra of A(R)° generated by the functionals l +3-i and 1 7 , j , j = 1, 2, - • • , N. Now let us suppose that Gq is one of the quantum groups GL q (N), SL q (N), 0 q (N) or Spq (N) and that A is the coquasitriangular Hopf algebra 0(G q ) with universal r-form r, (see Theorem 9) and fundamental corepresentation u = (uii ). Let U(G q ) denote the bialgebra U in the case A = 0(Gq ). By Proposition 11 (ii), U(G q ) is a Hopf algebra. All the above formulas remain valid with R = zR(q), where z and R(q) are as in Theorem 9. In addition, the L-functionals / ±ii associated with the fundamental corepresentation u of 0(G q ) have the following properties.

Proposition 17. (î) l = = 0, i > j, and l+kk l -: = E, k = 1, 2,• • - N. 00 1 +11 1 +22 .. .i+NN = E for Gq = SLq (N). ( iii) L+ ct (L+)t (c-i)t = = el for Gq = 0q (N), Sp q (N). Proof. (i ): Suppose that i > j. The matrices R given by (9.13) and (9.30) are lower triangular. Thus, by (41), 1(unm ) = zR -= 0 for all n, m = 1, 2, • - - , N. Since 1 ±ii (ab) = l(a)l(b) and l±ii (1) = 6ii by (40), we conclude that 1 +1 = 0 on the whole of A for Gq = SLq (N),0 q (N), Sp q (N). In the case of GL q (N) this is also true, since LI- (DV) is the inverse of the upper triangular matrix L±(Dq ). The proof of l7 = 0 is similar. Since el = 1T= as just shown, we have A(irii ) = irii 0/rii again by (40). Hence is a character of the algebra A. By (41), the characters 1ri i and E are equal on the generators and hence on A. (ii) follows from the fact that 1+11 • • • / 4-1 and E are both characters which coincide on the generators u m n by (41) and the assumption z N =q 1 . (One may also verify that E = 1 1 (;) 1111 • • • 1 ±:). (Hi) is obtained from (3) and the relations (9.34). 1=1

Let A be A(R) or 0(Gq ). Then the L-functionals l for the fundamental corepreseqation u of A are uniquely determined by the equations (40) and (41) or equivalently by the requirement that the matrices LI = (1 ±1) are algebra homomorphisms of A to MN(C) satisfying (41). For most applications this characterization of the functionals i is sufficient to work with. The theory of L-functionaLs for 0(Gq) developed above is closely related to that for Uh(g) and Uq (g) in Sect. 8.5. Both L-functionals have similar

10.2 Crossed Product Constructions of Hopf Algebras

349

properties (see, Propositions 17 and 8.28 and formulas (42), (40), (41) and (8.83), (8.84), (8.89), (9.40), respectively) and even some proofs are analogous (compare, for instance, the proofs of Propositions 15 and 17 with those of 1,i(G q ) are Propositions 8.27 and 8.28). However, the L-functionals l defined as linear functionals on 0(G q ), while the L-functionals iti and iti in 8.5 belong to the abstract algebras Uh(g), Ut( g ) and UqL (g), respectively. By Propositions 15 and 17 and formulas (40), there is a surjective = Hopf algebra homomorphism 9 : UqL (2) ti(G q ) such that VW; and V(e7) = l , i < j, where g --= glN , slN, soN , sp N and G q = 3 GL q (N),SL q (N),0q (N),Sp q (N), respectively. Likewise, there is a Hopf algebra homomorphism 0 of the algebra Urt (g) generated by l , Ç onto U(G q ) which maps /ti to rEji and /7,i, to l , i < j. In general, both homomorphisms V and 0 are not injective. (For instance, if g = s12 and q is a root of unity, one can verify that (/ ±i l ) = E in U(SL q (2)) for some n c N.) Obviously, if we identify Ugext (g) and U(g) by the isomorphism from Theorem 8.33, V and 0 coincide.

10.2 Crossed Product Constructions of Hopf Algebras Crossed products or semidirect products are useful tools for the study of covariance problems and for the construction of new objects from old ones. Most of such constructions occurring in the literature can be thought of as generalizations or combinations of two fundamental concepts developed in this section: crossed product algebras or coalgebras and twisted cocycle algebras or coalgebras. In the course of this the quantum double will be treated as a guiding example. If not specified otherwise, A and B are bialgebras in this section. 10.2.1 Crossed Product Algebras Recall from Subsect. 1.3.3 that an algebra X is a left A-module algebra if X is a left A-module (that is, there is a bilinear map a : Ax X 3 (a, z) at>s c X satisfying al> (bc. x) (ab)i> x and 1 i> x = x for a, b c A, s c X) such that

a (xy) =

(a(1) x)(a( 2)› y), a >1 = E(a)1.

(49)

Proposition 18. Let X be a left A-module algebra. Then the vector space X OA is a (unital associative) algebra, called the left crossed product algebra and denoted by X >l a A or simply X >1 A, with multiplication defined by

(ac a)(y b) =

E s(a( i ) t> y) 0 a( 2)b,

c, y c X, a, b E A.

(50)

Proof. It suffices to show that this product is associative. Using the formulas (49) and (50) and the left A-module property of X we obtain

350

10. Coquasitriangularity and Crossed Product Constructions

(x ®a)((y ®b)(z c)) =

Es(a() t (Y(b (i) z))) a( 2)b( 2)c x(a(i) r> Y)((a(2)b(1)) z)

a(3)b(2)c

> y) a( 2)b) (z e) ((x

a)(y

b))(z

111

c).

Clearly,themapsX s-4s01 EXmAandADa-4 10aEXA A are injective algebra homomorphisms. Therefore, by identifying x with x 0 1 and a with 1 0 a, we can consider X and A as subalgebras of the algebra X >i A. Then definition (50) of the product yields the commutation relations ax =

(a(1) x)a( 2) ,

a

E

A, s E X.

(51)

The algebra X )41 A may be thought of as the universal algebra generated by the algebras X and A with respect to the commutation relations (51). If cp is a representation of the algelSra X >1 A on a vector space V, it follows at once from (51) that the restrictions (p x :=-- (prey and (to A := [ A are representations of X and A on V such that the compatibility condition (pA(a)c,ox(x) =

E ,(a (1) x)c,oA(a( 2)),

a E A, x E X,

holds. Conversely, if cpx and cpA are representations of X and A on the same vector space V satisfying the latter condition, then the equation (p(ax) = cpA(a)9o x (x) defines unambiguously a representation of X >1 A on V. For instance, by (49), the representations cp x (x)y = sy and A(a)y = a > y of X and A on V = X have this property. In this case the corresponding representation cp of X xi A on X acts as (p(xa)y = x(a> y), a E A, s, y E X. It is called the Heisenberg representation of the crossed product algebra X xi A. Before we discuss a number of examples, we briefly mention the corresponding right handed version A x X. Let X be a right A-module algebra, that is, X is an algebra equipped with a bilinear map a : XxA (x, a) —3 xaa E X such that (x a a) ab=sa (ab), s i A is the "ordinary" tensor product algebra X 0 A with product (x

a) (y 0 b) = xy 0 ab.

A

10.2 Crossed Product Constructions of Hopf Algebras

351

Example 5. Each Hopf algebra A is a left A-module algebra with respect to the left adjoint action a b adL(a)b = E ambS(a(2)). Then the corresponding left crossed product algebra A xi ad L A has the product (a

b)(c 0 d) = E ab( i)eS (b( 2)) b( 3)d,

a, b, e, d c A.

If A is quasitriangular, then the algebra A x ad i, A is even a Hopf algebra for d some appropriate coproduct, see Example 25 in Subsect. 10.3.6 below.

Example 6. Let A be the universal enveloping algebra U(g) of a Lie algebra g, see Example 1.6. Suppose that 7r is a homomorphism of the Lie algebra g to the Lie algebra of smooth vector fields on a, C'-manifold M. (An important special case is when M is a Lie group G with Lie algebra g and the elements of g act as left invariant vector fields on G.) Then 7r extends uniquely to an algebra homomorphism of U (g). For simplicity let us assume that this extension is injective and identify A = U(g) with its image. Then the algebra X = Ccic (M) (or any other subalgebra which is invariant under the action of g) is a left A-module algebra. Indeed, it suffices to check (49) for elements a of g. Since Z1(a) = a 1 +1 Oa, the first equation of (49) reduces to the Leibniz rule which holds because a c g acts as a vector field by assumption. In this case the elements of the crossed product algebra X A act as differential operators on M with coefficients in X. Let us specialize to the case where g = 111n, M = R X = C[Xl, ' ' • xn] and 7r(e) = 0/a, i = 1, 2, - • • , n, for some basis fed of g. Then X >4 A is just the Weyl algebra and its action on Rn is the realization as the algebra ,

of differential operators with polynomial coefficients.

Example 7. Let G and H be groups such that H acts on G by group automorphisms, that is, h c:;. (gg') = (h g)(h g') and 1 > g = g. The semidirect product G >i H of G and H is the set Gx H with group operation (g, h)(g' , h') := (9 (to. g'), hh'). Extending the actions of H and G by linearity to the group algebras X = CG and A = CH, X becomes a left A-module algebra and the crossed product algebra X xi A is just the group algebra C(G >1 H) of the semidirect product G >4 H. Important crossed product algebras are the Heisenberg doubles. Let us suppose that (•, •) is a dual pairing of two bialgebras B and A. Define

b>a := E(b,a(2) )a(i) ,

a E A, b c B.

(52)

Using the properties of the pairing (1.31) and (1.32) it is easy to check that the map (b, a) b > a defines an action of B on A such that A is a left Bmodule algebra. The corresponding crossed product algebra A >1 B is called the Heisenberg double of the pair A, B and denoted by H (A, 8). When the bialgebra B is the dual A° of A, then H(A) := B) is said to be the Heisenberg double of A. Inserting the definition (52) of the action into (50), we get the following expression for the product of the algebra .1/(A, B):

352

10. Coquasitriangularity and Crossed Product Constructions

(a 0 b)(a'

(b(i) , a'(2) )ad 1)

b')

b( 2)b',

a, a' E A, b, b' E B.

We consider this general construction in two examples. Example 8. Let G be a Lie group with Lie algebra g, B = U(g) and let A be a bialgebra of C°°-functions on G. It is well-known that any b E B acts as a left invariant differential operator b on G. The evaluation of the function 6(a), a E A, at the unit element of G defines a dual pairing of the bialgebras B and A (see Example 1.6). The corresponding Heisenberg double 11(A, 13) is an abstract algebra of differential operators Ei ab, ai E A, b, E B, and the Heisenberg representation of H(A, 13) ---- A >a B describes the action Ei ab of these operators on functions of A. That is, the elements of B and A act as differential operators and multiplication operators on G, respectively. This picture is the reason for the name "Heisenberg double" . It also motivates the interpretation of (52) for a general Heisenberg double as the action of the A "generalized left derivation" b E B on the "function" a E A. Example 9. Let R be an invertible solution of the QYBE and let A(R) and it(R) be the corresponding FRT bialgebras defined in Sects. 9.1 and 9.4. By Proposition 9.20, these bialgebras are dually paired with pairing described by (9.44). Suppose that A is a quotient bialgebra of A(R) and 13 is a quotient bialgebra of .11(R) such that the pairing of it(R) and A(R) passes to the quotients. Then, by (9.44), the actions of the generators tti of B on the generators unn, of A are written in matrix form as ziF u2 = u2 R21 and ° u2 = 112Ri21 . Hence the commutation relations (51) between both sets of generators in the algebra II (A, B) are = u2R21eil- and

,C i- u2 = u2 R1-21 ,C IT.

(53)

That is, the Heisenberg double I/(A, B) has 3N2 generators ej, u j j, n, m = 1, 2, • • • , N. The defining relations are those of A and B and the cross relations (53). A ,

,

10.2.2 Crossed Coproduct Coalgekoras

This subsection is concerned with the dual notion to crossed product algebras. A coalgebra X is called a right A-cornodule coalgebra if X is a right Acomodule such that the comultiplication A x and the counit Ex are comodule maps. The latter means that

E(x (0) )(1)

(x(0))(2)

s( i ) _ E(x(1)) (0)

(x(2))( 0)

( x( 1) )(1) ( x(2) ) (1)

= ex (x)i.

Here and in the following we use the Sweedler notation O(x) = E so) ® s(1), Ax(x) = s(i) x(2) and 46.4(a) = E a(1) a(2), where : X X A is the coaction of A on X.

10.2 Crossed Product Constructions of Hopf Algebras

353

Proposition 19. Let X be a right A-comodule coalgebra. Then the vector space .4 0 X becomes a coalgebra, called the right crossed coproduct coalgebra and denoted by Af3 D< X or A X X, with comultiplication and counit given by ZA(a 0 x) = E a(1) 0 (x(1)) (°) 0 a(2)(x(0) (1) 0 x(2), E(a 0 x) = EA(a)Ex(x),

(54)

a E A, x E X.

Proof. The proof of the coassociativity of ZA is an advanced exercise in the D use of the Sweedler notation. We omit the details, see [Mon], 10.6.3. Let us turn to the corresponding left handed notion. A left A-comodule coalgebra is a coalgebra X which is a left A-comodule such that Zl x and Ex are comodule maps, that is,

® (x(0))(1) ® (x( 0) )(2) _ E(x(i)) " )(x(2))( _1 ) 0 (x(1))( 0) 0 (x(2))( 0) , = Ex(x)i l where we write fi(x) = E x (-1) 0x (°) for the coaction ,(3 : X —* A® X. For such a left A-comodule coalgebra X, the vector space X 0 A becomes a coalgebra with comultiplication and counit defined by Z1(x 0 a) ---=

Ex°, 0 (x( 2)) (-1) a( 1) 0 (x (2)) (°) 0

E(x g a) =-- Ex (x)E-A(a), x E X, a E A.

This coalgebra is called the left crossed coproduct coalgebra. It is denoted by X >03 A or X >1 A. Example 10. Each coalgebra X is a right A-comodule coalgebra with respect to the trivial coaction f3(x) = x 0 1. Then A)3 Ix X is just the tensor product coalgebra A ® X with coproduct d(a 0 x) --= E a( i) ®X (l) øa(2) ø x(2 ). A Example 11. The right adjoint coaction AdR(a) = E a( 2) 0 S(a( 1 ))a( 3) turns each Hopf algebra A into a right A-comodule coalgebra. Then the coproduct of AAd R ix A is given by 46(a 0 x) = E a(i) o x (2) 0 a(2)S(x( 1))x(3) 0 For a coquasitriangular Hopf algebra A, we shall show in Subsect. 10.3.6 (see Example 24) that the coalgebra A AdRix A equipped with a suitable product becomes a Hopf algebra. A

10. Coquasitriangularity and Crossed Product Constructions

354

10.2.3 Twisting of Algebra Structures by 2-Cocycles and Quantum Doubles In this subsection we construct new bialgebras from old ones by twisting the product by 2-cocycles and keeping the coproduct unchanged. Definition 3. A bilinear mapping y : Ax A C is said to be a left 2 cocycle on A (with values in C) if it satisfies the condition -

E 7(a( 1 ) b( 1))7(a( 2)b(2),e) = E ry(b(i) , c( 1 ))7(a,b(2)c( 2))

(55)

for a,b,c e A. It is called a right 2 cocycle on A if it satisfies the equation -

E 7(amb( i),c)7(a( 2), b( 2)) =

y (a, b( i)c( i))1 /0( 2) , c( 2))

(56)

for a,b,c E A. The mapping y is said to be unital if y(a,l) = y(1, a) --= a e A.

: A x C is a bilinear mapping which is convolution invertible, then (55) for 7 is equivalent to (56) for its inverse -7. That is, 7 is a left 2-cocycle on A if and only if -7 is a right 2-cocycle on A. If 7

Example 12. Any universal r-form r of a bialgebra A is a imitai left 2-cocycle on A and its inverse 1 is a unital right 2-cocycle on A. Let us verify (55) for r. Indeed, if we apply (3) to the term r(a(2)b(2), c) and (4) to r(a,b(2)c(2))) then (55) reduces to the QYBE which holds by Proposition 2(i). Thus r is a left 2-cocycle. By the preceding remark, 1 is a right 2-cocycle. A Proposition 20. (i) Let y be a unital left 2-cocycle on a bialgebra A which

is convolution invertible. Let y be its inverse. Define a new product on A by

a...y b := E y(a(1) ,b(1) )a(2) b(2)7(a(3) ,b(3) ),

a, b E A.

(57)

Then the vector space A equipped with the product . 1, and with the coalgebra structure of A becomes a bialgebra. It is denoted A(7) and called the twist of A by the cocycle 7. (ii) If A is coquasitriangular with universal r-form r, then ri.), := 721r7 and j; are universal r-forms of the bialgebra AN. In particular, A(y) is also coquasitriangular. (iii) If A is a Hopf algebra, then so is .A(7) with antipode given by SA(.. y )(a) = S(a( 2))) and f (a(OSA(a(2))7(a(3)), where f(a) := J(a) := E-7(S(a( 1)),a( 2) ), a E A. Moreover, rf- = Jf = e in A'.

Proof. All assertions follow by direct verifications of the corresponding axioms. As an example, we prove the associativity of the product Inserting the definition (57) of the product ••y we obtain

10.2 Crossed Product Constructions of Hopf Algebras

E

355

b( 1 ))-0( 2)b( 2), c( o )a( 3)b(3)c(2) x Py(a(4)b(4), c(3))e7(a(5), b(5) ) 7

Eey (b(1) , C(1))7(a(1) b(2)c(2) )a(2)b(3)c(3) x 77(a(3) 7 b(4)c(4) )70(5) 1 c(5)).

Since -y is a left 2-cocycle,'7 is a right 2-cocycle. Using (55) for -y and (56) for it follows that the right hand sides of the preceding relations coincide. Hence the product • is associative. For the last assertion, see also the proof of Proposition 3. CI

Example 12 (Continued). If the left 2-cocycle ey in Proposition 20 is the universal r-form of a coquasitriangular bialgebra, then, comparing (57) and (2), A we see that A(7) is just the opposite bialgebra A°I3 Let us retain the assumptions and the notation of Proposition 20. Then the twisting procedure can be extended to A-comodule algebras and to Acomodule coalgebras as well. Let X be a left A-comodule algebra. Define a new product • by x-7 .y =

-Y(X ( -1 ) 7 Y(-1))X(0)Y(0),

XI

Y E

X.

The product • makes the vector space X into an algebra / X which is a left comodule algebra for the twisted bialgebra A(-y). (To prove the associativity of the product we note that the scalar factors appearing in the expressions (x l .y).y .z and x-y -(y l -z) are just the two sides of left cocycle condition (55). Since 7 is unital, the unit of the algebra X is also a unit for the product „y •.) Similarly, a right A-comodule algebra X is a right A(7)-comodu1e algebra with respect to the new product defined by X2J = Es (o) yor-y(s( 1)1y( 1) ),

z, y E X.

If X is a right A-comodule coalgebra, then

,6 -y(x) = E(x( )(0) ® (x(2 (0)7( ( x (1) )

))

)

(I)

(x(2))(1)),

X E

X,

gives a new coproduct 461 on the vector space X such that X becomes a right A(-y)-comodule coalgebra. Next let us consider cohomologous cocycles. Let u be an invertible element of the dual bialgebra A° with inverse 77,. If 7 is a left 2-cocycle on A, then one verifies that -y' :=ri,)-y(u), that is, (a,b) = Eri(a (o )ri(b(1) )-y(a (2) , b(2))u(a( 3)b(3)),

a, b E

A,

is also a left 2-cocycle on A. Then we say that 7 and 7' are cohomologous. A left 2-cocycle -y is called a coboundary if 7 is cohomologous to the trivial

356

10. Coquasitriangularity and Crossed Product Constructions

cocycle 70(a, b) = c(a)c(b), that is, if 7 is of the form (77/ 0 Ti)70,6(u) for some convolution invertible element u E .

Proposition 21. If 'y and 1, are two cohomologous invertible unital left 2cocycles on A with 0 17)74(u), then the map 0 -= u * id * i (that is, 0(a) = u(a(1))a(2)%a(3))) is an isomorphism of the twisted bialgebras A(7) and A(Y). In particular, if if is a coboundary, then the twisted bialgebra A(7) is isomorphic to A.

E

Proof. Using the relations

(u0u)ey' -=

7d(u) and 7' ( 7, 0 'FL) = A(Tiry, we get

0(a). 0(b) = (Eu(a(1 ))a(2)%a(3)))

(Eu(b(o)b( 2)u(b(3)))

E u(a ( ou(b( 1 ) ),,f(a ( 2) , b(2))a(3)b(3)7i (a(4), xri(a( 5) )7EL(b( 5) ) =

h )11(

h

1

h

h 57(

,, (1),. ‘ a(2)-(2),a(3)-(3).,a(4)-(4),

E*7(a ( i ) , b(1)) 0 (a(2)b(2))Va(3) b (3)

17-v( 7 -(5), h ,a(5)

0 ((a'7b))•

It is easy to check that 0 preserves the coalgebra structure as well. We conclude this subsection by showing how the quantum double (see Subsect. 8.2.1) can be obtained by the twisting procedure. Proposition 22. Let A and B be bialgebras which admit a skew-pairing : AX B C. Suppose that a has a convolution inverse a. We define bilinear mappings : B A x 13 0 A —+ C and 5, :A0BxA0B —+ C by

(b 0 a, b'

a') = (b)Tr (a, b')E(a' ) and '5'(a

b, a' 0 b') = E (a)a (a' , b)e

.

Then 7 and '5/ are invertible unital left 2-cocycles on the tensor product bialgebras BOA and A0B, respectively. The corresponding twisted bialgebras are the quantum doubles D(A,B; a) and 1 3(13,A; 721), respectively. That is,

B; a) = (B A)(y)

and

1)(13, A;

=

(A 0 1 ) ( 1).

Proof. We carry out the proof for ey. The assertion concerning follows then by interchanging the role of A and B and of a and 7721. The left 2-cocycle condition (55) for 7 is equivalent to the equality E (a( i), bi )Tr(a( 2 )d, b)

b(1))5(al Vb(2))-

Therefore, since 5-21 is an invertible skew-pairing of B and A, -y is an invertible unital left 2-cocycle on BOA. Comparing the formulas (8.21) and (57) we see that the' products of the algebras D(A, B; a) and (B A)(7) on the vector space B 0 A coincide. By definition, the coalgebra structures are the same. Corollary 23. If A and B are coquasitriangular bialgebras with universal r-forms rA and r 8 , then the quantum double 13(A, B; a) is coquasitriangular with universal r-form i = (741(r13)13(rA)24(723, that is,

10.2

Crossed Product Constructions of Hopf Algebras

357

f(boa,bioal).E er- (a (1), b(o)r8(b(2)7/1(1))rA(a(1), a/(2) )17 (a(2) N2)). Proof Obviously, r = (1.8) 13(rA)24 is a universal r-form for the bialgebra B A. Since D(A, 8; cr) = (B A)(Py), i = ty = 7201', = d41(r8)13(rA)244 723 I=1 is a universal r-form of D (A , B; cr) by Proposition 20(4 Example 13. Let A be either the FRT bialgebra A(R) for some invertible solution of the QYBE or the coordinate Hopf algebra 0(G q ) for Gq = GL q (N), SL q (N),0q (N), Spq (N). By Theorems 7 and 9, A is coquasitriangular with universal r-form r such that r(ui,u2) = R. Therefore, by Corollary 23, the quantum double D(A, A; r) with respect to the skew-pairing r of A and A admits a universal r-form I such that T(111 0 112 1 113 0 114) = R4-11 R13R24R23.

(58)

Since 17.2 1 is a universal r-form of A as well, we also get three other universal r-forms of D (A , A; r) in this manner. They are obtained if the term R13R24 in (58) is replaced by R3-11- R 24 R13R4-21 and R3-11 R4-21 , respectively. Remark 4. There is a more general concept of crossed product algebras that contains the constructions from Propositions 17 and 19 as special cases. Suppressing technical details, it is obtained if the action of the Hopf algebra A on X. the A-module algebra X is twisted by an X-valued cocycle a : A A

Such crossed products have been introduced and studied in [DTI.] and [B CM], see also [Mon], Chap. 7. 10.2.4 Twisting of Coalgebra Structures by 2-Cocycles and Quantum Codoubles

In this subsection we dualize the main considerations of the preceding subsection. That is, we construct new bialgebras by twisting the comultiplication and preserving the multiplication. Definition 4. An element yEAOA is called a 2-cocycle for A if "Y12 • ( 2i 0 id)(PY) = PY23 •

The element y is called counital if (E it has an inverse, denoted by Py -1 , in

(id 0 2i) ("Y)•

(59)

(id 0 E)Fy = 1 and invertible if the algebra A&A. id) -y =--

Example 14. The universal R-matrix R, of a quasitriangular bialgebra A is an invertible counital 2-cocycle for A. (Formula (8.7) means that R., is counital and (8.2) and (8.6) imply (59).)

The next proposition is the counterpart to Proposition 20. Proposition 24. (i ) Let y be an invertible counital 2-cocycle for a bialgebra A. The algebra A equipped with the comultiplication ZI- f (a) := Fy.ZIA(a)-7-1,

10. Coquasitriangularity and Crossed Product Constructions

358

a E A, and the counit EA becomes a bialgebra, denoted by A(-y) and called the twist of A by the cocycle (ii) If A is quasitriangular with universal R-matrix R, then Ri : 721R-7 -1 is a universal R-matrix of A(7) and so A(7) is also quasitriangular.

(Hi) If A is a Hopf algebra, then the bialgebra A(7) is also a Hopf algebra with antipode given by Sy (a)=v5A(a)v' , where v := mA(id SA )( )) E A and v -1 := mA(SA id)(y -1 ) E A is the inverse of v in the algebra A. Proof. We only sketch the proof of the coassociativity of A-y . From the defi-

nition of

2i-r

we obtain

id),y (a)

(id 0

)

(a) =

VY12(ZA 0 id)(y)] •

id)f(a)] -

id)(y -1 )yi-21 ],

[y23(id A)(-y)] • [(id 0 21)2i(a)1 • [(id 0 2i)(y -1 ) -yÇ31 ].

The corresponding expressions in the squared brackets of both equations coincide, the first and the third ones by the cocycle condition (59) and the middle ones by the coassociativity of = AA. Remark 5. If yE A0A does not fulfill condition (59), then the comultiplication Ay = y • ZAA • y-1 is no longer coassociative in general. However, P := y12 • ( 2i id)(7) • (id 0 20(7 -1 ) 72-31

1

is an invertible element of A 0 A 0 A such that (id 0 2i-y )A,r (a) =--- 0 -1 • ((Ay id)tly (a)) • 0. That is, Li.1 is coassociative up to conjugation by 0. Such a weakening of the coassociativity axiom leads to the notion of a quasi-Hopf algebra, see [Dr51. Note that (59) implies that 0 = 1 and hence that Zly is coassociative, but the latter property is weaker than (59) in general. Our next aim is to define a dual notion to the quantum double. In order to do so, we first dualize the notion of a skew-pairing (see Definition 8.3). Definition 5. A skew-copairing of tWo bialgebras A and B is an element e A0B such that ( 6

A 0 id)cr

17

(id 0 ez3 )o- =1,

(60)

(61) (AA 0 id)(u) = 0130237 (id 0 218)(0) = 0'130'12. Note that if a e A 0 B is invertible, then the two equations (61) already imply (60 ) (see the proof of formula (8.7) in Subsect. 8.1.1).

Example 15. The universal R-matrix R. of a quasitriangular bialgebra A is a skew-copairing of A and A itself (see (8.7) and (8.2)).

A

Proposition 25. Let a E A0 B be an invertible skew-copairing of two bialE B A0 B® A is an invertible gebras A and 13. Then the element -y :=

10.2 Crossed Product Constructions of Hopf Algebras

359

counital 2-cocycle for the tensor product bialgebra B 0 A. The corresponding twisted bialgebra (13 0 A)(y) has the algebra structure of the tensor product 13 0 A, the counit e(b 0 a) = e(b)EA(a) and the comultiplication .60 0 a) =

E b(i) O cr -1 (a( i) 0 bp) )o Øa( 2).

(62)

If A and 13 are Hopf algebras, then so is (13 0 A)(y) with antipode SO 0 a) = cr21 (S8(b)

SA(a))0-2-11 .

(63)

Proof. From (61) one easily derives that 11 is indeed a counital 2-cocycle for B 0 A. Thus, Proposition 24 applies. The formula tl,), = 'y • .6 130.4 11- 1 for the twisted bialgebra (13 0 A)(1) now reads as (62). To prove the formula (63), it suffices to show that the element y in Proposition 24 is C121 • Let = . Then we have

Ei

i (1 a')(913(bi) 0 1)

-= m/3 0,A(id 0 id 0 SsigA)(-y) =

= (8 B id)(cr2) = an, where the last equality follows similarly as the second formula in (8.8).

ID

Definition 6. The bialgebra (13 0 A)(y) in Proposition 25 is called the (generalized) quantum codouble of the bialgebras A and 13 with respect to the skew-copairing a- and denoted by D"(A,13;cr).

10.2.5 Double Crossed Product Bialgebras and Quantum Doubles In this subsection we are concerned with a generalization of the crossed product algebra construction by considering two bialgebras which act on each other. This situation is precisely described by the following definition. Definition 7. A pair t13, A} of two bialgebras is called matched if A is a right 13-module and B is a left A-module with actions a(a b) = a < b of 13 on A and )3(a b) = a> b of A on B satisfying the following conditions: 46A (a 46/3(a c

E a( 1 ) < b(i) O a( 2) b( 2), b) = E a( i) > b( i ) O a( 2) > b( 2), b)

(ad) (bb') =

EA(a 1 = EA(a),

(67)

(a < (d(1) r b(1)))(a(2) < b(2))1

E(a(1) t b(1))((a(2) < b(2)) D

E a( l ) 1)( 2) =

Ea(2) 4 b(2)

a( i) >

(68)

Equations (64) and (65) mean that the actions a and )3 are coalgebra homomorphisms. They can be expressed by saying that A is a right 13-module

360

10. Coquasitriangularity and Crossed Product Constructions

coalge bra and B is a left A-module coalge bra. Equations (66)—(68) are compatibility conditions of the two actions. Note that (68) is always fulfilled if A and B are both cocommutative. Proposition 26. (i ) Let {B, A} be a matched pair of bialgebras. Then the vector space B 0 A becomes a bialgebra with the tensor product coalge bra structure of B 0 A and with the product (b a)(b'

a') =

E b(a (i) > Wo.) )

(a (2) V(2) )ai .

(69)

This bialgebra is called the double crossed product bialgebra of the pair {B, A}. It is denoted by B a k' 0A or simply by B m A. (ii) If A and B are Hopf algebras, then B ki A is also a Hopf algebra with antipode S (b 0 a) = (1 0 SA(a))(SB(b) 0 1), a E A, b G B. Proof. The proof follows by direct verification using the conditions in Definition 7. As in the case of similar constructions (see Subsects. 8.2.1 or 10.2.1) 7 we can consider B and A as subbialgebras of the double crossed product bialgebra B 0.1 A by identifying b E B with b01EBN A and a E A with 10aEBN A. We then have the following cross commutation relations between elements a E A and b E B in the algebra B txi A: ab =

E a( i) >

a( 2) 1)( )

E a( 2) 1)( 2)

a( 1) b( 1) .

We now show that the quantum double D(A, B; a) can be considered as an example of double crossed product bialgebras. Let us assume that A and B are bialgebras equipped with an invertible skew-pairing a (see Subsect. 8.2.1). For a E A and b E B, we set aab=E a( 2) a(a( 1)7 b( 1 ))47(a( 3) ,b( 2) ),

(70)

E b(2) cr(a( 1)7 b(o)a(a (2) , b(3) ),

(71)

a> b =

where 6; denotes the inverse of a. With some work one verifies that (70) and (71) define actions of B on A and of A on B such that { 5, A} is a matched pair of bialgebras. Let B txi 0. A denote the corresponding double crossed product bialgebra. By (70) and (71), we have

E a( i ) t> V( 1 )

a(2) ci crA on the vector space 13 A is just the product (8.21) of the algebra D(A, B; a). Recall that both This shows that the product

(69)

10.2 Crossed Product Constructions of Hopf Algebras

361

B 1>ci A and D(A, B; a) have the tensor product coalgebra structures. Thus we have proved the following equality of bialgebras:

D(A, B; o-) = B

o. A.

(72)

Let us suppose in addition that A and B are Hopf algebras such that the antipode SB of B is invertible. Then the two actions < and c defined above can be nicely expressed as duals to the right adjoint coactions of the Hopf algebras A and /3°P. First a few preliminaries are needed. Using the formulas (8.20) and (8.19), we rewrite the equations (70) and (71) as (73)

aab=E a c> b =

b(2)a(a,b( 3 )85-1 (b( 1 ))).

(74)

Let (•, •) be a dual pairing of two Hopf algebras g and 71. By Proposition 1.15, any right 7-t-comodule V becomes a left G-module with action given by g> V :--= E V(0) (g, v( i)) , g E G, V E V. Likewise a right G-module is a left N-module with action h> y = E v(0) (v(1) , h). For the right adjoint coactions AdR (a) = E a (2) S(a (i ))a( 3) of g and 71 on itself, the associated left actions of G and 7-t on each other read as

h>g = g

h=

E g(2) (8 (g( 1) )g (3) , h),

(75)

h( 2 )(g S (h (o )h(3 )).

(76)

-) of A and /3°P. We now apply the preceding to the dual pairing (-, -) Then the left action (75) of B°P on A yields the right action (73) of B on A. In terms of the product and antipode of B the right adjoint coaction of the Hopf algebra /3°P can be written as AdR(b) = E b(2) ®b (3) s,3-1 (b ( ,) ), b E B°P. Hence the left action (76) of A on B°P gives just the left action (74). We close our study of the quantum double by treating an instructive example. It is based on the FRT bialgebra A(R) which has always been our guiding example.

Example 16 (The quantum double D(A(R), A(R); r)). Let R be an invertible linear transformation on C N satisfying the QYBE. Since the canonical universal r-form r of the FRT bialgebra A(R) (see Theorem 7) is a skewpairing of A(R) and A(R) itself, the quantum double D(A(R), A(R); r) is a well-defined bialgebra. Let u and y denote the fundamental matrices of two copies of A(R). By (11), we have r(ui,v2) = R12 and 1(U1, v2 ) = .R 1-21 . Inserting these expressions into the formulas (70) and (71) we see that in matrix notation the actions i and > of A(R) on A(R) are written as ul v2 = Ri-21v2R12.

362

10. Coquasitriangularity and Crossed Product Constructions

Similarly, the commutation relations (8.22) of the entries of u and v in the algebra D(A(R),A(R); r) are u1v2 = R-1 v2u1ft. Hence, by the discussion after Definition 8.4, the algebra 73(.A(R), A(R); r) is generated by the entries of the matrices u and v subject to the defining relations RU1 112=----- U2 111R ) RV1V2 = V2V1R 1 U1V2 = R-1 V2V1R.

On the other hand, the algebra it(R-1 ) has the defining relations (see (9.43))

RC-IF 4 = .4Z-2E R, RZre = £jeR , RCi",4 = Therefore, the assignments v v 0 1 bialgebra isomorphism

R.

Z+ and u 1 0 u -0 Z - define a

D(A(R), A(R); r) = A(R) 3r A(R) 11(R -1 ) = it(R)°".

(77)

We will give two other interpretations of the latter result. Let /4(4 =-- ±, -, be the algebra with matrix generators ef and defining relations = It is clear that the map £C --0 u is an algebra isomorphism of it, (R) to A(R -1 ), so it(R) is a bialgebra with comultiplication LC) =-..CC 0 ,C€ and counit E(2,€) = I. Under the isomorphisms it_ (R) A(R 1 ) and it+ (R) A(R 1 ) the universal r-form r of A(R 1 ) goes into a skewpairing i of the bialgebras it_(R) and 14(R) determined by the equation , = Ri-21 . Thus, (77) yields the bialgebra isomorphism

(R) , it+ (R) ; ) = U+ (R) m it_ (R) 11(R)

(78)

This representation of the bialgebra it(R) as the quantum double of it_ (R) and .LWR) explains the source of the third matrix relation ZIT ;FR = . It is just the cross relation of the quantum double. Likewise, we could have expressed it(R) also as the quantum double of it+ (R) and it_ (R) built over the vector space it_ (R) 0 it+ (R) rather than it + (R) LL (R). To give still another view of the isomorphism (77), we recall from Proposition 9.20 that there is a dual pairing (•, •) of the bialgebras A(R) and it+ (R) such that (ui, = R12. The corresponding skew-pairing cy of A(R) and U+ (R)°" is just the canonical universal r-form r of the bialgebra A(R) under the isomorphism U+ (R)°P = £t+ (R 1 ) A(R), so (77) means also that

D(A(R), it+ (R)°P o-) it.(R - 1 ) . 10.2.6 Double Crossed Coproduct Bialgebras and Quantum Codoubles In this very short subsection we only state dual versions to the main results of Subsect. 10.2.5. We consider two bialgebras which coact on each other. Definition 8. A pair {A, B} of bialgebras is said to be comatched if ,8 is a A and A is a left right A-comodule algebra with right coaction a : 8

10.2 Crossed Product Constructions of Hopf Algebras

363

B-comodule algebra with left coaction 0:A --BOA such that for a E A and b G B the following compatibility conditions hold: (Z1 0 id) o c(b) =

((id 0) 0 ex(b (1) ))(1

(id 0 Z1) o ,3(a) =

(Nam)

a(b(2)) 7

1)((a 0 id) o

a(b)0(a) = 3(a)a(b). Proposition 27, Suppose that {A, B1 is a comatched pair of bialgebras. Then

the tensor product algebra A013 becomes a bialgebra, called the double crossed coproduct bialgebra and denoted by Aa 3 B or A M B, with counit e = EA 0 es and comultiplication Z1(a 0 b) =

E a( 1 )

Proposition 28. Let a-

oe(b (0) )3(a ( 2)) 0 b ( 2) ,

a

C A, b C B.

c A® B be an invertible skew-copairing of the bialge-

bras A and B. Then the pair {B. A} is comatched with respect to the coactions a(a) = a-1 (a 0 1» and OW = a-1 (1 0 b)a and the corresponding double crossed coproduct bialgebra B 14 A is just the generalized quantum codouble vco (A , B ; 10.2.7 Realifications of Quantum Groups

Let A be the coordinate Hopf algebra of a quantum group G q . Roughly speaking, a realification of Gq should be a Hopf *-algebra B which contains A as a Hopf subalgebra and which is generated as an algebra by A and A*. Here A* denotes the set of adjoint elements a* for a E A with respect to the involution of the larger *-algebra B. The elements of A are then interpreted as "holomorphic polynomials" and the elements of A* as "anti-holomorphic polynomials" on Gq . The purpose of this subsection is to show that the quantum double D(A, A; r) = A m i. A provides a general procedure for the construction of a realification of a coquasitriangular Hopf *-algebra A equipped with a real universal r-form r. The involution of A will be used in order to inherit an involution on the algebra D(A, A; r). Definition 9. A Hopf *-algebra B is called a realification of a Hopf algebra A if A is a Hopf subalgebra of B and if the linear map tz : A0 A* B defined by 1.1,(a O b*) = ab* is bijective, where A* is the set {a* I a E A} in the *-algebra B.

Remark 6. Some authors call the passage from A to B a complexification and B a complex quantum group., A Suppose that A is a coquasitriangular Hopf algebra with universal r-form r. By Proposition 8.8 and Corollary 23, the quantum double 1) (A, A; r) is also a coquasitriangular Hopf algebra. Because this Hopf algebra is essentially needed in the following, we briefly repeat its structure (see Sects. 8.2,

364

10. Coquasitriangularity and Crossed Product Constructions

10.2.3, and 10.2.5 for details). As a coalgebra, D(A, A; r) is the tensor product coalgebra A 0 A. As an algebra, D(A, A; r) is the vector space A 0 A with product (a

b)(c 0 d) = E a c( 2)

b(2)d Ob( i ), c( i ))10( 3), c( 3)).

(79)

Further, D (A , A; r) is the free algebra generated by two copies, denoted by A and À, of the algebra A with cross commutations relations =

Eiim aoo r(a (2) , 6- ( 2)),

Here a E A is identified with a®1 and 6- E

a E A,

A with 1®

E

A.

(80)

.

Proposition 29. Suppose that A is a Hopf *-algebra and r is a real universal r-form of A. Then the quantum double 13 := D (A , A; r) is a Hopf *-algebra with involution defined by (a 0 b)*

b* 0 a* ,

a, b E A.

(81)

B is a realification of the Hopf algebra A. There is a homornorphisrn 1 of the Hopf *-algebra B to the Hopf *-algebra A such that 0(a b) = ab, a, b e A. Moreover, r' := N1r13r24r23 is a real universal r-form and r" := f'41k31r24r23 is an inverse real universal r-form of 13. Proof. Since r is a real universal r-form of A, r is a real universal r-form of A°P . Using these facts and equations (79) and (81), we obtain ((a g b)(c 0 d))* = (b(2)d)*

= d* b* (2)

, c(o)r(b( 3 ), c(3)) (ac(2))* c* ( 2)a* r(c* ( 1 ) b* ( 1 ))r(c* (3) 7 b* (3 )) at) = (c d)* (a b)* =(d*c)b

The other requirements for 13 = A; r) being a Hopf *-algebra are clear. Obviously, A becomes a Hopf subalgebra of B if we identify a E A with a 0 1 E B. Then, because of (79) and (81), it follows that E i a 0 b = Ei (ai 1)(bi 0 1)* = ab. Hence the mapping p, : A 0 At B is bijective and B is a realification of A. By Corollary 23, r' and r" are universal r-forms of B. The reality of r', the inverse reality of r" and the fact that 0 is a homomorphism of Hopf *-algebras are easily verified. CI

Ei

Let us emphasize that A is not a *-subalgebra of B because the adjoint a* in B of an element aag1E AC B is the element 1 g a*. We now apply Proposition 29 in order to construct realifications of the coordinate Hopf algebras 0(G q ), G q GL q (N), SL q (N), 0q (N), Spq (N), for real q. In doing so we use the Hopf *-algebras A = O(G) for the corresponding real quantum groups G eirte = Uq (N), SUq (N;u), 0q (N;u), Spq (N;u) and the real universal r-forms r z from Proposition 10 (i) If we consider G q as

10.3 Braided Hopf Algebras

365

a complex quantum group (which means that we "forget" the involution of 0(GRe q )), then Proposition 29 yields a realification of G q . By the discussion before Proposition 29 the structure of this realification B = D(A, A; rz ) can be explicitly described as follows (see also Example 16 in Subsect. 10.2.5). Let u = (ui.3 ) and y = (vi.) be the fundamental matrices of two copies of the Hopf algebra A = 0(G q ). Then, the algebra B = D (A , A; rz ) admits the generators j = 1, 2, • • • , N. The defining relations of B are the defining relations of 0(G q ) for each of the sets fu jil and {v } together with n k / = Rnklm or in matrix form Ru 1 v2 = v2 u 1 R. the cross relations R uvm Note that the latter are just the relations (80) for the generators of the two copies of A. The involution of B is determined by the equations (74)* = v. As noted above, we consider the uii as holomorphic generators and the vni as anti-holomorphic generators (or as complex conjugates to uii ) of the algebra B. The coalgebra structure of B is that of the tensor product AoA. It is given by the formulas di and e(u) = = 60 . = u ® 74 AO) = Clearly, one can also verify directly that the preceding description gives a realification B of A. However, an advantage of the above approach is the explanation of the cross commutation relations Ru 1 v2 = v2 14 R as the defining relations for the quantum double. Further, the general theory of the quantum double provides the universal r-forms r' and r" of B.

gut

10.3 Braided Hopf Algebras A braided Hopf algebra X is in general not a Hopf algebra according to Definition 1.4. The main new feature is the requirement that the comultiplication is an algebra homomorphism of X to X 0 X, where the latter carries a particular product, the so-called braided product. 10.3.1 Covariantized Products for Coquasitriangular Bialgebras

The right adjoint coaction AdR(a) = E a(2 ) S(a(o)a( 3) of a commutative Hopf algebra A is obviously an algebra homomorphism of A to A 0 A. In the general case this is no longer true, so the algebra A is not necessarily a comodule algebra for the Hopf algebra A with respect to the coaction AdR• For coquasitriangular Hopf algebras one can remedy this defect by introducing a new "covariantized" product on A such that AdR is a homomorphism of the corresponding algebra B(A) := (A,) to B(A) B(A). Further, there is another "natural" product • on B(A) I3(A) such that the comultiplicadon 44 is an algebra homoniorphism of B(A) to (B(A) B(A),$). In this manner B(A) becomes a braided Hopf algebra. In this subsection we consider a similar passage from A to B(A) for bialgebras. This requires some additional technical preliminaries. The Hopf algebra case will be treated in the next subsection.

10. Coquasitriangularity and Crossed Product Constructions

366

Definition 10. A universal r-form r of a coquasitriangular bialgebra A is called regular if there is a linear form s : A0 A C such that for a,b E A,

E r(a( 1 )0b(2))s(a( 2)01)( 1)) = E s(a( i ) b(2) Wa(2) 0b(0) =-- e(a)e(b).

(82)

Recall that A°P (resp. A"P, A°P'"P) is the bialgebra built on the vector space A with opposite product (resp. coproduct, product and coproduct) of A. Then equation (82) says that s is the convolution inverse of r when r and s are considered as linear forms on A"P 0 A. The universal r-form r of a coquasitriangular Hopf algebra A is always regular. Indeed, one easily checks that the linear form s(a b) r(a 0 8(b)) satisfies (82). In the bialgebra considerations below the form s is used, roughly speaking, as a substitute for the possibly missing antipode. Example 1 7 ( The FRT bialgebra A(R)). Let R be an invertible solution of the QYBE. Then the canonical universal r-form (see Theorem 7) of the FRT bialgebra A(R) is regular if and only if R is regular, that is, if Rt2 is invertible. (The sufficiency was stated in Proposition 8. The necessity follows from (82) applied to a = 4 b = u lic and combined with (14) The R-matrices for the quantum matrix groups SL q (N),0q (N) and Sp q (N) are all regular, because they lead to coquasitriangular Hopf algebras by Theorem 9. The flip R = T is a solution of the QYBE which is not regular. A ,

Throughout this subsection we assume that A is a coquasitriangular bialgebra and r is a regular universal r-form of A. Since r is a dual pairing of the bialgebras A and A°13 (see Subsea. 10.1.1), it follows easily that s is a dual pairing of A°P and A and of A and A"P. Hence s is a skew-pairing of the bialgebras A and A0P ' 013 7 so the quantum double D(A, A°P ' "P ; s) is well-defined by Proposition 8.8. By (72), it coincides with the double crossed product bialgebra A°P'"P N. A from 10.2.5. In order to avoid notational confusion, let us denote the identity map from A to A°P'"P by i and assume that the Sweedler notation always refers to the comultiplication of A. That is, i(ab) = i(b)i(a) and ,AAop,cop (0)) = ® i(b( 1 )) for a, b E A. Then, by Proposition 8.8, the bialgebra

o(2))

D(A)

A'P' cop ;

s) _

Aop,cop

s

A

(83)

is built on the vector space A°P ,c°P 0 A with product

(i(b)

a)(i(b')

a') =

E i(1/(2) b) g a(2)a( r(a( 1 ), V(3) )s(a( 3),b1(1) )

(84)

and coproduct ,

AD(A)(i(b) 0 a) -= E i(b(2)) g a(i) O i(b(1))

a(2).

Proposition 30. Let A be a bialgebra equipped with a regular universal rform r. The vector space A equipped with the new product

10.3 Braided Hopf Algebras

a•b :=

367

E a( 2) b(3)r(a( 1) , b(2) )s(a(3)7b(i) = E b(2) a(i) r(a(2) ) boos(a(3) , boo (85)

is an algebra, denoted B(A). The algebra B(A) is a right comodule algebra for the bialgebra D(A) with respect to the coaction : B(A) —) B(A)0D(A) given by (P(a) : => a(2) i(a(1)) a(3), a E A. The product of A can be recovered from that of B(A) by the formula -

ab =

a( 2) J0( 3) f(a(i) 7b(i))r(a(3), b ( 2 )),

a , b G A.

(86)

Proof. We begin with the associativity of the new product. Using first the

definition (85) of : , then formula (5) and finally the facts that r(-, -) and s(-, • ) are dual pairings of A and A°P resp. A°P and A, we compute (a=b) :e =

E a(3)b(4)c(3)r(a(2)b(3), C(2) )s(a(4)b( 5 ) , c (l) Ma (l ), b(2))s(a(5), b(i) E a( 3)b( 4)c(3)r(b( 2) a( 1 ) , C(2) )s(a(4)b(5) c (i) Ma(2)7 b(3) )s(a(5)7 b(i) E a( 3 )b( 4)c( 5 )r(b( 2 ), coOr(a(i), C(4))s(a(4)7C(2))s(b(5)7c(i)) X r(a( 2) , b(3) )s(a(5), b(i) )7

E a(2)b(4)c(5)*(1) b( 3)c( 4 ) )s(a(3)7b(2)c(a) ) 10(1)7c(2))s(b(5)1c(i) = E a( 2)b(4)c(5 )r(a( 1 ) 7b(3)e(4) )s(a(3) , c(2)b(i) )r(b(2)7C(3) )s(b(5)7 C(i))

a•(b•c) =

=

a(3)b(4)c(5)*(1),c(4)*(a(2), b( 3))s(a( 4 ), c(2))s(a(5), No) x r(b( 2 ), c(3))s(b(5), c(i)).

Proposition 2 (u i) implies that 1 :a = a:1 = a, a E A. Thus, we have shown that B(A)=(A, :) is indeed an associative unital algebra. It is clear that (i) is a coaction of D(A) on B(A). Using first the definitions (85) and (84) of the products and then the identity (82) we get

(p(a):(p(b) =E a( 2) lb(2)

(i(a(1))

a(3))(0(i))

a( 3)b(6 ) r(a( 2), b(5 ))s(a( 4) b(4) )

b(3)) i(b( 2) a( 1) )

a( 6)b(7)

r(a( 5 ), b(3))s(a(7)7b(i))

E a( 3)b(4)

i(b(2)a(1)) 0 a(4)b(5) r(a(2)7b(3))s(a(5)7b(i))7

where : = :0• . On the other hand, applying (85) and (5) we obtain

Ea(3) b(4) = E a( 3)b(4)

p(a•b) =

i(a(2)b(3)) 0 a( 4 )b(5 ) r(a(i), b(2))s(a(5) NO i(b(2)a(1)) 0 a(4)b(5) r(a(2)7b(3))s(a(5)7/9(i)).

Moreover, (p(1) = 1 0 1 0 1. This proves that (p is an algebra homomorphism of B(A) to B(A)®D(A).

368

10. Coquasitriangularity and Crossed Product Constructions

Inserting (85) into the right hand side of (86) and using (82) one obtains I=1 the equality (86). The second equality in (85) follows from (5). The product is called the covariantized product and the algebra B(A) the covariantized algebra of the coquasitriangular bialgebra A, because they turn the right D(A)-comodule A with coaction cp into a right D(A)-comodule algebra. Note that the algebra A itself does not have this property in general. Let us describe the covariantized algebra for the FRT bialgebra A(R). Example 18 (The FRT bialgebra A(R) - continued). Let R be a regular invertible solution of the QYBE. By Proposition 8, the canonical universal r-form r of A(R) is regular. Recall that r(u, u ilc) = kill' and s(ujj , = ) -1 . Hence formula (85) for the new prodby (11) and (15), where k 2 n u(Rnirs . In matrix n utfirrk kr, and so uiilulicki.rk = um uct yields u i=tegic --= u m notation the latter reads as U1R12112. = R12 11 1112.

(87)

Each of the three last formulas expresses the products of B(A(R)) and of A(R) by each other. From (87) we obtain u2:R21 u 1 R 12 - R21 u2 u 1 R 12 and

R21 u 1LR 12 u2 = R21 R 12 u 1 u2 . (88)

Therefore, since R 1 2u 1 u2 = u2u1R12 by (9.2), the matrix u satisfies the equation u2•12 1 u 1 R1 2 '= R211-11.:R12u21 (89) which is called the reflection equation. Recall that we have met this equation already as formula (44); see also (43). An equivalent form of (89) is u2:fli2u2.fii2 = iii2u2j112u2-

Since R is invertible, it follows from (88) that the reflection equation for the new product is equivalent to the equation R 12 u 1 u2 = u2 u 1 R 12 for the old one. Thus, the covariantized algebra B(A(R)) for the FRT bialgebra A(R) has generators uji , i , j = 1, N, with defining relations (89). This algebra is denoted 13(R) and called the braided matrix algebra associated with R. Z\ Example 19. Let R be the matrix (3.40). Using the notation a u , b c --= ui, d --= ui for the matrix entries of u, the defining equation (89) of the braided matrix algebra I3(R) is equivalent to the following six relations:

= (q- 2

-

as- c:a (1- q-2 )ds, b:c-c:b = (q-2 -1)Ca- d), a:4 = da b•d = q -2 0, c:d = q2 d:c.

A

The comultiplication ZAA is not necessarily an algebra homomorphism of B(A) to B(4)0B(A) if the latter carries the "ordinary" tensor product multiplication. However, there is another product • on the vector space

10.3 Braided Hopf Algebras

B (A) such that ZAA is a homomorphism of B (A) to (B (A)

B (A)

369

13 (A) , •)

This product is derived from the following more general lemma. Lemma 31. Let a:Ax.B C be a skew-pairing of two bialgebras A and B

and let X and y be right comodule algebras for A and B, respectively. Then the vector space X ® y is an algebra with product defined by

(x0y)•(x' Oy i ) := E xx/(0) 0y(0)y' o- (y( i), x4 ) ), x, x'

X, y,y 1 G

y.

(90)

If A = B and a is a universal r-form of A, then this algebra is a right A-comodule algebra. Proof. Using the assumption that X and Y are comodule algebras and the properties of the skew-pairing a we get

((x

y) • (x'

V)) • (x"

y") = E(xxi(o) Ø y(o)v) • (x"

y") cr(y(i), X1(1))

E xx/( ,) x ) y(0)y6y" cf(y ( i)yi, x") (y( 2), x /(1) ) E xxlm x (0) y(o)v(0) y" a(y(i), 41) ) cr(V 7 6)) cr(y( 2), x4 ) ). The same expression is obtained for (x0y) • ((x' y')• (x" y")). Hence, the product • is associative. Clearly, (x y) • (1 0 1) = (1 0 1) • (x 0 y) = x 0 y. We prove the last assertion of the lemma. By property (5) of the universal r-form o• , the expressions

ciox

cloY((x

Y) • (x i 0 YI ))

= E x (0) x/(0) (PAX 0 OPX 0

(19

y(0)y o)

x(i)x /(1) y(04 ) 0-(Y(2), X1(2)),

(PAX / 0 y l )

E x (0 ) x/(0)

y(0)yo,

xo,y(2)x/(2 ) y/( 1 ) a(y(i) 7 xl(1))

coincide, so tpx cpy is indeed an algebra homomorphism. Proposition 32. Let A and B(A) be as in Proposition O. The equation

a- 0. (b )0a, i(b1 )0d) :=

ET (b (* d(i) )r(b( i ), W(2) )r(a (i) , d(2) )s(a (2) , b4 ) ) (91)

defines a skew-pairing of the bialgebra D(A) with itself. With the product (a 0 b) • (CO d) = E a,c (2) b (2):dc,(0 (1) )

b (3) , i(c(i) )

C(3) )

(92)

the vector space B(A)®13(A) _becomes an algebra, denoted B(A)0B(A), such that 2IA is an algebra hornomorphism of B(A) to B(A)0B(A). Proof. We omit the lengthy verification that a is a skew-paring. By Proposition 30, B(A) is a right D(A)-comodule algebra with respect to the coaction w(a) = a(2) ®i(a( 1 )) a(3), a E A. The product (90) specializes to (92) in

E

370

10. Coquasitriangularity and Crossed Product Constructions

the case when A and B are the bialgebra D(A) and X and Y are the comodule algebra B(A). Therefore, by Lemma 31, (B(A)0B(A),•) is an algebra. It remains to prove that AA is an algebra homomorphism of B(A) to BA0B(A). Applying first the definitions (92) (combined with (91)) and (85) of the products of B(A)0B(A) resp. B(A) and finally (82), we get 46.4(a) • AA(b) =

E(a,(1) a( 2)) • (b(i) 0 bp))

E a( 2)b(5) 0 a( 7)b( lo ) *(l) 7 b(4))S(a(3)

b (3)) r ( a (6) 7 b(9) ) S ( a (8)

b(8))

xiqa,( 5),b(6 ))r(a( 9 ), b(7))*(4)7b(2))s(a(10)1b(1))

= E a(2)b(3)

I=1

a(3)b(4) r(a(i),b(2))s(a(4),b(1)) = AA(aJ)).

Note that the skew-pairing o- defined by (91) is not a universal r-form of the bialgebra

MA).

Example 20 (The FRT bialgebra A(R) - continued). Recall that the covariantized algebra B(A(R)) of A(R) is the algebra B(R) with product Then the product • on the vector space B(R) 0 B(R) is given by

(a 0 ti i ) • (ulic

b) = afulr'n, tirs:b (1?-1 ):in' RpiY7i R zs f

,

a, b E B(R).

In matrix notation this may be rewritten as

(a cs) R i-21 u 1 ) • (R12u2 0,b) = a=u2R 1-21 0 ui.LbRi2-

A

10.3.2 Braided Hopf Algebras Associated with Coquasitriangular Hopf Algebras We now assume that A is a coquasitriangular Hopf algebra and show that then the above results take a much simpler form. Some necessary technical facts for that are collected in the next proposition. The first three assertions give the motivation for the use of the forms s and a and of the bialgebra D(A) in the preceding subsection. Proposition 33. For any coquasitriangular .Hopf algebra A with universal r-form r, we have: (i) r is regular and s(a,b) = r(a, S(b)) for a,b E A. (ii) o- (i(b)0a,i(b')0d) --= r(S(b)a, SOW) for a, b, a' ,b' E A. (Hi) There is a Hopf algebra homomorphism 0 : D(A) A such that 0(0) 0 a) = S(b)a. (iv) Define 57(a) := E S(a( 2))r(S2 (a(3))S(a( i )),a( 4)) for a E A. Then the linear mapping S : A A intertwines the right adjoint coaction AdR of A (that is, S E Mor (AdR)) and satisfies the condition

E awl i, (a(2)) = E Na(i))La(2) = e(a)1,

aE

A.

(93)

10.3 Braided Hopf Algebras

371

Proof (0 and (ii) are verified in a straightforward manner using the properties of r. In the case of (ii) we apply (1) and the identity r(S(a), = Y(a, b). (iii): Using the formulas (3) and (5) and the assertion (i), we obtain

0((i(b)

a)(i(W)

d))

E 0(i0/(2) b) a(2)d) r(a(i),N 3) )s(a(3), b'(1) ) .= E S (b ) (S(11(2) )a( 2))a' r(a( i),N3) )*( 3), S( 1/(1))) = E S(b)a( 3)5(N 1) )ai r(a(i), W 3 ))r(a(2), S(1/( 2))) = E S(b)a( 2), (11(1) )al r(a( i), S(b'(2) )N3) ) = S(b)aS(V)ai = ONO 0 a) 0(0) a').

It is easy to check that (0 0 0) o ,6 D(A) = ,6A o 0. Thus, 0 is a Hopf algebra homomorphism. (iv): Using the functionals f and f from Proposition 3 we compute

E a( l) : ,i(a(2)) E a(2) S(a(5) ) r(a( i ), S(a( 6)))r(a( 3), S2 (a( 7)))r( 52 (a( 8)), a( 9)) x r(S(a(4)), a(io))

— E a( 2)S(a(5)) r(a( i ), S(0,(6) ))*(3), S 2 (a(7)) f (a(8)))r(S(a(4)), a(9)) E a( 2)S(a(5)) r(a( i), S(a( 6)))*( 3), S2 (a( 7))) xr(S(a(4)),S 2 (a(8)))7 (a(9))

E a( 2)S(a( 4)) r(a( i) , S(a(5) ))r(a (3) , S(a( 7))52 (a(6)))f (a( 8)) E a( 2)5(a( 3)) r(a(i), S(a(4)))1 (a(5) E r(a( 1 ), S(a(2)))7 (a(3))1 = E f(a(1) )7(a (2) )1 = E(a)1. Here the first equality follows from the definitions of and S and assertion (i ). The second and third equalities are obtained by inserting and applying formula (8), respectively. Similar (slightly simpler) reasoning yields the second equality of (93). In order to prove that S E Mor (AdR), we compute

7

E S(a(3)) 0 5 2 (a(4))5(a(2)) r(S 2 (a(5)) ,*(1)), a(6)) = E S(a(3)) 52 (a(4))51 (a(2)) f (a(5))r(S(a(1)), a(6)) = E S(a)) O a( 5)S(a( 2)) f (a(4))r(S(a(i)) a(6)) = E s(a (3) ) O S(a(1))a(6) f (a(4))r(S(a(2)) a(5))

AdR(S(a)) =

= E S(a(3) ) O S(a(1) )amr(S2 (a(4))S(a( 2) ), a(5) ) = E 8 (a(2))

S(a(1))a(3) = (ii 0 id)AdR(a).

372

10. Coquasitriangularity and Crossed Product Constructions

Here we used the formulas (8), (4) and (5).

I=1

The next two propositions restate some key results and formulas from the preceding subsection in the case of a coquasitriangular Hopf algebra A with universal r-form r. Proposition 34. (i) The vector space A equipped with the product

a•b =

E a( 2)b( 2)r(8(a( o )a(3), S(b( l ))) = E b( 2)amr(a( 2), S(b(o)b( 3))

is a right A-comodule algebra, denoted by B(A), with respect to the right adjoint coaction MR. (ii) Recall that r(a) is the linear functional on A defined by (21). Then the map a —> r(a) is a homomorphism of the algebra B(A) to the subalgebra f(A) of the dual Hopf algebra A°. Proof. (i ): From Propositions 32(i) and 2(v) and formula (3) we get r(S(a)al, S(b)) =

E r(S(a), S(b( 2)))r(a l ,S(b( 1 ))) = E r(a, b( 2))s(a 1 ,

and r(a, S(b)b') = E r(a (1 ), bi)s(a (2) , b), so the product (85) takes the above form. Since obviously AdR = (id ® 8) o, B (A) is a right A-comodule algebra with respect to the coaction AdR by Propositions 29 and 32(iii). (ii): Since s(a, b) = r(a, (0) by Proposition 33 (i), a comparison of the formulas (30) and (85) shows that i(a:b) = i(a)T(b) for a, b E A. Let us say that a coquasitriangular Hopf algebra A (or more precisely a pair (A, r) of a coquasitriangular Hopf algebra A and a universal r-form r on A) is factorizable if the bilinear form q given by (9) is nondegenerate. Since q(S (a) , (b)) = q(b, a) and the antipode of a coquasitriangular Hopf algebra is bijective, it is clear that A is factorizable if and only if 1(a) = 0 implies that a = 0 or equivalently if [(a) = 0 is only possible for a = O. Therefore, by Proposition 34(ii), for a factorizable Hopf algebra A the map a t(a) is an algebra isomorphism of the covariantized algebra B (A) to the subalgebra i(A) of A°. Note that cotriangular Hopf algebras are those coquasitriangular Hopf algebras for which the form q is trivial, that is, q = E 0 E. The factorizability of a coqua.sitriangular Hopf algebra is just the other extreme where the form q is far from being trivial. The following proposition is only a reformulation of Lemma 31. Proposition 35. Let X and y be right A-comodule algebras. Then the vec-

tor space X®Y becomes a right A-comodule algebra, denoted by respect to the tensor product coaction and the product

(x y) • (x' y') := (x 0 1)r y x (y 0 s')(1. 0 y'),

E

xoy,

X, y y'

E

with

y, (94)

where the product on the right hand side is the tensor multiplication and

10.3 Braided Hopf Algebras

ryx (y

s') := Ex/(0) y(o) r(Y(1), si(1))•

373

(95)

The product (94) on X 0 y is called the braided product and the mapping ryx : Y X X 0 Y is called the braiding of the A-comodule algebras y and X. Recall that ryx is just the mapping defined by formula (6). Some properties of the braiding maps can be found in Proposition 1. In particular, it was shown therein that each map rxx satisfies the braid relation. Applied to the case when X = y is the right A-comodule algebra B(A) with coaction AdR, the product (94) of the algebra B(A)0B(A) writes as

(a

b) • (c 0 d) =

as(2)

b(2) :d r(S (b ( 1 ))b ( 3) , S (c(o)c( 3)).

(96)

By Proposition 33(ii), it coincides with the product (92) of the algebra B(A)0B(A). Now let us summarize the main facts established above. By Proposition 34, B(A) is a right A-comodule algebra. Obviously, B(A) is a right A-comodule coalgebra with comultiplication ,AA and counit EA inherited from A. By Proposition 32, A4 is an algebra homomorphism of B(A) to C is an algebra homomorB(A)0B(A). It is obvious that EA : B(A) phism. Proposition 33(iv) says that the mapping S : B(A) B(A) defined therein intertwines the right adjoint coaction Ad R and satisfies the antipode axiom. Thus, the algebra B(A) and the mappings 4,4, EA and S fulfill all Hopf algebra axioms except for one modification: the algebra B(A)0B(A) carries the braided product (94) rather than the usual tensor multiplication. We shall express all that by saying that B(A) is a braided Hopi' algebra associated with the coquasitriangular Hopf algebra A. The precise definition of this concept is Definition 11. Let A be a coquasitriangular bialgebra equipped with a fixed universal r-form r. A right A-comodule algebra and right A-comodule coal-

gebra X is called a braided bialgebra associated with A (briefly, a braided bialgebra) if Ex : X C and ,Ax : X X0X are algebra homomorphisms, where X0X carries the braided product (94). A braided bialgebra X is a braided Hopf algebra associated with A (briefly, a braided Hopf algebra) if X admits an antipode that intertwines the coaction of A on X. In other words, a braided bialgebra X associated with A is a right Acomodule X which admits a product and a coproduct such that:

(i) X satisfies the bialgebra axioms with the modification that the vector space X0X is equipped with the braided product (94); (ii) the product and the coproduct of X intertwine the corresponding coactions of A, that is, X is a A-comodule algebra and coalgebra. 'It should be emphasized that the definitions of the braided product (94) and of a braided bialgebra depend on the choice of the universal r-form of A.

374

10. Coquasitriangularity and Crossed Product Constructions Recall that the antipode of a Hopf algebra is an anti-homomorphism of

the algebra and of the coalgebra. For a braided Hopf algebra X there is a similar result which says that Sx 0 mx ---= mx 0 rxx 0 (Sx 0 Sx), 2ix 0 Sx = (Sx0Sx ) 0 rxx 0 .6x1

Sx(i) =--- 1 7

(97)

Ex 0 Sx =- Ex-

(98)

Let us turn to examples of braided Hopf algebras. Each "ordinary" Hopf algebra X is, of course, a braided Hopf algebra associated with any coquasitriangular bialgebra with respect to the trivial coaction (3(x) = x 01, x E X. A nontrivial simple but still instructive example is the following Example 21 (The braided line). Let A = CZ be the group Hopf algebra (see Example 1.7) of the additive group Z of integers. That is, A is the Hopf algebra of all complex polynomials in a single group-like generator, say x, and its inverse x'. As noted in Example 1 above, for any nonzero complex number q there exists a universal r-form r = rq on A such that rq (xn, km) = q - ", m,n E Z.

(99)

The algebra X = C[zj of all polynomials in one variable is a right A-comodule algebra with coaction 0 given by f3(z) = zn 0 xn , n E Z. By (95) and (99), the corresponding braiding map r xx acts on the vector space X 0 X as rxx(zn 0 en) = CT' el 0 zn , m,rt E No .

Therefore, the braided product (94) of the algebra X 0 X is given by (z 2 0 zi) • (zn 0 el) = q -inz i+n 0 zi+m In particular, we see that the two generators x := z 0 1 and y := 1 0 z of the algebra X 0 X satisfy the relation yx -= q -l sy. Hence the algebras X 0 X and O(C) are isomorphic. Now we define algebra homomorphisms d x : X --- X0X and ex : X —p C such that 2ix(z) =---- z 0 1 + 1 0 z and Ex (z) = O. From Proposition 2.2 we obtain a n-k 0Z k l nE No. 2,x(zn) = En [72] k =0 k q1 Then the vector space X becomes a right A-comodule coalgebra (see Subsect. 10.2.2) with comultiplication Zix, counit Ex and coaction )3 as given above. (Indeed, since all structure maps are algebra homomorphisms, it suffices to check the axioms on the generator z which is easily done.) Rather, one verifies that the linear map Sx : X -- X defined by n E No, S x( z n ) : = (_ 1)Tz q - (n-1)7q2 z n , „

satisfies the antipode axiom. Obviously, we have 0 o Sx = (Sx 0 id) o 0. Thus, we have shown that X is a braided Hopf algebra associated with the coquasitriangular Hopf algebra A. It is called the braided line algebra.

10.3 Braided Hopf Algebras

375

In this example the first two formulas of (97) and (98) read as

Sx(Zin+n )

A x (Sx(zn)) =

Z—dic=0

7n q- infSx( Z)Sx(Zn), [n] -k(n-k) s x (z n-k) k q-.., n

sx ( z k) .

A

Obviously, Example 21 remains valid if the Hopf algebra A = CZ is replaced by the subbialgebra C[x] of polynomials in x. If we interpret this bialgebra C[x] as the FRT bialgebra A(R) (with R being the 1 x 1 matrix (q-1 )), then the braided line algebra becomes a special case of an important class of examples which will be constructed in Subsect. 10.3.5. Nevertheless, the braided Hopf algebras B(A) developed above are still a main source of interesting examples. In fact, the same arguments of proof as in the case of B(A) yield the following more general result. Proposition 36. Let p : 1-t —> A be a Hopf algebra homomorphism from an arbitrary Hop! algebra N to a coquasitriangular Hop! algebra A. Then the vector space 7-t becomes a braided Hopf algebra associated with A, denoted by B(7-(; A, p), with structures given as follows. The product of the algebra B(H; A,p) is defined by

hIg =

E h( 2)g (2) r(p(S (h( i))h( 3)) ,

(g ( 1)))),

h, g c

The coalgebra structure and the unit element of B ( H; A, p) coincide with those of N and the antipode S of 13(7i; A, p) acts as

(h)=

E S(h( 2)) r(p(52 (h( 3 ))S(h( i))), p(h( 4))).

Further, B (fl ; A, p) is a right A-cornodule algebra with respect to the coaction

O(h) :=

h(2) p(S(h( 1))h(3 )),

h E H.

Clearly, if H = A and p is the identity map, then B (Ii; A, p) is just the braided Hopf algebra B(A) defined above. At the other extreme, if p is the trivial Hopf algebra homomorphism (that is, p(h) = e(h)1 for h E H), then obviously B('1-t; A, p) = H. In this case the coaction of A on H is trivial, so 7-t is a braided Hopf algebra with respect to the trivial coaction. Following S. Majid, the passage from the Hopf algebra A resp. the Hopf algebra homomorphism p : H A to the braided Hopf algebras B(A) resp. B(Ii; A, p) will be called transmutation.

376

10. Coquasitriangularity and Crossed Product Constructions

10.3.3 Braided Hopf Algebras Associated with Quasitriangular Hopf Algebras

In this subsection let B be a quasitriangular Hopf algebra with universal Rmatrix R. -= Ei x i ® yi . We briefly state (without giving proofs) the dual versions to the main results of the preceding subsection. Proposition 37. Let a> b := adL(a)b = E ambS(a( 2 )), a, b C B, denote the left adjoint action of B on itself. With the new comultiplication 46 defined by

41(b) = E.

2

Eb(l) S(y ) ® xi › bp),

b E B,

the vector space B becomes a coalgebra, denoted by B(B), such that B(13) is a left 13-module coalgebra. Moreover, 4 (1) = 1 01.

and the counit

68 1

Proposition 38. If X and Y are left B-module algebras, then the vector space X®Y is a left B-module algebra, denoted by xoy, with the tensor product action of B and the braided product .'defined by

(x0y)•(x'00 = E(x01)7,,,x( y 0x1)(100, x, xi E X, y, y' E y, (100) where Ry,x(y 0 x') :=

E i yi › x' 0 x i > y.

The algebra B itself is also a left B-module algebra with respect to the left adjoint action. Therefore, the braided product (100) makes the vector space BOB into an algebra denoted by BOB. It can be shown that 4 : B B0B is an algebra homomorphism. Further, the mapping S : B —> B defined by S(b) := E i y i Ss(x i > b),

b E B,

intertwines the left adjoint action (that is, a › S (b) = S (a > b)) and satisfies the equation E b(1) ,5(b (2) ) = > (b(1) )b(2) = 6 (b )1, b E B. Here 4 (b ) = E b( ,) ® b(2) is the Swee-dler notation for the new comultiplication A. The left B-module coalgebra 8(B) is also a left B-module algebra with the algebra structure inherited from B. We call B(Bj equipped with these structures a braided Hopf algebra associated with the quasitriangular Hopf algebra B. The general definition is easily guessed from Definition 11 above. A braided bialgebra associated with 13 is a left B-module algebra and a left B-module coalgebra X such that ex : X —> C and 4&v : X —› X0X are algebra homomorphisms, where X0X is equipped with the braided product (100). If X has an antipode which intertwines the action of B on X, then X is called a braided Hopf algebra. Finally,, we want to make more precise the sense in which the preceding constructions of B(A) and B(B) are dual to each other. In order to do so, we consider in additiOn to the quasitriangular Hopf algebra B a coquasitriangular Hopf algebra A with universal r-form r. We suppose that there is a dual pairing (•, .) of the Hopf algebras A and 5 such that r(a 0 a') = (a 0 a', R.)

10.3 Braided Hopf Algebras

377

for a, a' E A. The next proposition shows that there is a close link between the two new structures (the product • of B(A) and the coproduct 4 of B(8)) and the two old ones (the product of B and the coproduct LtA of A). Proposition 39. Retain the preceding assumptions on A and B and define a linear mapping Q : A —* B by Q(a) = E ir (a, ai )bi , a E A, where 1Z211Z•12 = ai bi E BOB. Then, Q is an algebra hornornorphism, of B(A) to 8 (that is, Q(a:b) =-- Q(a) • Q(b)) and a coalgebra homomorphism of A to B(B) (that 4Q(a) (Q CA(a)).

Ei

Idea of proof. Some computations show that Q(a:b) and Q(a).Q(b) are both equal to (a0b, 112R21 7Z13R31) and that the expressions ,Q(a) and (Q0Q)zA(a) coincide with (a, 7Z137Z317Z23R.32)• See the proof of Proposition 7.4.3 in [Mai] for the details. LI 10.3.4 Braided Tensor Categories and Braided Hopf Algebras

In this subsection we shall see how both braided products (94) and (100) and both braided Hopf algebras B(A) and B(13) can be considered from a unique point of view. The common notion behind these concepts is that of a braided tensor category. In the following treatment we shall concentrate on the essential points and suppress mathematical subtleties. A category C is a collection of objects X, Y Z • • • and of sets Mor (X, Y) of morphisms between two objects X, Y such that a composition of morphisrns is defined which has similar properties to the composition of maps. That is, given morphisms f E Mor (X, Y) and g E Mor (Y, Z) there always exists a morphism gof E Mor (X, Z) and the composition of three morphisms has to satisfy the associativity law. Moreover, each set Mor (X, X) has to contain a morphism idx such that f o idx = f and id x o g = g for any f E Mor (X, Y) and g E Mor (Y, X). A mapping f E Mor (X, Y) is called an isomorphism if there is another morphism g E Mor (Y, X) such that g o f = idx and f 0 g = idy. An obvious example of a category is obtained by taking the vector spaces as objects and the set of all linear mappings f : X Y as Mor (X, Y). For our purpose the categories MA and MA of a bialgebra A are most important. The objects of MA and MA are the right A-comodules resp. the right A-modules and the morphisms are the linear intertwiners of the corresponding coactions resp. actions. In what follows we consider tensor (or rnonoidal) categories. These are categories C that have a product, denoted 0 and called the tensor product, which admits several "natural" properties such as associativity and existence of a unit object denoted 1. The associativity of the tensor product requires that there is an isomorphism qS X ,Y,Z (X 01 7 )0 Z X0(YOZ) for any triple X ,Y , Z of objects in C. We will not list the axioms of a tensor category here (see, for instance, [Kas], Sect. XI.2). For our subsequent discussion it suffices to know that the categories MA and MA for any bialgebra A become such tensor categories if we take the tensor product of comodules resp. modules as ,

378

10. Coquasitriangularity and Crossed Product Constructions

the tensor product 0 and the trivial one-dimensional comodule resp. module as the unit object 1 in the category. The categories we are interested in are tensor categories which are equipped with some kind of transposition of tensor products. A tensor category (C, 0) is called a braided tensor (or monoidal) category if for any pair 17 0X such that X, Y of objects in C there is an isomorphism Wx ,y : XOY (g f) 0 WX,Y = WX 1 ,171 0 f. g) for arbitrary morphisms f E Mor (X, X') and g E Mor (Y, Y') and the hexagon axiom holds. For simplicity we will suppress writing the isomorphisms Ox,y,z which govern the associativity of the tensor product. Then the hexagon axiom is the validity of the two conditions (

WX,Z ° WY,Z = WX0Y,Z7 WX,Y ° WX,Z = WX,YOZ

for all objects X, Y, Z in C. A family W of mappings Wxy having the above properties is called a braiding (or a quasisymmetry) in the tensor category (C, 0). Obviously, the flip gives a braiding in the category of vector spaces. But in contrast to this example, the -mapping Wy, x need not be the inverse of Wxy in general. The two hexagon identities imply the relation (PY,zOidx) 0 (idyetPx,z)0(Wx,Y0idz) = (idz0Wx,y)0(Wx,zOidy)0(idx®Wy,z)•

Indeed, there are two ways to go from (X0Y)0Z to ZO(Y0X) by applying W, one over Z0(X0Y) and another one over (Y0X)0Z. The hexagon conditions for these ways give the above 'relation. In the case X=Y-Z it is just the braid relation for Wx ,x and this explains why W is called a braiding. From now on let A be a coquasitriangular bialgebra with universal rform r and B a quasitriangular bialgebra with universal R-matrix R. Then, by Propositions 1 and 8.4, the families of mappings Wx,y := rxy and WX,Y := Rxy are braidings in the tensor categories MA and MB, respectively. Thus, MA and MB are braiding tensor categories. Let C be a braided tensor category. By an algebra in the category C we mean an object X in C together with morphisms mx : X0X X and 77 : 1 X from C (!) satisfying the usual associaivity and unit properties. From the corresponding definitions we conclude that the algebras in the categories MA and MB are precisely the right A-comodule algebras and the right 8-module algebras, respectively. Now let X and Y be two algebras in the category C. Then it can be shown that the object X®Y in C becomes an algebra in C, denoted by X®Y and called the braided product algebra of X and Y, with product morphism mxoy := (mx 0my ) o (idx 0Wy,x0idy)

and the tensor product unit morphism. If Wy,x is the flip, then X031 is just the tensor product algebra X 0 Y. In the case of the category MA with braiding Wx,y = rxy, the algebra X®Y in the category MA is nothing

10.3 Braided Hopf Algebras

379

but the right A-comodule algebra X®Y from Proposition 35 with braided product (94). Likewise, for the category MB the algebra X®Y in MB is the right 13-module algebra X®Y from Proposition 38 with product (100). In a similar way, a number of other concepts from "ordinary" Hopf algebra theory make sense in the category C. In addition to the usual axioms one has to require that the corresponding object and all structure maps belong to the category C. In particular, a Hopf algebra X in the category C is defined in this manner, whereby the comultiplication LX x has to be an algebra homomorphism of X to the braided product algebra X0X. It is clear that the Hopf algebras in the braided tensor categories MA and MB are precisely the braided Hopf algebras associated with A and B, respectively. Let us illustrate the preceding by a simple example. It can be generalized to any abelian group rather than Z2 equipped with a fixed bicharacter. Example 22 (A = CZ2 ). Let A be the coquasitriangular group Hopf algebra with universal r-form r(,3) CZ2 of the group Z2 = Z/2Z {0, j G {0, 1 } . Then any right A-comodule X with coaction ço splits into a direct sum of an even part XT, :=--- Ix G X I (}0 ( x) = x0-61 and an odd part XT = {X E X 1 (P(x) = x0I}. Therefore, the objects of the category M A are the Z 2-graded (or super) vector spaces and the morphisms are the Z 2-graded linear mappings. The braiding obtained from r is determined by the formula Wx,y(x7 yy) = (-1) ijyy 0 x7 ,

E

y.7 E

j E {0,1}. (101)

As noted above, the algebras in MA are the right A-comodule algebras. If X is an algebra equipped with a coaction ço of A on X, then ‘,61:X--X0A is an algebra homomorphism if and only if X6 X6 C Xo, X6X1 C I , XIX() C X

and Xi-Xi C X.

Thus, the algebras in MA are precisely the Z 2-graded algebras. Similarly, the Hopf algebras in MA or equivalently the braided Hopf algebras associated A with A are just the super Hopf algebras (see Subsect. 1.2.9). We close this subsection with some comments on the physical interpretation of the braided product algebras. For this purpose we assume that the right A-comodule (or 13-module) algebras X and y are algebras of observables of two physical systems. Let us consider the braided product algebra xoy as an algebra of observables of a joint system which is built from the two systems. First let A = C. Then the braiding of the category MA is the flip and the algebra X®Y carries the product (x0y) • (xi0V) = xxiOyy', so the subalgebras X and y of xoy mutually commute. This means that the two systems are independent • in the joint system. Next we set A = CZ2 . As usual in "super physics", we consider Xr) and Y6 as algebras of observables of bosonic systems and Xi: and y, as algebras of observables of fermionic systems. From the formula (101) of the braiding it is clear that bosons and bosons as well as bosons and fermions of the two systems commute, while

10. Coquasitriangularity and Crossed Product Constructions

380

fermions and fermions of the two systems a.nticommute. That is, in this case the braiding contains the statistics between the two systems. In the general case the braiding W describes the commutation relations between observables of the two systems. Having the above example A = CZ2 in mind, one says that the systems obey the braid statistics W. This picture opens an important conceptual view of the braiding and the braided Hopf algebras and their possible future role in physics. 10.3.5 Braided Vector Algebras

From Proposition 9.4 we know that the algebra X R( f ; R) is a right quantum space (that is, a right comodule algebra) for any quotient bialgebra A of the FRT bialgebra A(R). Recall that X R(f R) is the unital algebra with generators x 1 , x2 , . xN and relations f(h) iii x k x i = 0, i j = 1,2, • - • , N, and that the coaction çart of A on XR(f;R) is given by the equations (pR(x) = xi 0 u,ji , see (9.10). The purpose of this subsection is to show that under suitable assumptions the algebra X R ( f ; R) becomes a braided Hopf algebra. ,

Proposition 40. Let R be an invertible solution of the QYBE and let f be

a complex polynomial such that + I) f(ii) = 0.

(102)

Suppose that A is a quotient bialgebra of A(R) such that the canonical universal r-form r of A(R) passes to a , universal r-form of A. Then the right A-comodule algebra X := XR (f;R) is a braided Hopf algebra associated with the coquasitriangular bialgebra A with comultiplication 2i x , counit e x , antipode Sx and braiding rxx determined by the formulas

/-X(Xi) = x i 01+10 x i ,

Ex(x i )

rxx (xi 0 xi ) = Xk 0

0, Xi

S(x) = —xi ,

(103) (104)

.

Proof. As in similar earlier proofs (for instance, of Proposition 9.1), we first show the corresponding result for the free algebra C(xi ) and then we pass to the quotient algebra X of C(xi ). Because C(xi ) is a right A-comodule algebra with coaction PR (xi) = Xi 0 tr i: 1 the braided product algebra C(x i )0C(x i ) is well defined by Proposition 35. Since C(xi ) is the free algebra with generators xi , there are unique algebra homomorphisms : C(x i ) —> C(xi )0C(xi ) and E : C(Xi) —> C such that 2i(xi) = xi 0 1 + 1 0 x i and E(xi ) = 0. The set {1, xii • • • xi n } is a vector space basis of C(xi ). Hence there is a unique linear map S of C(xi ) such that 5(1) = 1, 5(x) = —x and -

S(Xi• - -x n ) =

(-1) 1 x..

• *Rln R23' "R2n• • •Rn-1,n

7

n? 2, (105)

in the usual matrix notation. That is, we define 8 (1) = 1, S(xi) = xi and extend S to a linear map of C(xi) according to the formula (97). One easily —

10.3 Braided Hopf Algebras

381

verifies that C(xi ) equipped with these mappings becomes a braided Hopf algebra associated with A. In order to complete the proof we have to show that these structures pass to the quotient algebra XR(f;R)=C(x i )1J, where J denotes the twof(11)i xkx/. For sided ideal of C(xi) generated by the ./V2 elements the coaction ço R this holds by Proposition 9.4. Formula (104) for the braiding rxx is obtained by repeating the reasoning of (13). By the definition of E we have E(J) = {0). Using the formulas (103), (94), (104) and finally (102), we obtain

= f (14) ikjiA(xkxi) = f(ki kj(xk ® 1 + 1 ® x k ) • (xi 01 + 10 x i ) f( fo iki xm xr, = f(ft)ikjxkxi el + f (Ncjxk ® x l + +10 f(14) ikjxkx1

® + ((k+ 1- )f (M)Ti.r; n Trol Xn +10

1+ 1

and hence .A(a/ii b) = 2i(a) •

01+10

• 2i (b) E J C(x i ) + C(xi) 0

for a, b C C(xi). From equation (105) it follows that

S(I) = f(1ViciS(xkx1)= f(k) icjx,x n Rircir = RjIkl C J. By (97), this implies that S(aIti b) E J for a, b E C(x2 ). Thus we have shown that J is a Hopf ideal of the braided Hopf algebra C(i). Hence the quotient X=C(xi )/J is also a braided Hopf algebra. Note that (105) also hold for the antipode Sx of the braided Hopf algebra XR(f; R) and allows us to compute Sx on products of generators. In order to discuss the applicability of Proposition 40, let us assume in addition that the transformation it admits a spectral decomposition le = Ei A,Pi , where Pi are nonzero projections such that Ei P = I and Pi Pi = 0 and A i Ai if i j. It is well-known from linear algebra that the projections Pi are then polynomials in ft, say, Pi = Mk). If an eigenvalue A i is —1, then fi satisfies (102) and hence, by Proposition 40, XR( fi ; R) is a braided Hopf algebra associated with A. In the case A = A(R) the transformation R can be replaced by any nonzero complex multiple, so each eigenspace of it gives rise to a braided Hopf algebra. However, if A is one of the coquasitriangular Hopf algebras 0(G q ), Gq = SL q (N),O p (N),Sp q (N), and R = zR as in Theorem 9, such a scaling is impossible and k cannot have the eigenvalue —1 unless q is a root of unity. Our next proposition shows how to overcome this difficulty: XR(f;R) becomes a braided Hopf algebra associated with the coquasitriangular bialgebra A. which is obtained from A by adjoining an additional central invertible group-like generator. Recall that the group algebra CZ of the group Z of integers is the algebra of Laurent polynomials in a single generator x. It is a Hopf algebra with

382

10. Coquasitriangularity and Crossed Product Constructions

comultiplication Zi(xn) = xn xn, n E Z. For each nonzero A E C, there exists a universal r-form r), on CZ such that r),(xn g xm) = A -nm (see Example 1 in Subsect. 10.1.1). One easily verifies that the tensor product bialgebra A := A g CZ is coquasitriangular with universal r-form i defined by î(a xn

b xm) = r(a g b)A - nm,

a, b E A,

n, m E Z.

(106)

Clearly, each algebra XR(f; R) is also a right kcornodule algebra with coaction given by c,oR (xi ) = xi 0 u3i: g x. Proposition 41. Let R, A and r be as in Proposition 40. Suppose that A E C, A L 0, and f is a polynomial such that

(il + All f

--= 0.

(107)

Then X :--= XR (f R) is a braided Hopf algebra associated with the coquasitriangular bialgebraA defined above. The Hopf algebra structure of X is determined by (103) and the braiding map kxx is given by

x(s i 0 x i ) ==

(108)

Proof. Clearly, (106) implies (108). The term + ilf(k) of the expression Zi(iii ) in the proof of Proposition 40 now becomes (A -1 k + .00) which vanishes by (107). The rest of the proof follows by the same reasoning. The braided Hopf algebras XR(f ; R) in Propositions 40 and 41 are called braided vector algebras. Note that Propositions 40 and 41 are also valid for the coquasitriangular Hopf algebra 0(Gid q (N)) with the same proofs verbatim. Finally, let us consider the case when the number A in (107) is a root of unity, say, Ak = 1 for k E N. Then, rx gives a universal r-form of the group algebra C(Z/kZ) of the group Z/kZ. Hence the assertion of Proposition 41 holds also for the coquasitriangular bialgebra A := A 0 C(Z I kZ). In particular, in the case A = 1 we can take k = 1 and recover Proposition 40. 10.3.6 Bosonization of Braided Hopf Algebras In this subsection we show that the crossed product of a braided Hopf algebra X by the coquasitriangular (or quasitriangular) Hopf algebra A which X is associated with becomes an ordinary Hopf algebra. In order to avoid possible confusion with the comultiplications of X and A, let us write the Sweedler notation for the coaction @ of A on X as a(s) = E so) 0 so). This notation will be kept in the rest of this section. To begin with, suppose that X is a braided bialgebra associated with a coquasitriangular bialgebra A. Then, by Definition 11, X is in particular a right A-comodule coalgebra with respect to the coaction0 of A on X. Therefore, by Proposition 19, we have the crossed coproduct coalgebra A x X built

10.3 Braided Hopf Algebras

383

on the vector space A0 X. On the other hand, X is also a right A-comodule algebra by Definition 11. Hence, by Proposition 4, the right coaction @ induces a right action 0' of A on X by xaa=-E x (°)r(x (1 ), a), a E A, x E X, such that X becomes a right A-module algebra, where 0(x) = E x ) ® x('). Thus the vector space A® X carries also the structure of the crossed product algebra Ap, tx X (see Subsect. 10.2.1). The next proposition says that these two structures (algebra and coalgebra) on the vector space A0 X indeed fit nicely together.

(°

Proposition 42. Let X be a braided bialgebra associated with a coquasitriX is a bialgebra, denoted angular bialgebra A. Then the vector space A by AS x X or Ax X , with the coalgebra structure of A 8 x X and the algebra structure of Aoi tx X . That is, the bialgebra A 3 tx X has the coproduct (54) and the product (a 0 x)(b y) =

E ab( l)

x (°) y r(s (I) , b ( 2 )),

a, b E A, x, y E X. (109)

The subalgebra1®X of AD< X is a right Ax X -comodule algebra with coaction PR(10X) := 2■ Apc x (1®x) =--

E 1® (x (1) )(0)®(s (1) )(1)0x (2) , X E X. (110)

If X is a braided Hopf algebra associated with the coquasitriangular Hopf algebra A, then the bialgebra A Lx X is a Hopf algebra with antipode S (a 0 x) = E(1 0 S x (x (1) ))(S A(ax (°) ) ® 1) ,

a

E

A, z E X.

(111)

Proof. Being the restriction of the comultiplication to an invariant subspace, (p R is a coaction. One has to show that (10 R 1 ZA and E are algebra homomorphisms and that S satisfies the antipode axiom. These technical verifications are carried out in [Wei] and in the dual situation of quasitriangular Hopf algebras in [Maj10]. Formula (110) follows from (54). The construction of the "ordinary" Hopf algebra A x X from the braided Hopf algebra X is a process called bosonization by S. Majid. It is in a sense dual to the process of transmutation which converts the Hopf algebra A or more generally any Hopf algebra homomorphism p : A into the braided Hopf algebras .13(A) resp. B(Ii; A; p), see Subsect. 10.3.2. Let us see how A and X are related to A tx X. From the formulas (54), (109) and (111) it is clear that the map a a01 is an injective Hopf algebra homomorphism of A to A x X, so we may consider A as a Hopf subalgebra of A x X by identifying a with a 0 1. Also, the map x 1 tg x is an algebra isomorphism of X to the subalgebra 1 tg X of A x X, but as seen from (110) and (111) it does not preserve the comultiplication and the antipode. Being a crossed product algebra (see Subsect. 10.2.1), Aix X is the universal algebra generated by the algebras A and X with cross commutation relations xa =

E a( l )x ° )r(x (1) , a(2)),

a E A,

X

E X.

(112)

384

10. Coquasitriangularity and Crossed Product Constructions

We now illustrate the bosonization procedure by two examples. The Hopf algebras A y X appearing in these examples are also of interest in themselves without referring to braided Hopf algebras. Example 23 (The braided line - continued). Let us retain the notation of Example 21. By Proposition 42, A(3 ix X is a Hopf algebra. It is generated by the elements x x01 and z 10z. Since 0(z) = zox and rq (x0x) = the cross relation (112) with x := z and a := x reads as zx xzq -1 . Since AA A, (z) = z 1 1 z, formula (54) for the coproduct of A y X yields (

X = X X7 igz) )

z

x 1

(113)

z.

From these formulas (or from (97)) we easily derive e(x) = 1, e(z) = 0, S (x) = x -1

(z) =

-

zx -1

(114)

That is, the Hopf algebra A ix X has two generators x and z with defining relation xz = qzx and structure maps determined by (113) and (114). For q =1 the Hopf algebra (A Ix X)"P is just the coordinate algebra of the group of affine transformations of the real line (the so-called ax + b-group). This suggests the use of the bosonization for the construction of inhomogeneous quantum groups. This idea will be realized in the next subsection. A Example 24 (The Hopf algebra A Ad R y13(A)). Let X be the braided Hopf algebra B(A) associated with a coquasitriangular Hopf algebra A. Recall from Subsect. 10.3.2 that B(A) has the same coalgebra structure as A and that the product of B(A) is given by (85). The coaction of A on B(A) is

the right adjoint coaction Ad R (a) = E a( 2) S(a( i) )a( 3 ). Inserting these data into the formulas (109) and (54), the product and coproduct of A D.ti 13(A) read as (a 0 s)(b y) =

E ab(i ) 0 x(20 r(S(x(1))x(3), b(2))

= E ab( i ) O 5 (3)Y(2) r(S(s(2))x (4)1 S (Y (1))*(S (x (Os (5) (a Ø s) = E a( 1 ) X(2) a(2)51 (5(i))5(3)

11 (2) )

X(4)-

The following interesting result emphasizes once more the fundamental importance of Drinfeld's double construction. Statement: The linear map 0 defined by 0(a 0 E ax (1) Ox (2) is a HoPf algebra isomorphism of the quantum double D(A, A; r) = A ix, A (see Subsects. 8.2.1 and 10.2.5) to the Hopf algebra AAd R x B(A). Proof. Using the definitions (115) and (8.21) of the corresponding products

and some properties of the universal r-form r we compute

10.3 Braided Hopf Algebras _

0(a0x)0(bOY)

385

Ek ax( 1) 0 x(2))(bY(i) ® Y(2))

E ax( l )b( l )y( l ) 0 x()y(4) r(S(x(3))x(5)

S(Y(3)))

xr(S(x(2))/(6)7b(2)Y(2))

E ax( i )boo( i )

x(5 )y( 4 ) s(S(x(4) ) x (6) 7 Y(3) )

xr(S(x( 3 ) )x (7) Y(2)*(S(x(2))x( 8),b(2)) =

ax(i)b(i)Y(1)

x(3)Y(2) r(S(x(2))x(4), b(2))

n rr h --(1)-(1)y(1)

X(3) Y(2)

au(2)x(2)y(1)

x(3)y(2) r(x(i) 7b(o)r(x(4), bo

r(x(2), b(2))*(4), b(3))

E e(ab (2)

x(2 ) 0 r(x(i)lb(1))r(x(3)

0 ((a 0 x)(b

y)).

o

b (3)

Here the third equality follows from Proposition 33(0, the fourth from (82) and the sixth from (5). Recall that the quantum double has the tensor product coalgebra structure. Hence, by (116), we obtain

61 (a 0 X) =

=

E ,i(ax( i ) 0 x(2)) E a (l )x (l ) .T(4 )

a(2)x(2)S(x(3))x( 5 )

x(6)

a( l )x( l) 0 x( 2 ) 0 a( 2 )x( 3 ) 0 x( 4 ) = (0 0 0) o ,i(a

x).

Since obviously 9(1 0 1) = 1 0 1 and E o8 = e 0 is an algebra and a coalgebra homomorphism. It is easy to check that 8 -1-(a x) := E aS(x( i )) 0 x( 2 ) is the inverse of 0. Therefore, 0 is a Hopf algebra isomorphism. El A ,

We briefly turn to the dual situation of a quasitriangular Hopf algebra B and state the counterpart to Proposition 42 as Proposition 43. Suppose that X is a braided Hopf algebra associated with a quasitriangular Hop! algebra B. Then, by definition, X is a left B-module algebra. Let a : (b o x) bi> x denote the corresponding left action of B on X and let R. = x 0 yi be the universal R-matrix of B. Then X is a left B-comodule coalgebra with respect to the coaction a'(x) := yi 0 x t x. The vector space X 0 B equipped with the left crossed product algebra structure of X m a B (see Subsect. 10.2.1) and with the left crossed coproduct coalgebra structure of X Na' B is a Hopf algebra, denoted by X >1„B or simply X >1 B. The product, the coproduct and the antipode of this Hopf algebra are given by

Ei

Ei

(x 0 ar)(y Zi(x 0 a) =

Ei

=

E x(a(i) L> y) 0

x ( i ) Yia(i) 0 xi f> x(2) 0 a(2) 7

386

10. Coquasitriangularity and Crossed Product Constructions

sB(Y ) a (2) )xi (x i t> Sx(x)) 0

S(x 0 a) =

where LA(a) = am a( 2) and Ax(x) = notations for the comultiplications of A and X.

S13 (Yia(i) )1

x(i) ® x (2) are

the Sweedler

Example 25 (The Hopf algebra B(8)>4 ad,8). Let B and R, be as in Proposition 43 and let X be the braided Hopf algebra B(8) from Subsect. 10.3.3. In this case a is the left adjoint action adi, (Os = E b ( i ) xst3(b ( 2) ). The product and the coproduct of the Hopf algebra B(8) >ad,, B are determined by (s

,A(x ® a) -=

b) = E xamySB(a( 2))

a) (y

E. E x ( i ) yia(i)

a( 3 )b,

ximx(2)SB(Xi(2))

a(2).

10.3.7 *-Structures on Bosonized Hopf Algebras

The definition of * - structures on quantum spaces of a coquasitriangular *bialgebra depends essentially on the reality type of the universal r-form. In this subsection we treat only the inverse real case which is much simpler than the real case. Proposition 44. Suppose that r is an inverse real universal r-forrn of a *bialgebra A. Let X and y be right A-comodule *-algebras. Then the vector space xoy equipped with the braided product (94) and the involution

(x

y)*

x(0)* ® y (°) *r(y (1) *, x (1) *),

becomes a right A-comodule *-algebra, denoted by

s E X, y E

(117)

3),

x(gy.

Proof. By Proposition 35, xoy is a right A-comodule algebra, so only the assertions concerning the involution have to be proved. Using the assumption that r is inverse real, we compute ((x0y)(z0w))* =

E

z (0)* s (0)* •

(z w)* and

(

* * _

Y )

w (o)*0)* r(w (i)* , x (1)* ) r (w (2)* , z (1)*)

xr(y (1) *, x (2) *)r(y (2) *, z (2) *)f(y (3) *, z (2) *) w (0)* )(x (0)* y (0)* ) r(w (i)* z (1)*) r ( y (1)* 7 s (1)*)

= E(z(0)*

(s

yMw r(y (1) , z (1) ))*

z(0)*x(0)* • w (0)*0)* r (w (1)* y (1)* 7 z (i)* x (i)*) r (y (2) , z (2))

—

(

E(xz(0)

®

y)

*

y (o) ,. (y (1 ) , x (1 )) ,.. (y(2) , y(2) )

y.

)

Hence we have an involution of the algebra xo_y. Next we show that the coaction cpR of A on ev(Iy is *-preserving. Indeed, by (117) and (5), we get

387

10.3 Braided Hopf Algebras

(pR((s

y)t) =

x( o) *

y( o ) *

x (ot y(i)*

so )*

yo) *

y (2)* x(2)* r (y (1)t , x (1)* )

E (s (o)

y(0) ) *

r(y (2)* , x (2)* )

(x (l)0.) ) * = ipR(x 0 y)t.

I=1

The next proposition shows that the involution (117) turns the bosonized Hopf algebra A x X from Proposition 42 into a Hopf *-algebra. Proposition 45. Let r be an inverse real universal r -form of a *-bialgebra A. Let X be a braided bialgebra associated with A such that X is a A-comodule

*-algebra and ,(x*) = ,ix(x)* for x e X, where X0X carries the involution (1 1 7). Then the bialgebra A Ix X from Proposition 42 equipped with the involution (a

x )t = E(a (i) )*

x (°) * r(x (1) *, (a( 2))*),

becomes a *-bialgebra, where dA(a) =

E a( i)

a E

A, x

E X,

(118)

a(2).

Proof. Since the involution (118) is a special case of (117) (by considering A as a right A-comodule *-algebra with respect to the comultiplication), A x X is a *-algebra by Proposition 44. It only remains to show that the counit and the comultiplication of the bialgebra A x X are *-preserving. For the counit this is obvious. For the comultiplication this requires some technical computations which will be omitted here. They are carried out in [Weil. 1:1 As usual, let us identify x E X with x 0 1 E X03) and y c y with 1 o y E xoy. Then it is clear from the definition (117) of the involution that X and y are *-subalgebras of the *-algebra X®Y from Proposition 44. Conversely, formula (117) already follows from that property, since

(x

y)* =

((x 0 1)(1 0 y))* = (1 0 y)*(x 0 1)* = (1 0 y*)(x* ® 1)

E x(0)*

y (°) * r(y(1) *, x (1) *).

In other words, there exists a unique involution of the algebra X®Y which extends the involutions of its subalgebras X and Y. The last assertion is no longer true if the universal r-form r is real rather than inverse real. More precisely, if A is a Hopf *-algebra with real universal rform r and X is as in Proposition 45, then there is no well-defined(!) involution of the algebra A x X which coincides with the involutions on its subalgebras A and X. An example will be seen in Subsect. 10.3.9 below. One possibility of handling this difficulty is to work with the realification 13 = A; r) of A (see Subsect. 10.2.7) rather. than with A itself. Recall that, by Proposition 29, 13 always admits an inverse real universal r-form r", so that Propositions 43 and 44 apply to 13 and r". Details can be found in [Wei].

388

10. Coquasitriangularity and Crossed Product Constructions

10.3.8 Inhomogeneous Quantum Groups

In this subsection we combine Propositions 41 and 42 in order to construct coordinate Hopf algebras of inhomogeneous quantum groups. In what follows let Gq be one of the quantum groups GL q (N), Sid q (N), 0q (N) or Spq (N). By Theorem 9, the Hopf algebra A = 0(Gq ) possesses a universal r-form rz such that rz (u1, u2) = R12 with R := zR, where the parameter z and the matrix R are as stated there. Let us suppose that 0 for q2 + 1 0 for Gq = GLq (N),SId q (N) and (q2 + 1)(€ + Gq = 0q (N),Sp q (N). Then the spectral projector P_ of the matrix I to the eigenvalue -q -1 is of the form f(k) for some polynomial f and the right quantum space X := X(f,R) of A has the defining relations P_x1x2 = O. Note that X = O(C) if Gq = Glaq (N),SL q (N) and X = 0(0,2N) if Gq = 0q (N).

+ zq-1 )P_ = -zq -1 P_ + zq -1 P_ = 0, Since (it + zq-1 )f(ft) = Proposition 41 applies to X with A = zq -1 . Hence X is a braided Hopf algebra associated with the coquasitriangular Hopf algebra ./1- = A 0 CZ. If (zq -1 ) k = 1 for some k G N, it is possible to take the quotient group Z/kZ instead of Z (see the remarks at the end of Subsect. 10.3.5). By Proposition 42, there exists the (ordinary) Hopf algebra .A y X. Definition 12. The Hopf algebra A D< X is called the coordinate algebra of the inhomogeneous quantum group IGq and denoted by O(IGq ).

We are now going to desribe the structure of the Hopf algebra 0(1.GO explicitly in terms of generators. We shall do this 'first in the case where = A 0 CZ and recall briefly the structure of the Hopf algebra A- Y X. As a vector space, .4 y X is A 0 CZ 0 X. We can consider A as a Hopf subalgebra ofAv X by identifying a c A with a 0 1 0 1 E Av X. Likewise, CZ becomes a Hopf subalgebra of .,21 x X and X a subalgebra of „.1 y X by identifying g E CZ with 1 0 g 0 1 and x E X with 1 0 1 0 x. As an algebra, A- ix X is a crossed product algebra of .A and X. Therefore, as noted in Subsect. 10.3.6, it is the algebra generated by the algebras A (and so by A and CZ) and X satisfying the cross relations (112). The coalgebra structure of A y X is that of a crossed coproduct coalgebra of .4 and X and described by Proposition 19. Finally, the coaction (pR and the antipode of A ix X are given by (110) and (111), respectively. Inserting the data of the coaction (0(s) = x i Oulx), of the universal r-form (i.(u i , u2) = zR12, x, x) = r(uji , 1)A -1 = 5ijqz -1 ) and of the braided Hopf algebra (2ix (xi ) = xi 0 1 + 1 0 xi , Ex (x i ) = 0, Sx (x i ) = -xi ) in all these formulas, we obtain the following description of the Hopf algebra 0(IG q ) for the quantum groups Gq SL q (N),0 q (N),Sp q (N). Algebra structure of 0(IGq ):

0(IGq ) has N2 + N + 2 generators uii , xil x, x---1 ) j)./ = 1,2, - • , N. The defining relations are the defining relations of C9(Gq ) for the generators u3i-

(see Sects. 9.2 and 9.3), the defining relations of X for the generators x i (that is, (P_)ijxkx/ = 0 for i, j = 1,2, • • - N) and the following relations:

10.3 Braided Hopf Algebras

61',niklUkX1 i X n Uii = ZuN 7 XUji -- Uji X,

XiX = qZ -1 XXil

XX

-1

389

= X 1X = 1.

Coalgebra structure of 0(/G q ): ,A(uii ) =

4 ®i4, ,i(xi ) = 1 ® xi + xi 0 ti3, x , ,A( x ) = x 0 x , E(u) = Su, E(xi ) = 0,

E(x) = 1.

Antipode of 0(IG q ):

8(u) = 51 0 ( G ,)(uii ) , 51(xi ) = -xiSo( G )(u)x - , S() = x -1 .

Coaction 'PR of 0(1G q ): X is a-right 0(/G q )-comodule algebra with coaction (pR(xi) = Let us emphasize that the preceding formulas already give a complete description of the structures of the Hopf algebra 0(/G q ). One may also work alone with these formulas combined with the fact that they define a Hopf algebra and ignore the braided Hopf algebra approach presented above. For the quantum group G q = GL q (N) it suffices to add a generator D-1 and to require that it is the inverse of the quantum determinant Dq . The corresponding formulas for Dq-1 then follow from the above structures. For example, the commutation relations for x n and uii imply that xn Dq, z NqDq xii , so we obtain Dq-1- x n = z N qxnD q-1 . The new phenomenon for the quantum inhomogenous groups IG q is the appearance of the additional element x which is not present in the classical case. It can be interpreted as a scaling (or dilation) generator which is adjoined as a central element to the homogeneous part. The generator x cannot be avoided if the parameter A = zq -1 is different from 1. Let us give two ar1. First, tracing back the above approach guments for its necessity when A without x or more precisely with x = 1, it would require that X is a braided Hopf algebra associated with A. However, the proof of Proposition 41 is not therein valid for A 1, because the term (R 1 )1(R) =-- (1 - ) )P_ of does not vanish. Secondly, having the above set of formulas, one can set = 1 and try to verify directly that they define a Hopf algebra. Computing (recall that (P_) ilci(xkx/) = 0 by using the formulas ,A(xi ) = 1®x +xi = 1) and xnuii = zk lj uik x / leads to the relations (A - 1)(P_ )flurk'x i = 0 for 1. all j, j, n = 1, 2, • • • , N. This is not reasonable when A Next let us discuss the case when A k Z k q -k = 1 for some k E Z. Then, in our above approach, the group Z can be replaced by ZikZ. In terms of generators this means that we have to add the equation )( k = 1 to the defining relations. For the quantum groups SL q (N), 0q (N) and Spq (N) an equation Ak = 1 with k E Z is only possible if q is a root of unity. However, in the case of GL q (N) the parameter z 0 is arbitrary, so it may be choosen as q. Then we have A = 1 and X is a braided Hopf algebra associated with A 0(GL q (N)) by Proposition 40. Thus, in this case the generator x can indeed be avoided.

390

10. Coquasitriangularity and Crossed Product Constructions

That is, for A = 0(GLq (N)) there exists a Hopf algebra with N 2 + N +1

generators uji , x i , Dq-1 and the defining relations and structure equations as listed above with x := 1. This Hopf algebra can be taken as the coordinate algebra of the inhomogeneous quantum group IGL q (N). The Hopf algebra structure of 0(/Gq ) can be nicely kept in mind if we use the matrix = The above formulas for the comultiplication are obtained from the entries of the product of û by itself and for the antipode from the entries of the inverse matrix S(u) 0

-xS(u)x -I 1) • 10.3.9 *-Structures for Inhomogeneous Quantum Groups As in earlier subsections, we shall distinguish between two cases of parameter values and reality types. First we assume that 11 = izl =--- 1. Let Gq be one of the real forms GL q (N;IR), SL q (N;R), 0q (n,n), 0q (n,n + 1) or Sp q (N; R) from Case 1 in Subsects. 9.2.4 or 9.3.5. Then, by Proposition 10(i), the universal r-form of the Hopf *-algebra A = 0(Gq ) is inverse real. It is easily checked that the Hopf algebra À = A 0 CZ becomes a Hopf *-algebra with the involution defined by (a0xn)* := a* 0 xn, a E A, n E Z, and that X is a right _A1, the universal r-form r), of A is comodule *-algebra. Since IA = also inverse real. Therefore, by Proposition 45 and the subsequent remarks, the Hopf algebra 0(IGq ) = A D.< X is a Hopf *-algebra with respect to the involution determined by the equations (14)* = i4 (x i )* = x i and x* = x. From now on we suppose that q and z are real and we turn to the real forms Uq (N), SUq (N;v), 0 q (N;v) and Sp q (N; v) treated in Case 2 of Subsects. 9.2.4 and 9.3.5. Then the universal r-form r, of A is real (by Proposition 10(ii)) and Proposition 45 does not apply in general. In order to point out the difficulties that occur in this case; let us consider the Hopf *-algebra A = 0(0q (N ; 0) with vi = • = UN = 1 and its right comodule *algebra X = O(0(1; v)); see Subsect. 9.3.5, Case 2. As in the inverse real case above, we define the involution on A by (a xn)* := a* 0 xn. Then A is a Hopf *-algebra and X is a right sa-comodule *-algebra. The universal r-form r), of A is real, because A = zq -1 is real. However, there is no involution of the algebra A y X which extends the involutions of and X. Assume on the contrary that such an involution exists. Recall that by .definition ()* = 8(u-ii ) and (xn )* = q - P-sni. Applying the in,

volution to the equation xnuii = A nkli ulxi, then multiplying by tir on the left and by t4 on the right and finally summing over j and i, we get sniq-Pnuran = Atturxilq -P1 - Comparing the latter with the relation

10.3 Braided Hopf Algebras

xretesn = A 37.41,18 urx1i (note that the elements urx/t

391

ur 0 1 0 xi/ E A 0 X

kt8li en- =

for all j,1, n, s. This are linearly independent), we obtain contradicts formula (9.30). Thus, the involutions of the *-algebras A and X cannot be extended to an involution of the algebra A y X. One way to circumvent this difficulty and to still get a Hopf *-algebra is to "double" the generators of the translation part. This procedure will be carried out in the rest of this subsection. Let XR, denote the algebra X( f , g, h, 7; R) from Subsect. 9.1.2 with f (t) g(t) = t - q, h(t) = qt, = 0. That is, XRe has 2N generators xi, • • •/ yN and the defining relations (9.11) of XRe read as y17 • • • 7

P lk

Irki ii XkXi = (1XiX j

qyiyi,

Pi k

xkyi = q

-1

YiX j •

(119)

Let X and Y denote the algebras with generators • • , xN resP• yi, • • • , Y N and defining relations klxkx/ = qx jx j resp. Wykyt = qyiyi• Clearly, X and 3) are right A-comodule algebras with coactions given by the formulas çoR (xi ) xj x and ço R (yi ) = yi S(uii ) 0x -1 . Then the algebra XR, is nothing but the braided product algebra X®Y with product defined by (94) with respect to the universal r-form (i.) 21 of „4, where î is given by (106) and is the convolution inverse of f. Indeed, the cross commutation relations of the braided product algebra are Yij

= xicYi (021(S(t4) x -1 , ui; x) = qz -l xkY/

uit ) = qkixkY/ •

This is just the third group of relations in (119). The algebra XRe becomes a *-algebra with involution determined by x i* := viyi, where vi are the numbers in the definitions of St/q (Ar, v), 0q (N; v) and Spq (N; v) and vi := 1 in the case of Uq (N). To prove this assertion, it suffices to check that the relations (119) are invariant under this involution. Using the facts that ki = this is easily done. In the special case = • • • = vN = 1 the elements x i and yi are interpreted as holomorphic and anti-holomorphic generators, respectively, and the *-algebra XRe may be considered as a realification of X (see Subsect. 10.2.7). Obviously, the algebra A is also a *-algebra with involution (a a* 0 a E A, n E Z. Recall that, by Proposition 9.5, XRe is a right quantum space of A. Hence XRe is also a right quantum space of A with coaction çoR given by

ki

(PR(xi) =

0

u3i

X/ 'PR(Yi) = yj 0 S(u) 0 x -1 .

From the relations x = viyi ,.(q)* = vi vj S(uii ) and x* --= x' we conclude that çoR(x) = (p R (si )*. Therefore, XRe is a right À-comodule *-algebra. Next we turn XRe into a right À-module algebra. Since i and (ii )21 are universal r-forms of it follows from Proposition 4 that X and y are right À-module algebras with actions defined by

392

10. Coquasitriangularity and Crossed Product Constructions

s V by Ai(ei x) = 6iix, x E V. Then the restriction Bi of Ai to /C0 belongs to Mor (îP, cp), so Bi is either zero or bijective by Schur's lemma. Since ko {0}, Bk 4 0 for some k. Then BiBk-1 E Mor (ç9), so that BiB k-1 = Ai/ with A i E C again by

op)

11.1 Corepresentations of Hopf Algebras

401

Schur's lemma. Hence any nonzero y E Ko is of the form y = Ei Bi (y) O. Since IC is invariant (Ei Ai e'i ) Bk(y) =: x with x' O and x under (p R and (pL , it follows that IC D x f V and iC D VI x 7 so /C =-V' V and T = O. The latter implies that = 0, which is a contradiction. Thus, we have proved that IC = ker T = 101. Hence T is bijective and provides an equivalence of a direct sum of n corepresentations ço and the corepresentation 2i e ( 1 ). Moreover, the elements vii constitute a basis of C(cp). Therefore, the dual algebra C(cp)' is isomorphic to Ma (C) (in fact, the dual basis to {yii } corresponds to the matrix units). Thus, C((p)' is simple and hence C(cp) is cosimple. Corollary 9. If c,o and V) are irreducible corepresentations of A, then either C(cp) = C(tP) and ço and 1,b are equivalent or C(v) n C(p) = {0} and cp and V)

are not equivalent. {0}. Since C(cp) n c p) is a subcoalgeProof. Assume that C(o) n bra of the cosimple coalgebras C(c,o) and C(P), it follows that C(v) = Thus, 2ic ( ip) and 2ic(p) are equivalent corepresentations, hence cp and 1,b are equivalent by Proposition 8(ii). (

Corollary 10. Let {çoa

Ic

E I} be a family of pairwise nonequiva-

lent irreducible corepresentations of A with matrix corepresentations {v = E I}. Then the set {v i j = 1, 2, • • • , dcr , a E I} of matrix elements is linearly independent. ,

Proof. The proof follows from Proposition 8(ii) and Corollary 9. 11.115 Unitary Corepresentations In this subsection we suppose that A is a Hopf *-algebra.

Let V be a finite-dimensional Hilbert space with scalar product (•, •) conjugate linear in the first variable and linear in the second one. Then L(V) and L(V) 0 A are *-algebras. Let cp : V V 0 A be a corepresentation of A on V and let y = (vii ) be the corresponding matrix corepresentation with respect to an orthonormal basis fed of V. Set y* ((e) ii ) = ((v3i)*). Proposition 11. The following conditions are equivalent: (x(i)) * x, y E V. (i) E(x, NON') = E(x(o), (ii) E(x ( 0 ) , y(0))x(i ) * y( i) = x, y E V. (iii) * = * I in the *-algebra L(V) A. (iv) `0*(,O = I (resp. cPc,a* = I) in the *-algebra _C(V) 0 A. (v) v*v = vv * = I (that is, S(v ii ) = evii r for all i, j). (vi) v*y = I (resp. vy* ).

Proof. If eki E L(V) are the matrix units defined by eki(ei) = bikei, then (P can be written as = E i eii vii. From this we immediately obtain that (Hi) (v) . and (iv) (vi). Setting x = ei ,y = e• into (i) yields vij = so v*v = vv* = I. Conversely, the relations v u = S(vii)* imply (i). Thus,

402

11. Corepresentation Theory and Compact Quantum Groups

(v). Similarly, (ii) is equivalent to the equality vv = I. Multiplying (i) v*v = I (resp. vv* = I) by 8(v) on the right (resp. on the left), we obtain v* = 8(v) and so vv = vv* = I. This proves (N) 4-+ (vi). Definition 5. A corrpresentation cp of A on a finite-dimensional Hilbert space V is called unitary if it satisfies the conditions in Proposition 11. A corepresentation cp on a finite-dimensional vector space V is said to be unitarizable if there exists a scalar product on V such that (p is unitary.

Let v =(vii ) i,j , i ,...,n, be the matrix corepresentation of a corepresentation (p of A on a vector space V. Then y is unitarizable if and only if there exists an invertible matrix A E Mn (C) such that w := AvA. -1 satisfies the equality S(w) = w* or, equivalently, if BS(v)B -1 = v* with B = A t A. It is clear from Proposition 11 that direct sums, tensor products and subcorepresentations of unitary (resp. unitarizable) corepresentations are again unitary (resp. unitarizable). However, contragredient and conjugate corepresentations of unitary corepresentations are not necessarily unitary. Proposition 12. Any unitarizable (in particular, any unitary) corepresentation is completely reducible, that is, it is a direct sum of irreducible corepresentations.

Proof. It suffices to show that for a unitary corepresentation y the orthogonal complement W I of a y-invariant subspace W in V is also y-invariant. To prove this, let x E W and y E WI. Write (p(y) = yi ai such that fail is a linearly independent set. Since W is y-invariant and y E , the right hand side of the equation in Proposition 11(i) vanishes, so E(x, yi )ai = O. Hence, (x, yi ) = 0 for arbitrary x E W which implies yi E

11.2 Cosemisimple Hopf Algebras This section is concerned with Hopf algebras which are spanned by the matrix elements of their irreducible corepresentations. 11.2.1 Definition and Characterizations

First we put the subsequent considerations in the proper algebraic context and give Definition 6. A coalgebra is called cosemisimple if it is the sum of its cosimple subcoalgebras (that is, of coalgebras which have no nonzero proper subcoalg eb ras

Example 1. Any group algebra CG is obviously cosemisimple.

A

Example 2. The universal enveloping algebra U(9) of a Lie algebra g {0} is not cosemisimple, because its only cosimple subcoalgebra is C•1.

A

11.2 Cosemisimple Hopf Algebras

403

Example 3. For the Hopf algebra of polynomial functions on an algebraic group G cosemisimplicity means that G is linearly reductive, that is, the finite-dimensional polynomial representations of G are completely reducible (see [Ho], Sect. V.3). A large number of classical groups such as SL(N, C), SO(N, C), Sp(N, C) has this property. A Example 4. If q is transcendental, the Hopf algebras 0 (G L q (N)) , 0 (S L q (N)) 0 (0q ( N)) , 0 (Sp q (N)) are cosemisimple (see Subsects. 11.2.3 and 11.5.3). A

A be a coalgebra. Denote by Â. the set of equivalence classes of irreducible corepresentations of A. For any a E A , we fix a corepresentation (,oa Let

from the class a and a basis of the underlying space. Then (pa is given by a matrix corepresentation ua =-- (UZ)i,j,_1,...,d a,. Recall from Subsect. 11.1.4 that irreducible corepresentations are always finite-dimensional and that the coefficient coalgebra C(cp') = C(P) = Lin fuZ I j, j = 1, 2, • • • , c/} depends only on the class a, but not on the choice of the representative (pa E a and of the basis. Definition 7. A linear functional h on A is called left-invariant (resp. rightinvariant) if for all a e A,

(id 0 h)A(a) = h(a)1 (resp.

(h

id)A(a) = h(a)1).

(1)

Equation (1) means that h intertwines the left (resp. right) comodule A with respect to the comultiplication and the trivial comodule C - 1. Clearly, a functional h on A is left-invariant (resp. right-invariant) if and only if f h = f (i.)h (resp. h f = f (1)h) for all functionals f e A'. In the Hopf algebra literature such functionals h are called left (resp. right) integrals on -

A

(see [Mon], Definition 2.4.4). A number of characterizations and properties of cosemisimple Hopf algebras are collected in the following theorem. Theorem 13. For any Hopf algebra A the following statements are equivalent:

(i) A is cosemisimple. (ii) Every corepresentation of A is a direct sum of irreducible corepresentations. (Hi) There exists a left-invariant linear functional h on A such that h(1)=1. (iv) There exists a unique left- and right-invariant linear functional h on A satisfying 141) =1. (y) A is equal to the sum of all C(u), a E A. (vi) The set {uZ I a G A, j=1,2, • -•,d 4,} is a vector space basis of A. If A is a cosemisimple Hopf algebra, then the antipode S of A is bijective and we have S 2 (C(u")) = qua) for any a G Â.

Proof.

The equivalence of (i) (iv) is Theorem 14.0.3 in [aw]. By Proposition 8(1), the cosimple subcoalgebras of A are precisely the coefficient spaces —

404

ii. Corepresentation Theory and Compact Quantum Groups

C(0), c E Â. Hence we have (i)4-qv). (v)—(vi) follows from Corollary 10. (vi)— ( v) is obvious. The last assertion is proved in [La] , Theorem 3.3. The crucial implications of the theorem will be proved in the next subsection. ID Let A be a cosemisimple Hopf algebra. The unique left- and right-invariant linear functional h such that 1 ( 1) = 1 from Theorem 13(iv) is called the Haar functional of A. By (v) and (vi), we have

A =

e

aE..4

(2)

C ( ua).

The decomposition (2) is referred to as the Peter-Weyl decomposition of A.

11.2.2 The Haar Functional of a Cosemisimple Hopf Algebra The existence and uniqueness of the Haar functional is a crucial property of a cosemisimple Hopf algebra. Because of its importance we carry out the proof of the relevant implication (vi) --+ (iv) of Theorem 13 and obtain additional information in this way. (In the above proof we referred only to [Sw].) Suppose that (vi) is fulfilled. Then A is the direct sum of the coefficient coalgebras C ( P). Let a = 1 denote the trivial one-dimensional corepresentation of A with matrix element 1. We define a linear functional h on A

by h(1) = 1

and

h(a) = 0 for a E C(0), a E

:Al a

I•

(3)

Since ,I(C(u)) C C(0) 0 C(0) for each a E Â, h is obviously left- and right-invariant. This proves the existence assertion of (iv). For the uniqueness, let h denote a left-invariant linear functional on A such that h(1) = 1. Then we have E awit(a(2)) = h(a)1 for a E C(0). Since LX(a) E C(0) g C(ua) and 1 C(u) if a 1, it follows that iii(a) := 0 for a E OM, a 1. That is, h is of the form stated in (3). The same reasoning proves the uniqueness of a right-invariant linear functional h on A satisfying /41) = 1. This finishes the proof of the implication (vi) (iv) in Theorem 13. Since the antipode S of A is injective, one easily verifies that h(S) := WO) is also a left-invariant functional on A. Since h(1) = 1, the uniqueness of h yields h = h. That is, we have

h(S(a)) = h(a),

a C A.

(4)

Note that the condition 141) = 1 in Theorem 13(iv) is essential. Indeed, for any finite-dimensional Hopf algebra there exists a unique (up to complex

multiples) left-invariant linear functional hi (see [Mon], 2.1.3), but hi is not right-invariant in general (see Example 5). But if kW 0, then the Hopf algebra is cosemisimple by Theorem 13(iii) and hi is right-invariant as well. Example 5 (Sweedler's Hopf algebra). Let us retain the notation of Examples 1.9 and 10.2. One easily verifies that the functionals 140 := (x + gx,.) and (x — gx, -) are left- resp. right-invariant on A. Note that h1(1) = 14)

11.2 Cosemisimple Hopf Algebras

405

hr (1) = 0. If A would possess a Peter-Weyl decomposition (2), then we have hi(a) = h r (a) for a E C(0) by the above reasoning and hence hi = hr . Therefore, A is not cosemisimple, because hi hr . In the rest of this subsection we assume that A is a cosemisimple Hopf algebra. Let h denote the Haar functional of A. Our next aim are the Schur type orthogonality relations (7) and (8) below. A key step for this is Lemma 14. If v = (NJ) and w = (wii ) are matrix corepresentations of A, then

E h(vii s(w,n)),,, =

En

( 5)

Vinh(VnjS(Wki)),

E h(s(wkov) % = E wkn h(S(wra)Yii)).

(6)

Setting iLl 3n''k) := h(viiS(wk n )) and k l 'i) h(S(wkOvi n ), the equations (5) and (6) can be written as A ( I'k) w = vA 5,k ) and B( 1,i)v = wB( 1,i) , respectively. That is, A ( i , k ) E Mor (w, y) and B (1,0 E Mor (y, w). Proof. By the left invariance (1) of h, we have

h(viiS(wkn ))1 = (id 0 h)(vii8(wkn)) =

h(yri S(wks ))yir S(wsn ).

Inserting this into the left hand side of (5) and using the identity S(wsn)wni = 6 .9/ 1 , we obtain (5). Applying the right invariance (1) of h with a = S(wki)vin) (6) is derived in a similar manner. Proposition 15. Let v = (vii ) and w = (wki) be irreducible matrix corepresentations of A. Then we have:

(i) If v and w are not equivalent, then for any j, j,k,l, h(v ii S(wki)) = h(51 (wki)v ii ) = 0.

( 7)

w is equivalent to its bicontragredient corepresentation w". If F = (Fin ) is an invertible intertwiner of w and w", then Tr F 0, Tr F -1 0

and h(wk/S(wii)) =

Tr

F

and

h(S(wki)w 2,3 ) = (5k3

(F-1)i1

Tr F -1-

(8)

,Proof. (i ): Since y and w are not equivalent, their intertwiners AO' k) and are zero by Schur's lemma (Proposition 6(0). This gives (7). (ii): First we apply Lemma 14 in the case y = w. By Schur's lemma (Proposition 6(ii)), the intertwiners A ( i , k ) and B (1,i) are scalar multiples of the identity, so there are complex numbers aik and Oti such that A U' k) = h(wiiS(wki)) = aik6ii

Setting j

and B k(13:4) = h(S(wki)wii) =

1, j = k and summing over j and j, respectively, we obtain

(9)

406

11. Corepresentation

Theory and Compact Quantum Groups E aii = E 022=1.

z

(10)

Next we apply Lemma 14 with y = wcc. Then yii = S2 (wii ). Combining (5) and (6) with (4) yields E h(wkns(wi i ))wrii=E 82(wirt)mwo(wni )),

‘--dn

h(s(w inwk1 )s2(wni ) =

wk n h(S(w ii )w ni).

(11)

(12)

With Cle ) := h(WknS(Wij)) and D k( in'i) := h(S(win)wki) the equations (11) and (12) mean that C (i' k) E Mor (w,wcc) and D (l 'i) E Mor (w", w). Moreover, by (9), we have Ci(ni'k) = an ibkj and .13)c( in'i) = finkbli which gives (8) for k j. From (10) together with the preceding expressions we obtain Tr C (i,j ) = Tr = 1. Since the antipode S is bijective by Theorem 13 and w is irreducible, wcc is irreducible as well. Because CU'i) E Mor (w, tucc) is nonzero, w and wcc are equivalent by Schur's lemma. Let F be an invertible element of Mor (w, wcc). Since such an F is unique up to a complex factor (again by Schur's lemma), there are complex numbers ai and Oi such that ai F = C(i,i ) and f3i F -1 = D(i , i). Thus, ai Tr F = Tr CUM = 1 and 3Tr F' =Tr B(i ,i) = 1, and (8) for k = j follows.

Let us restate formula (8) in the case where S2 = id. Recall that, generally speaking, all Hopf algebras coming from groups have this property. Then we have wii = ( w "), so (8) applies with F = I. If d, is the dimension of the representation space of w, then (8) reads as h(wk/S(w ii )) = h(S(wki)wii) = bki bi/c/V.

Next we apply the Schur type orthogonality relations (7) and (8) to the study of characters. Definition 8. Let cp be a corepresentation of A on a finite-dimensional vector space. The element x,p := (Tr 0 id» of A is called the character of cp. If (vii ) is a matrix corepresentation of cio , then x‘p = Ei vii . For any f E A', we have f•x v, = x(p .f,, where f .x,i; and x,p .f are given by (1.65). Note that T O 46(X(p) = 46 (X49) x(Pelb = 740 ± )(lb X(p00 = Xy0X10 and Xwe = Moreover, x,p = xv, if y; 0. Of course, Definition 8 and the preceding simple facts are valid for any Hopf algebra A. Let x,„ a E A, denote the character of 0. An immediate consequence of the formulas (7) and (8) is Corollary 16.

h(x„S(xs)) = h(8(x4 ) xs) = b as for a, E A.

Proposition 17. Let c,o be a corepresentation of A on a finite-dimensional vector space V. For a E Â, we set na := h(S(xa)x 0 for all a C A, a

O.

Proof. (i) (ii) is trivial. (ii) —> (i): Let fuil be a set of finite-dimensional unitarizable corepresentations such that their matrix elements generate A as an algebra. Let R. be the smallest family of corepresentations of A which contains all u i and is closed under direct sums and tensor products. These operations preserve unitarizability, so all corepresentations in R. are unitarizable. Since c ((,0 zp) = ( w ) + c (' p) and C(ço zP) = C(v)C(0), the union of C(v), is an algebra which contains all C(u). By assumption (ii), this algebra is A. For any (p C R. there is a unitary corepresentation ço such that CM = C(ç). Hence A is a CQG algebra. (iv) —> (Hi): The proof is an adaption of the proof for the corresponding result on representations of compact groups. Let (p be a corepresentation of A on a finite-dimensional vector space'V. Let (., .) be an arbitrary scalar product on V. Define another Hermitian sesquilinear form (., •) on V x V by (x, y) :=

(x(0) , y(0))h(x( 1 )*y( l )),

x, y G V.

Let v be the matrix elements of (it) with respect to an orthonormal basis {xi } of (V, (a, -)). Then, we have (xi , x i ) = SO

E k,I (xk ,x0h((vki)*vii ) = E

h((Vkir Vki),

11.3 Compact Quantum Group Algebras

417

0, the left hand side vanishes only for any ai E C. Since h(a*a) > 0 for a when ajVki = 0 for all k. But then ak = e(E i ctivki) = 0 for all k. Hence (-, .) is a scalar product on V. Using the right invariance of the functional h, we conclude that

Ei

E(x (0 ) ,y(0 ) )X 0) *y ( 1 ) =E(x (0) ,(0 ) )h(x ( i ) * N ox (2 ) *y (2 ) ___,_,x (o) ,y (o) )(hoz id)A( (1) * y(0 )=E(x (o) , y(0) )h(x (i) * y( 1 ) )1 .E(x

Therefore, by Proposition 11, the corepresentation c,o is unitary with respect to the scalar product (. 7 .). (iii) 4 (i ): Let a E A. By Proposition 4, there is a finite - dimensional subcoalgebra C containing a. The restriction c,o of z to C is a finite-dimensional corepresentation of A and hence is unitary with respect to a certain scalar product on C. Since (e id)(p(a) =-- a, we have a E C(cp). The proof of the implication (i ) -4 (iv) is given in Subsect. 11.3.2 along with the proof of Proposition 29 below. -

Proposition 28. For a Hopf *-algebra A, the following are equivalent:

(0 A is a CMQG algebra. (ii) Them is a finite-dimensional unitary compresentation 'a of A whose matrix elements generate A as an algebra. (iii) There is a finite-dimensional corepresentation u of A such that u and uc are both unitarizable and the matrix elements of u and uc generate A as an algebra. Proof. (i)--)(ii): Take a finite set of generators of the algebra A. Since A is a CQG algebra, each generator is a linear combination of matrix elements of finite-dimensional unitary corepresentations of A. The direct sum of these corepresentations has the desired property. (ii)--(i) follows by the proof of implication (ii)-40 of Theorem 27 applied to a singleton ful. (ii)-4(iii): By the implication (ii)-4(i), A is a CQG algebra, so uc is unitarizable by Theorem 27(iii). (Here the more involved part of Theorem 27, which contains the implication (i) --> (iii), is needed. This can be avoided by an alternative proof based on Proposition 12.) (iii) —(i) is obvious.

Let A be a CMQG algebra and let u be a corepresentation of A as in Proposition 28(iii). By choosing an appropriate basis, we can assume that u is given by a unitary matrix corepresentation u = Then we have u*u = uu* = / and hence uc --= j Since uc is unitarizable by Proposition 28(iii), there is an invertible matrix A E MN(C) such that v := Auc.A.' is a unitary matrix over A. Setting E := A* A, one immediately checks that the relations vv = vv* = I are equivalent to the matrix equations ut EfLE -1

L

(23)

That is, any GANG algebra A is generated as a *-algebra (I) by the entries of a unitary matrix u = (uii)i,j=1,...N which satisfies the equations (23). Further,

418

11. Corepresentation Theory and Compact Quantum Groups

if V E MN(C) is a unitary matrix, then ii := Vt ul7 is another unitary matrix corepresentation of A such that the entries of el generate A as a *-algebra and 71 satisfies (23) with E replaced by .t := VEV -1 and u by ft. By the spectral theorem, the matrix È is diagonal with positive numbers on the diagonal for some suitable choice of V.

We close this subsection by considering some examples.

Example 6 (Universal CMQG algebras). Let E E MN(C) be an invertible positive Hermitian matrix. Let A(E) denote the free *-algebra with generators j = 1, 2, • • , N, subject to the 4N 2 relations u*u = uu* = I and (23), where u (u ji), u (Fiji ) := ((u ii )*) and u* := fit . Then A(E) is a

CMQG algebra with comultiplication given by Z1(u ii ) = uik 0 uki •

Proof. The free algebra qu ii , vii ) with 2N 2 generators u ii and vii is a bialgebra with comultiplication determined by L(u) = u i k 0 uki and ZI(vii ) = vik (gvki. Let J be the two-sided ideal of quii , Yii ) generated by the elements corresponding to the relations vu = uvt = utEvE -1 = EvE - lut = I, where u = (uii ) and y = (vii ). As in the proof of Proposition 9.1 we verify that J is v ii ) I J is a bialgebra. One easily checks that the a biideal, so that A := elements vii, v := (E -l ut E) ii of A satisfy the defining relations of the opposite algebra A°P. Hence there exists an algebra anti-homomorphism S : A A such that S(u) = and S(vii ) = v. Since S(u)u = uS(u) = I and S(v)v = vS(v) = I by construction, it follows from Proposition 1.8 that S is an antipode for the bialgebra A. Thus, A is a Hopf algebra. Clearly, C(u ii , vii ) is a *-bialgebra with respect to the involution given by u;ri = vii . Since the matrix E is Hermitian, .7 is a *-ideal and hence A is a Hopf *-algebra. Obviously, as a *-algebra, A is just the *-algebra A(E) defined above. From the defining relations u*u = uu* = I and (23) E1/2,Et-E1/2 it follows that u and w := Ei/2 ucE -112 are unitary matrix corepresentations. Since w uc, the corepresentation uc is unitarizable. Since the matrix entries of u and uc = ft generate A as an algebra, the Hopf algebra A is a CMQG algebra by Proposition 28(iii). CI From the remarks preceding this example it follows that any CMQG algebra is a quotient of some CMQG algebra-A,,,,(E) for a certain diagonal matrix E with positive diagonal entries. That is, the CMQG algebras A(E) are universal CMQG algebras. Note that the CMQG algebras A(E) are very large in general, because no relations among the entries u ii are required. A Example 7. Each of the Hopf *-algebras 0 (Uq (N)), O(SU q (N)), 0(0, 7 (N ; R)), 0(S0q (N; R)) and OW Spq (N)) (see Subsects. 9.2.4 and 9.3.5 for their definitions) for q E R is a CMQG algebra. In these cases condition (ii) in Proposition 28 applies, since the fundamental corepresentations u are unitary (recall that uu*, = I by the definitions of the involutions). Except for Uq (N), the matrix elements u generate the corresponding algebras. The algebra 0(Uq (N)) is generated by the matrix elements of the unitary corepresentation u631(;7 1 ). A Recall that Dq-1 is a unitary element of the *-algebra 0(Uq (N)).

11.3 Compact Quantum Group Algebras

419

11.3.2 The Haar State of a CQG Algebra Because any CQG algebra is a cosemisimple Hopf algebra, it possesses a unique Haar functional. In this and the next subsections we are mainly concerned with the Haar functional h of a CQG algebra A. Recall that a linear functional f on a *-algebra A with unit is called a state if f(1) = 1 and f (a* a) > 0 for a E A. A state f on A is called faithful if f (a* a) > 0 for all nonzero a E A. Proposition 29. The Haar functional h of a CQG algebra A is a faithful state on the *-algebra A. The key step for the proof of this proposition is the following Lemma 30. Let w be an irreducible unitary corepresentation of A on a (finite-dimensional) Hilbert space W. Then there exists a unique invertible intertwiner F E Mor (w, w") such that Tr F = Tr F-1 > O. This operator F is positive definite, that is, (Fx,x) > 0 for all x c W, s O.

Proof. We fix an orthonormal basis of W and consider w and w" as matrix corepresentations. Since wc is also irreducible, there exist an irreducible unitary corepresentation y of A and an invertible complex matrix A such that = A - lyA. Let  denote the matrix with entries A. Since y and w are unitary, we have (y9 ii = S(vii ) = vii*, (wc) ii wii * and (wcc) ii = S((wc) ii ). Therefore, we conclude from yA = Awc that ycA = Aw and Atyc = wCCAt and so AtAw = wccA t il. That is, At A E Mor (w, w"). Since A is invertible, AtA is positive definite. Since dim Mor (w, wcc) = 1, F E Mor (w, wcc) is uniquely determined by requiring that Tr F = Tr F-1 > O. Let A be a CQG algebra. By the second reformulation of Definition 9 given in Subsect. 11.3.1, we can choose a unitary corepresentation ua in each equivalence class a E A. Moreover, let us fix Fa, G Mor (ua ) ) such that Ma := Tr Fa = Tr F;1 > O. Since (4)* == S(4,3) and (4)* = S(4) by the unitarity of u)3 and ua, (7) and (8) can be rewritten as

h(uck'i (u)*) = 5,035kiMaT i (Fa)ii,

h((urk )* 14) = 84,06kimc-g-1 (Fa— I)

(24) (25)

These two equations are referred to as the Schur orthogonality relations for the CQG algebra A.

Proof of Proposition 29. Since A is a CQG algebra, any element a G A is a finite sum Acrij uZ with Actii E C. By (25), we have

h(a* a) =N--`

/ki 4—s a,,,

AcitikAalkM (F ct—i )il.

Since F;' is positive definite as well, h(a*a) > 0 and h(a* a) -= 0 implies that all coefficients Accik are zero, so a = O.

420

11. Corepresentation

Theory and Compact Quantum Groups

Because of Proposition 29, the Haar functional of a CQG algebra will be called the Haar state. The Haar state is a fundamental tool for the study of CQG algebras. It was essentially used in the previous subsection in order to prove the important result that any finite-dimensional corepresentation of a CQG algebra is unitarizable (and hence completely reducible). Also, the character theory in Subsect. 11.2.2 was based on the Haar functional. Other applications will follow below. Since the Haar state h is a faithful state on A, the equation

(a, b := h(a* b) ,

a, b E A,

defines a scalar product (., •) on A. Let 1-1(A) denote the Hilbert space completion of the pre-Hilbert space (A, (., .)). The left and right invariance of h are expressed in terms of the scalar product (., •) as (a, b) =

E 0, (1) *b(1) (a(2) b (2 ))

=

(a( i ), b ( o)a(2)*b (2)

a, b E A.

Equation (25) shows that for a [3, a, @ E Â, the coefficient coalgebras C(L') and C(u0 ) are orthogonal subspaces and the characters x,, and x0 are orthogonal elements of li(A). Therefore, for any CQG algebra A the Peter— Weyl decomposition (2) is an orthogonal sum in the Hilbert space 11(A).

Example 8. Let G be a compact topological group, j.t the Haar measure of G and A the CQG algebra Rep (G), see Example 1.2. Then the Haar state h on A is given by h(a) = f a(g)dp(g), a E A, and MA) is just the Hilbert space L 2 (G,p) of square integrable functions on G with 'respect to /I. The latter follows from the Peter—Weyl theorem which asserts that Rep (G) is dense in L 2 (G, 11.3.3 C*-Algebra Completions of CQG Algebras

In this subsection we show that each CQG algebra A admits a completion to a C*-algebra. In order to do so, we extend the considerations of the first two paragraphs of Subsect. 4.3.4 to general CQG algebras. First let us recall some standard terffiinology (see, for instance, [BR] or [Mu]). A C*-seminorm on a *-algebra A is a seminorm p which has the so-called C*-property p(a* a) = p(a) 2 , a E A. It can be shown that any C*seminorm p satisfies p(a*) = p(a)* and p(ab) < p(a)p(b) for a, b E A and *1) = 1 if p O. A C*-norm is a C*-seminorm which is a norm. A C*algebra is a *-algebra A equipped with a C*-norm 11'11 such that the normed linear space (AA II) is complete. A *-representation of a *-algebra A on a Hilbert space 7-1 is a *-homomorphism of A into the *-algebra of bounded linear operators on 1-1. We née-d a simple technical lemma. Lemma 31. Let V be a dense linear subspace of a Hilbert space with scalar product (.,.). Suppose that 7r is an algebra homomorphism of a CQG algebra

11.3 Compact Quantum Group Algebras

421

A into the algebra of (not necessarily bounded) linear operators on V such that (71- (a)x,y) = (x,r(aly) for all a E A and x, y E V. (i) Let y = (vii ) be a unitary matrix corepresentation of A. Then each linear operator 7r(vii ) on V is bounded and 117r(vii )11 5_ 1. (ii) 7r extends uniquely to a *-representation of A on Proof. (ii) follows immediately from (i). To prove (i ), recall that vv r= so E k Vki * Viej = 1 by Proposition 11(v). Hence, for x E V we have

I and

117r(vii )x11 2 <E k (7r(vk i )x,7r(vki )x) = E k (7r(vk j *vki )x,x) = 1411 2 .

El

We now apply Lemma 31 to the case V = A, H = H(A), 7r = 7rh, where rh is defined by 7rh(a)b = ab, a, b E A. Since (7rh(a)b, c) = h((ab)* c) h(b* a* c) = (b, h (a* )c) for a, b E A, 7rh satisfies the assumptions of Lemma 31. Hence there exists a unique extention of 7rh to a *-representation, denoted again by 7rh, of the CQG algebra A on the Hilbert space Ii(A). The *representation 7rh of A on H(A) is called the GNS representation of the state h (see [BR] or [Mu]). From the definition it is clear that rh(A)1 is dense in the Hilbert space IAA) and that h(a) = (7h(a)1, 1) for a E A. These two properties characterize the GNS representation up to unitary equivalence. Moreover, T-h(a) = O implies that a = O. (Indeed, rh(a) = 0 yields h(a*a) = 1lrh(a) 1 11 2 = 0, so a = 0 by the faithfulness of h.) Hence 117rh•)11 is a C*-norm on A. Proposition 32. Let A be a CQG algebra. For a E A, let Majj„,3 denote the supremum of p(a) for all C* -seminorms p on the *-algebra A. Then, 11 • II 00 is a C* -norm on A and the completion of A with respect to this norm is a C* -algebra A with unit element. The counit c and the Haar state h of A have unique continuous extensions, denoted by the same symbols, to the C* -algebra A. The comultiplication of A can be uniquely extended to a *-hornomorphism A : A -4 MM. Here A(gdzi denotes the C* -algebra obtained by completing the algebraic tensor product A g A in the least C* -norm on A 0 A. These extensions satisfy the relations (A 0 id) o A = (id 0 A) o zl on AAA, 0 id) o L = (id 0 6) o = id and (h id),6 = (id h )Z1 = hi on A. (c

Proof.

First we have to show that 011,0 is finite for all a E A. Since A is a CQG algebra, it suffices to do so for a = vii , where vii is a matrix element of some unitary matrix corepresentation of A. Let p be a C*-seminorm on A and let ip denote the canonical mapping of A to the quotient *-algebra A/ker p. Then p(ip (-)) := p • ) defines a C*-norm on A/ker p. Let .Ap be the C*-algebra which is the completion of A/Icer p with respect to p. The C*algebra flp has a faithful norm preserving *-representation (see [ 1311], 2.3.4), say frp , on some Hilbert space. lip . Then rp = ïrp o ip is a *-representation of A on lip . By Lemma 31(i), 117rp (vij )11 1 and hence p(vii ) =- 75(ip (vii )) = II Trp (ip (vii ) )11= II7rp (vii ) if _Ç 1. Thus, II vii 11,, 5_ 1 and II • II co is finite on A. Since if wh011 _is a C*-norm on A as noted above, • 11,,„ is a norm, hence a C*norm, on A. Clearly, the completion of (A, II 1100) becomes a unital

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11. Corepresentation Theory and Compact Quantum Groups

C*-algebra A. Let II • denote the least C*-norm of A 0 A (see [MO. Then 114A(•)iimin is a C*-seminorm on A. Also, WI is a C*-seminorm on A and Ih(•)1 1lrh (-)11. Therefore, II • Hoe) le(*)1 It • 1100 and I h()I < P • 1100 by the definition of II li c,Q . Since 4 e and h are continuous with respect to the norm • he) they have unique continuous extensions to A. The algebraic relations are preserved by continuity. Remarks: 5. The antipode of A is not necessarily bounded with respect to the norm • li„„, so it cannot be extended to the C*-algebra A in general. 6. Since • II 00 is the largest C*-seminorm on A, each *-representation of A on a Hilbert space has a continuous extension to a *-representation of the C*-algebra A. This property characterizes the C*-algebra A in Proposition 32. However, the largest C*-seminorm is not the only possible choice of a C*-norm on A for getting a C*-algebra completion . Â of A such that the comultiplication of A extends continuously to di (see also Subsect. 11.4.1). Further, it should be mentioned that in general there are several inequivalent C*-norms on the algebraic tensor product A 0 A (see [MO.

11.3.4 Modular Properties of the Haar State Throughout this subsection let A be a CQG algebra with Haar state h. A linear functional f on A is called central if f (ab) = f (b a) for a, b E A. It is called a character if f(1) = 1 and f (ab) = f (a) f (b) for a, b E A. We shall see below (Corollary 35) that the Haar staté h on A is not central if the square of the antipode of A is not the identity. In this subsection we investigate a family of characters and of related automorphisms of A which provide qualitative measures of the extent to which both properties fail. First we state a preliminary result on holomorphic functions. Let us say that an entire function g is of exponential growth on the right half-plane if there exist constants M > 0 and u E R such that Ig(z) l < MeARe Z for any z E C, Re z > O. Lemma 33. If g i and g2 are entire functions of exponential growth on the right half-plane such that gi (z) = g 2 (z) for z E N, then gi(z) = g 2 (z) for all z E C. Proof. This is in fact Carlson's theorem (see, for instance, [Ta]).

0

Proposition 34. There exists a family If zlzEc of characters on the CQG algebra A such that for all z, z' E C and a,b E A the following conditions are fulfilled: (i) The function z f(a) is an entire function of exponential growth on the right half-plane. (ii) fzfzi = fz+z/ and fo = E. (iii) fz (1) = 1. (iv) fz (S(a)) = f_(a) and fz (a*) = f_2-(a).

11.3 Compact Quantum Group Algebras

423

(v) S2 (a) = where is defined by (1.65). (vi) h((fz-i.a.fzi---1)0= h(b(fz.a.fzI)). In particular, (26)

h(ab) = h(b(fi.a.fi))-

The linear functionals h, z G C, are uniquely determined by the (0, (II) and equation (26).

conditions

Proof. Recall that by Lemma 30 the intertwiners Fa G Mor (ua, (u1") defined in Subsect. 11.3.2 are positive definite. Hence Fa has positive eigenvalues, say A 1 , • - • , Ada . For z G C, let Daz denote the diagonal matrix with diagonal entries Al , • • • , . By the finite-dimensional spectral theorem, there is a unitary matrix Uct such that Fa = (Ia ./Jai-tic:. Define the complex power z G C, of Fa by Tic', := Ua Daz Uc*,. We write klz, as the matrix (F ) ii with respect to the fixed basis of the representation space of ua. Now we define the linear functional fz , z E C, on the elements of the vector space basis fu'icrit la G A , i , j = 1,2, • • • of A (see Theorem 13(vi)) by setting f(u) := (Faz)ii. Then the properties (i)-(iii) are clear. For (ii) we use the

relation Plz,F,z: . By (ii), f_ z is the inverse of fz in the algebra A'. This implies the first equality of (iv). To verify the second one, it suffices to take a = u. Using the facts that Tjt- z = (FV)* and u is unitary, we compute fz((uZi r) = fz(S(ucli)) = f—zett7 i ) = (Fct— z) j i =

(1 1—Z \ ._ = f_2(u'icrti).

Thus, (iv) is proved. Since Fa G Mor (u", (0 ) ), we have S2 (u) = (F,t )ik4i (F; 1 )ii = (fi id 0 f_i)(L\ 0 id) 04(u) = f_i.uZ.fi. This proves (v) for a = uZ and hence for all a G A. It suffices to prove (vi) for elements a = u and b = (41 )*. If a )3, then both sides of (vi) vanish by (24) and (25). If a = 0, then using once more the Schur orthogonality relations (24) and (25) we compute that both sides F ta ik ( F zi) ii. of the equation in (vi) are equal to /UV- Lz-i) Next we show that the functionals fz are characters. Put p(a) := By (26), we have h(cp(ab)) = h(abc) = h(bcp(a)) = h(cp(a)p(b)) and hence h(e(p(ab) p(a)p(b))) = 0 for any c E A. Setting c = (p(ab) - p(a)p(b))* and using the faithfulness of h (by Proposition 29), we conclude that p(ab) p(a) p(b), so that h.ab. f = (h.a.h)(fi.b.fi). Applying e to this equality and using the relation fifi = f27 we get f2 (ab) = f2 (a) f2 (b). Since the convolution product of characters is again a character as is easily seen, the latter implies that fn(ab) = f(a)f(b) for n = 2,4, • • From Lemma 33 (M we derive applied to the functions g1 (z) = f22 (ab) and g2 (z)f \ay., 2z = J(2z)f that fz (ab) = Ma) f z (0 for all z G C. To prove the uniqueness assertion, let triatec be an arbitrary family of linear functionals satisfying (i), (ii) and (26). By a similar argument to that used in the preceding paragraph, (26) implies that fl.a.fi =

424

11. Corepresentation Theory and Compact

Quantum Groups

Therefore f2 (a) = /2 (a) and hence f(a) = J(a) by (ii) for all n = 2,4, • • •. Finally, Lemma 33 yields that h(a) = fz (a) for any z E C.

Corollary 35. The Haar state h of A is central if and only if 5 2 = id. This holds if and only if f, = E for all z E C. Proof. If S2 = id, then u" = Oil' and hence Fa = I for a E À. Therefore, h = E for all z E C. By (26), the latter implies that h is central. Conversely, if h is central, then fi.a.fi = a by (26) and so fn = E for n = 2,4, • • by the reasoning used in the above proof. Hence f„ = e for all z E C by Proposition El 34(i) and Lemma 33.

In terms of the functionals h the Schur orthogonality relations (24) and (25) can be expressed as h((4)*43.) =45,06ki M; l f_ i (uici).

h(4/ (4)*)

Since the functionals h, z E C, are characters, they belong to the Hopf dual A° of A. Thus, the first equality in (iv) can be rewritten as S(f) — f—z. Further, for any z, E C the equation

(27)

a E A,

Pz,zi(a) =

defines an automorphism pz , z , of the algebra A. From Proposition 34 we immediately obtain the following formulas for these automorphisms: Pz,z' °

Pw,w' = Pz-Fw,z' -Fw'

and

poi = id,

(28)

(p,, w ® p_ w , z ,) o Z = X o pz,z ,, Pz,z'

(e) =

P—

(29) (30)

(a) * 7

p,/(S(a)) = S(p_z , ,_z(a)),

(31)

h(p„,„,(a)) = 144

(32)

Set Oz piz ,i, and piz ,_ iz , Z E C, where i = Then Ot ItER and OthER are commuting one-parameter groups of *-automorphisms of A by (30). They are important tools for the study of the CQG algebra A. By condition (vi) in Proposition 34, for any a, b E A we have Nab) = h(b0_ 1 (a)) and h(O z (a)b) = h(b0,_ 1 (a)),

z E C.

(33)

That is, {Oz } measures the extent to which h fails to be central. Recall that, by the proof of Corollary 35, h is central if and only if 0, = id for all z E C. Further, by Proposition 34(i), the function z h(0,(a)b) is entire and bounded on any strip {z I a < Tm z < 0 } , a, 8 G R. Thus, by (33), for any a, b E A there exists a bounded function f on the strip {z 0 < Tm z < 1} which is holomorphic in its interior and satisfies the boundary conditions f(t) = h(O t (a)b) and f(t + i) = h(bO t (a)), t G R. The latter is expressed by saying that {Ot } tE R satisfies the KMS condition relative to the Haar state

11.3 Compact Quantum Group Algebras

425

h. The KMS condition plays an important role in the theory of operator algebras and in statistical mechanics (see [BR]). It characterizes the modular automorphism group of a faithful normal state on a von Neumann algebra. We call {Ot } tE R the modular automorphism group of the CQG algebra A. The second automorphism group {0,} is related to the behavior of the antipode S of A. First we note that each V, commutes with S by (31). Iterating condition (v) in Proposition 34 we obtain 82n

Set U(a) :=

p_n,n oin

n E Z.

19_0(a)), a c A. From this definition it is clear that =U o19112 =1900U.

(34)

The map U is a *-preserving anti-automorphism of A such that U oU = id. It is called the unitary antipode of the CQG algebra A. Note that U0/9, =19,0U, U 0 Oz = 0_ z oU, US = SU and h(U (a)) = h(a) for z E C and a E A. We close this subsection by considering once more the coordinate Hopf *-algebras of compact quantum matrix groups.

Example 9. Let Gq denote one of the compact quantum groups Uq (N), SUq (N), 0 q (N; R) or USpq (N), q E R. As noted in Example 7, the Hopf *-algebra A := 0(Gq ) is a CQMG algebra. In order to treat all four quantum groups Gq at once, we set i := 2i

-

N

-

1 for Uq (N), SUq (N),

-

2pi for 0q (N; R), USpq (N),

where pi are the numbers defined in Subsect. 8.4.2. Then the value of the character h at the entry u i of the fundamental matrix u = (u i ) of Gq is given by f2 (u) = for j = 1, 2, • • • , N and z E C. (35) Let us prove this formula. By the definition of the involution in (.9(Gq ), the fundamental corepresentation u is unitary. As noted in Sect. 9.4, the vector representation Ti of Uq (g) is the associated representation with u by Proposition 1.15. Therefore, since T1 is irreducible, so is u and we can assume that u is one of the corepresentations u', c c Â. Set F = (Fii ) := (q1 60 ). By Propositions 9.10 and 9.13, S2 (u) = qiu.;47 -3 . Hence F E Mor (u, u"). Since = -P with i' := N + 1 i by the definition of we have Tr F = Tr F -1 . Thus, F is the intertwiner from Lemma 30 and formula (35) follows from the definition of the functionals h given in the proof of Proposition 34. Further, let (., .) denote the dual pairing of the Hopf algebras Uq (g), g = glN ,s1N, 502n,5pN, resp. U0 / 2 (502 n+ 1) and 0(Gq ) from Theorem 9.18. If p is the half-sum of positive rots of g and K2 p the element of Uq (g) resp. Uq 1/2(5027, + 1) defined by (6.26), then fi (a) = (K2,,, a), a E 0(Gq ).

(36)

426

11. Corepresentation Theory and Compact Quantum Groups

Because fi is a character and K2p is group-like, it is sufficient to prove (36) for a By (35) and (9.39), formula (36) is then equivalent to the equaThe latter can be verified by using the explicit forms tion ti(K2) = of the vector representations T1 (see Subsect. 8.4.1). Under the additional assumption that Uq (g) resp. Uq 1/2(so2n+1) separates the elements of C9(Gq ) (see Corollaries 23 and 54), there is a more instructive proof of (36) given as follows. Since S2 (x) = K2px1(2-pi for z in U(g) resp. Uq 1 / 2(s02n+1 ) by Proposition 6.6, we have S 2 (U n i )) (K2p, Ujn ) = ( 82 (X)K2p, Ui)= (K2 091 U n )(X,

and hence S2 (uni )(K2p , (K2p , uni )u.7. Therefore, the linear transformation F = (Fii ) := ((K 2 , u)) is the intertwiner of u and ucc from Lemma 30. By the definition of fi , (36) follows for a = u. Finally, combining the formulas (27) and (35) we obtain the action of the algebra automorphism p,,,, of 0(Gq ) on the generators u as Pz,zi(ILD = qz 14 z 3 uii . 11.3.5 Polar Decomposition of the Antipode

With formula (34) a decomposition of the antipode as an anti-automorphism of A was given. The following proposition describes the polar decomposition of the closure of the antipode in the Hilbert space 7-t(A). Proposition 36. Let A be a CQG algebra. The antipode S of A, considered as a linear operator in the Hilbert space 'NA), is closable. Let S = Uolgi be the polar decomposition of the closure S' of the operator S. Then U0 is a unitary self-adjoint operator on 71(A) and we have

Uo(a) = fi.S(a)

and II(a) = a.fl,

a E A.

Proof. Using the formulas from Subsect. 11.3.4 and formula (4), we derive that (b, 51 (a)) = (p l , i (S(b)), a), a, b E A.,Therefore, the domain of the adjoint operator of S contains A, hence it is dense and S is closable. Define Uo (a) = .h.S(a) and So(a) = a.f1 , a c A. Then, by (31), we have U0S0(a) = p i ,o(S(po, i (a))) = S(p0,-1 0 po,i(a)) = 8(4

a E A.

Since S 2 = p_i,i by Proposition 34(v) and (27) and So* = * S' by (1.39), we obtain p i ,o (S(a))* --= po,_ 1 (S(a*)) by (31). Using Proposition 34(vi) and the formulas (32) and (4) we get (U0(a ) ,U0(b)) =

h(Pi,o(S(a)) * Pi,o(S(b))) = h(Po,-1(S(a * ))pi,o(S(b)))

h(f)1,0(S(b))Pi,o(S(e))) = h(Pi,o(S(a * b))) = h(S(a*b)) = h(a* b) = (a,

11.3 Compact Quantum Group Algebras

427

Therefore, since U0(.4) = A, U0 extends uniquely to a unitary operator, denoted again by U0, on 7-1(A). Further, by (32) and (30), we have (a, So (a))

h(a*p0 , i (a)) = 4,90,1/2(Po,-1/2(e)Po,i/2(a))) ()= h(P0,112(a) * k,112(a))

That is, S0 is a positive operator on A. Thus, by the uniqueness of the polar decomposition (see [RS], Sect. VIII.9), U0 and the closure of S0 are the two factors of the polar decomposition of S. From Proposition 34(v) and (31) it follows that Ug = I. Hence U0 is unitary and self-adjoint.

11.3.6 Multiplicative Unitaries of CQG Algebras First let us motivate the definitions given below. Let G be a unimodular locally compact group and let 7-1 = L 2 (G) be the Hilbert space of square integrable functions on G with respect to the Haar measure of G. The unitary operator V on the tensor product Hilbert space 7-107-1 = L 2 (G x G) defined by (V f)(s,t) -= f (s, st), s,t E G,fE L 2 (G x G), plays a fundamental role in the duality theory of locally compact groups. This operator, originally invented by W. F. Stinespring, is usually called multiplicative unitary or the KacTakesaki operator. In fact, V allows one to introduce a Hopf algebra structure on the von Neumann algebra generated by the left regular representation of G from which the group G itself can be recovered (see [ES] and [Tk] for treatments of this topic). The unitary operator W on 7-1 7-1 given by (Wf)(s, t) = f (st,t) is used in a similar manner. It is easy to check that V and W satisfy the following equations on 7-1 0 7-1 0 7-1: V12 1723 = V23 1713 1/121

(37)

W23W12 = W'12 W'13 W23.

(38)

Equations (37) and (38) are called pentagon equations. Obviously, V satisfies (37) if and only if V -1 satisfies (38). Now let A be a Hopf algebra with invertible antipode. We define linear mappings V, W:A0A-4 A0A by

V (a 0 b) = 46(b) • (a 0 1)

and

W(a 0 b) = .6(

• (1 ®b).

(39)

Clearly, these are generalizations of the mappings V and W mentioned above, but because of the commutativity of the function algebras we could have also taken qa 0 b) = (a 0 1)-(b) and 1-47 (a b) = (1 0 b). 46(a) . Proposition 37. (0 V and W are bijections of A0 A. Their inverses act as V - I (a(g)b) = (G9-1 0i46(b))•(a01) and W -1 (a0b) = ((id0S)41(a)).(10b). (it) V and W satisfy the pentagon equations (37) and (38), respectively. (iii) The comultiplication, the counit and the antipode of A can be recovered from the mappings V and W by

428

11. Corepresentation Theory and Compact Quantum Groups

(a)=V(1 0a)V - ', E(a)1=m0T0V -1 (1 0a), S -1 (a)=m(id0E)V -1 (1 0a), Zi(a) = W(a 0 1)W -1 ,

(a)1 = mW -1 (a 0 1) , 8(a) =--- m(e

id)W -1 (a 1),

where .A(a) is considered as a multiplication operator on A 0 A. Proof All assertions follow by straightforward algebraic manipulations. We verify (for instance) the pentagon equation (38) for W. In the Sweedler notation, we have W(a b) = a(1 ) a(2)b and hence

E

(E a 0 b(1) b(2)c) 1V12 (E a (1 ) 0 b(i) 0 a(2)b(2)c) = E a(1) 0 a(2)b(i) 0 a( 3 )b( 2 )c,

W12W13 W23 (a 01)0

W2314712 (a

b

= Wi2W13

= W23 (

= E a( i )

a( 2 )b( 1 )

a (i ) a( 2)b

c)

a( 3)b( 2 )c.

A be a CQG algebra. Then the mappings V and W defined by (39) have unique extensions to unitary operators of the Hilbert

Proposition 38. Let

space 1-1(A) 1-1(A). Proof By Proposition 37(i), V and W are bijective mappings of A 0 A. Therefore, since A 0 A is dense in 7i(A) 0 'H(A), it suffices to show that V and W preserve the scalar product of A0 A. We prove this in the case of W. Let a, b, c, d E A. Using the definitions of the scalar product on A 0 A and A and the right invariance of the Haar state h, we, compute (W (a 0 b)

=

W(C

0 d)) =

(a (i) O a(2)b,c( i) O c(2)d)

(a (i) , c( i ))(a( 2) b, c(2) d) =

E h(a(1) *c(o )h((a(2) b)*c(2) d)

= h (b* (E h((a* c)( 1 ))(a* c) (2 )) d) = h(b* h(a* c)l)d) = (a, c) (b d) = (a b, c d).

Let us denote the continuous extensions of V and W to 7-0A) 0 H (A ) again by V and W, respectively. These extensions (which also satisfy the pentagon equations (37) and (38)) are called multiplicative unitaries of the CQG algebra A. The unitary operators V and W are linked by the relation V* = T(U0 O 1)W(U0 0 1)T,

(40)

where T is the flip and U0 is the unitary self-adjoint operator from the polar decomposition of the closure of the antipode (see Proposition 36). Indeed, (40) is equivalent to the equation V-1 T(U0 01) = 0 1)W, which can be verified by inserting the corresponding formulas for V', U0 and W. Multiplicative unitaries play a crucial role in the C*-algebra approach to noncompact quantum groups (see [BS], [B], [Wor81).

11.4 Compact Quantum Group C . -Algebras

429

11.4 Compact Quantum Group C*-Algebras In Proposition 32 we constructed a C*-algebra completion of a CQ.G algebra. In this section we take the C*-algebra of a compact quantum group as the starting point and show that it always contains a unique dense CQG subalgebra.

11.4.1 CQG C*-Algebras and Their CQG Algebras Definition 11. A compact quantum group C*-algebra (briefly, a CQG C - algebra) is a C* -algebra A with unit element such that the following two conditions hold:

AA which satisfies the (i) There is a unital *-harnomorphism z : A id),6 = (id 0446 on AAA. relation is an index set and Iv E 11 (ii) There is a family fuv = d, E N) of matrices with entries in A such that: 7 dv • (ii.1) d(u) = uZ i for v E I, i,j =1,2, (ii.2) uv and its transpose (uv)t, v E I, are invertible matrices over A. (ii.3) The *-subalgebra A of A generated by the entries 'al) ." is dense in A. If the family fuv I v E 11 consists only of a singleton, then A is called a compact quantum matrix group C*-algebra (a CQMG C*-algebra).

As in Subsect. 11.3.3, A07;)A denotes the completion of the algebraic tensor product A 0 A in the least C*-norm (see [Mu], p.190). The invertibility of a matrix y = (vij ) over A means that there is another matrix w = (w ii ) with entries in A such that Ek VikWkj = Ek Wikkj = kl for all i, j. Let A be a CQG C*-algebra. Note that the *-homomorphism 46 maps A to the completion AA rather than to the algebraic tensor product A 0 A. Let us still call 46 the comultiplication of A. However, 46 maps the *-algebra A to the algebraic tensor product A® A by (ii.1) and satisfies the "ordinary" coassociativity axiom (46 0 id) o46 = (id o on A 0 A 0 A. By Theorem 39 stated below, A is even a Hopf *-algebra, but at the moment we do not yet have a counit and an antipode on A. Next we define finite-dimensional corepresentations of A. Definition 12. Let V be a finite-dimensional vector space. A linear mapping ço : V 7 A is called a corepresentation of the CQG C* -algebra A on V if (id 2i) = (y) id) ço and ker ço={0}.

By Propositions 1 and 2, the two conditions in Definition 12 characterize a corepresentation of a coalgebra. Let us emphasize here once more that a CQG C*-algebra is in general not a coalgebra according to Definition 1.2. The concepts and results on finite-dimensional corepresentations of coalgebras developed in Sect. 11.1 go over almost verbatim to corepresentations of CQG C*-algebra.s (except for the few statements where the counit occurs). In particular, the first condition (id 0 Z1) o ço =(cp 0 id) o cif) in Definition 12

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11. Corepresentation Theory and Compact Quantum Groups

is equivalent to the relation (id .6) o (`i6 012013 , where (;6 is the associated element to ço of L(V) A (see Subsect. 11.1.1). Let us recall the notion of a unitary corepresentation which plays a crucial role in the following. A corepresentation ço of a CQG C*-algebra A on a finite-dimensional Hilbert space V is called unitary if ço is a unitary element of the *-algebra L(V) 0A, that is, (P*0 -== Iv 01A. For a CQG C*-algebra A it is more important to treat finite-dimensional corepresentations of A as elements E £(17) 0 A rather than as mappings (p : V V 0 A. The technical reason is that L(V) A is itself a C*-algebra, so the C*-algebra theory applies to O. On the conceptual side, this view paves the way for the definition of infinite-dimensional corepresentations of CQG C*-algebras (see Subsect. 11.4.3 below). There is a rather close link between CQG C*-algebras and CQG algebras which we will now discuss. On the one hand, for each CQG algebra Ao the C*-algebra completion A of Ao from Proposition 32 obviously satisfies the conditions of Definition 11 (with A = A o and 1 = dio ), so A is a CQG C'algebra and ,6.40 = 2iAr.ito. Thus, each example of a CQG algebra Ao gives rise to an example of a CQG C*-algebra A in this way. On the other hand, for each CQG C*-algebra A there exists a unique CQG algebra Ao such that Ao is a dense *-subalgebra of A and 41A0 = 2■ ArAo. The latter assertion is the main content of the following theorem. Theorem 39. Let A be a CQG C* -algebra and let A be the *-algebra from Definition 11 (11.3) . Then A is the linear span of matrix elements of all finitedimensional unitary corepresentations of A and A is a CQG algebra with comultiplication 2■ A := LA[ A. It is the 'unique 'CQG algebra which is a dense *-subalgebra of A and whose comultiplication is the restriction of the comultiplication of A.

The proof of Theorem 39 will be given in Subsect. 11.4.3. It is essentially based on the existence of a Haar state on a CQG C*-algebra which will be proved in Subsect. 11.4.2. Let us continue the above discussion concerning the connections between the CQG C*-algebra A and the associated CQG algebra A. By Lemma 45 below and Theorem 27(iii), finite-dimensignal corepresentations of A and A are unitarizable. Therefore, the first assertion of Theorem 39 implies that there is a one-to-one correspondence between finite-dimensional corepresentations of A and finite-dimensional corepresentations of A. Clearly, this correspon dence preserves all standard operations and concepts from corepresentation theory such as direct sums, tensor products, equivalence, irreducibility and unitarity. That is, the CQG algebra A carries the full information on the finite-dimensional corepresentation theory of the CQG C*-algebra A. It might be necessary to emphasize that despite the close relations explained .above the correspondence between CQG C*-algebras A and their associated CQG" algebras A is not one-to-one. As already mentioned in Subsect. 11.3.3, for a given CQG algebra A there is in general no unique C*-norm on A such that the corresponding completion is a CQG C*-algebra.

11.4 Compact Quantum Group C*-Algebras

431

11.4.2 Existence of the Haar State of a CQG C*-Algebra

The main aim of this subsection is to prove the following

Theorem 40. Let A be a CQC C* -algebra. Then there exists a unique state h of A such that (id 0 h).6(a) = (h id)d(a) = h(a)1 for all a G A. The state h in Theorem 40 is called the Haar state of the CQG C*-algebra A. The proof of Theorem 40 is divided into several steps stated as lemmas.

Lemma 41. If A is a CQC C* -algebra, then the sets 46(A)(A 0 1) Lin 1,6(a)(1 b) I a,b E AI Lin 1,6(a) (b 0 1) I a,b E A} and ,6(A)(1 0 A) are dense in AA.

Proof. Let y = (yii ) i ,i =1,...4 be a matrix with entries in A such that 46(vii ) = Vik OVki • Let V be a d-dimensional vector space with basis {e j = 1, 2, • • , dI and define (,0 G (V, V A) by cp(e i ) = ei 0v ii. If vt is invertible, then ker = {0} and hence c,o is a corepresentation and y is a matrix corepresentation of A. If vt is invertible, then 7/ is also invertible and hence 3 is also a matrix corepresentation. From these remarks and conditions (ii.1) and (ii.2) it follows that and ---7 u are matrix corepresentations of A. Now let R be the smallest set of matrix corepresentations of A that contains all ?iv and uv, u E I, and is closed under direct sums and tensor products. Then A is the linear span of matrix coefficients of corepresentations in R. Let u = (u ii ) be an arbitrary matrix corepresentation in R. By induction one can easily verify that the matrices u and fi are invertible. Hence ut is also invertible. For y := u -1- and w (ut) -1 , we have Ei ,6(uni)(1 ® vim) = unj ® uj ivi„,, = u„,„, ® 1 and Ei 46(u,„)(wim ® 1) = 1 ® umn . Therefore, A 0 A. Since A is dense in A by 460)(1 0 A) A 0 A and 46 (A)(A 0 1) condition (ii.3), the assertion follows. El For the following three lemmas we assume that A is a unital C*-algebra such that condition (i ) of Definition 11 is satisfied and ,i(A)(A 0 1) in dense in AA.

Lemma 42. For any state f on A there is a state g such that gt=fg=g.

Proof. Define a state jrn on A by the Cesaro sum fn := (f + f 2 + • • ' fn), n E N. Since the state space of a unital C*-algebra is weakly compact (see [Mu] or [BR]), there exists a weak accumulation point g of the sequence Ifn l. From the inequality II jen f frLII = — fil we obtain g f = g in the limit. Similarly, fg = g. Lemma 43. Let f and g be .states on A such that fg = g. If (p is another positive linear functional on A such that (p < f, then (pg = (p(1)g.

Proof. Let a E A and put b := (id

g),6(a). Then we have

(id 0 ALI(b) = (id 0 fg),d(a) --= (id 0 g).(a) = b

11. Corepresentation Theory and Compact Quantum Groups

432

and so (id 0 f)((b) — b 1)*((b) — b 01) = (id 0 f)(b*b) b*b — (id 0 f)(b*)(b 0 1) — (id 0 f)(b* 1)(b) = (id 0 f)LX(b*b) — b*b. Now let be a state on A such that Of=7,b. Such a state always exists by Lemma 42. Applying to the preceding equation, we obtain b 1) = O.

1)*((b)

(0 0 f)(z(b) —

By the Cauchy—Schwarz inequality, this yields p® f)((cod)((b)—b01)) = for all c, d E A. Inserting the definition of b, we get (

g)((c

(0 0 f

d 1)(LX

id)46(a)) = f(d)(0 g)((c

1)(a)).

We have 1)W 0 idW(a) =

(cØ d

(c ®dØ 1)(id ,4(a)

= (1

d

1)(id

,)((c 0 1)(a))

Applying the involution to ,i(A)(A 0 1) we see that (A 0 1)z(A) is dense in AA. Therefore, we can replace (c 1)d(a) by 1 0 x, z E A, on the right hand side of the preceding equalities and obtain OP

f

g)((1

d

1)(id ,i)(1

x)) = f(d)(cp 0 01 0 x).

Setting yx := (id 0 g)46(x) and using the equality OM = 1, this reads as f (dy s ) = f (d)g(x), d, x G A. Now let 7rf denote the GNS representation of f on the Hilbert space 7-tf (see [Mu] or {BR] for this concept). Since 1(a) = (Cf, f(a)Cf), a E A, by the definition of 7rf , we have f(dy s ) = (71-f(d*)( f ,ir f (yx )(f ) = g(x)(1-f(d*)( 1 ,(1 ) = g(x)f(d), d,x E A.

Because 7rf (A)(f is dense in Hy, the latter implies (71,7rf(Y s )(f) = g(x)(77, (f) for all n E Nf . From the inequality (it) < f it follows that there exists a vector nv, E Nf such that ço(a) = (7,rf (a)(f ) for all a E A. Hence (10 (Yx) = g(s)(p(1). But ço(Ys) = ((pg)(x) by the definition of yx . Thus, 1=1 (pg(x) = g(x)c,o(1) for x E A.

Lemma 44. There exists a state hi on A such that (id 0 hi)(a) = h i (a)1 for all a E A. Proof For a positive linear functional f on A, let K 1 be the set of all states g on A satisfying fg = f(1)g. Clearly, Kf is a weakly compact subset of the dual space of the C*-algebra A. By Lemma 42, Kf is not empty. By Lemma 43 we have K f C K p if ço < f. Hence K1 1 +12 C K1 1 n K1 2 for arbitrary positive linear functionals fi and f2 on A. Since the state space

11.4 Compact Quantum Group C*-Algebra.s

433

of A is weakly compact, the intersection of all ICf is nonempty. If hi is an element of this intersection, we have fh 1 = f (1)hi for all positive and hence El for all continous linear functionals f on A. This implies the assertion. Proof of Theorem 40. By Lemma 41, the sets d(A)(A 0 1) and 2(A)(1 0 A) are dense in A(, so the preceding lemmas apply to the CQG C*-algebra A. Let hi be as in Lemma 44. By similar arguments (or by applying Lemma 44 to A equipped with the opposite comultiplication 2icIP = T 0 2i) there is a state h, on A such that (h, id)(a) = h r (a)1. Applying h, to the formula for hi and hi to the formula for hr., we get kJ-4(a) = hi(a) and hr hi(a) = hr (a) h and the existence of h is proved. The latter for a E A. Thus, hi --- h, 1=1 reasoning proves also the uniqueness assertion. C • 1. Remark 7. Let A be a C*-algebra with unit element such that A Define a linear mapping LX : A -4 JUM by 2i(a) = 1 g a. Obviously, 2i satisfies the coassociativity axiom and 2i(A)(A 0 1) is dense in AA. In this case, each state h on A is left-invariant, but there is no right-invariant state on A. Hence A is not a CQG C*-algebra. (If A is the algebra of the continuous functions on a compact topological Hausdorff space, the above comultiplication L corresponds to the semigroup structure on X defined by x.y = y for all x, y E X.)

11.4.3 Proof of Theorem 39 The following lemma is used in the proof of Theorem 39. It is the counterpart to assertion (iii) of Theorem 27 for CQG C*-algebras.

Lemma 45. Let ço be a corepresentation of the CQG C* -algebra A on a finite-dimensional vector space V. `(,6 is an invertible element of L(V) 0 A, then w is unitarizable. Proof. Let (•, -) be a scalar product on V. Then L(V) 0 A becomes a CY - algebra. Since (,`(,' is invertible, (0*(,6 > 0 1) for some 8 > O. Put E := (id 0 1)((;6* (0), where h is the Haar state of A. Then E is a self-adjoint operator on the Hilbert space V such that E > 61. Since (p is a corepresentation, we have (id 0 2 ) ((,6*(P) =-- 013 012 012013 (see Proposition gii)). Applying (id 0 h 0 id) to both sides of this identity and using the right invariance of h we obtain (0* (E 0 1)0 = E 0 1 and hence (,O(E -1 01)0* = `0(E -1 1)[0*(E 1)0145 -1 (E -1 0 1) = E -1 0 1. Let .iPj := (E 1 / 2 01)ç(E -1 / 2 01). Then is a corepresentation of A which is equivalent to (p (via E" 2 c Mor ((p, 0). By the two equalities at the end of the preceding paragraph we have IN = 17)11/)* /4 = I® 1, that is, V) is unitary. Therefore, (p is unitarizable.

Now we begin the proof of Theorem 39. Suppose that A is a CQG C*algebra. Let '14_ be the class of all finite-dimensional corepresentations of A which have a matrix u such that .1) and i3 are invertible. Since this

434

11. Corepresentation Theory and Compact Quantum Groups

property is preserved under direct sums, tensor products and conjugations, the linear span Ao of the entries of all such matrices y is a *-algebra. Any corepresentation from R o is unitary with respect to some scalar product by Lemma 45 and hence a direct sum of irreducible unitary corepresentations by Proposition 12. Since unitary corepresentations of A belong to Ro , Ao is spanned by the matrix coefficients of all irreducible unitary corepresentations of A. Let A denote the equivalence classes of such corepresentations. For each o E A, we fix a unitary corepresentation ( 19' of the class a and an orthonormal basis of the underlying Hilbert space, so that (p' is given by a unitary matrix E Â} V a = (VZ)i,j=1,...,d a • By Corollary 10, the set {yi'i I i j =1, 2, • • • , dc„ is a vector space basis of Ao . (Note that in the proof of Corollary 10 the counit was not used.) We define two linear mappings e : A0 --+ C and S : A0 -+ A0 by e(v) = kJ and S(y) = (qi )* for a E A, i,j = 1, 2, • • • , da . Using the relations = vagqi and Ei (v)*va Ei v7,40* = bik one easily checks that e and S satisfy the counit and antipode axioms, respectively. Therefore, Ao is a Hopf algebra. Since the conjugate (p a of vcc is irreducible and unitarizable by Lemma 45, va belongs also to some class of A, so that Ao is a *-algebra. By Definition 11(i), L is a *-homomorphism. Hence Ao is a Hopf *-algebra. Since Ao fulfills condition (ii) in Theorem 27, Ao is a CQG algebra. Now let 13 be an arbitrary CQG algebra such that 13 is a dense *= / A [B. Then the irreducible unitary corepresubalgebra of A and sentations of 8 are also irreducible unitary corepresentations of A. Hence B C Ao by the Peter-Weyl decomposition (2) of B and the definition of Ao We prove that B = Ao . Assume on the contrary that B A o . Then there exists a E A such that C(u0 ) n B = {0}. By the uniqueness of the Haar functional (see Subsect. 11.2.2), the restriction of the Haar state h of A to Ao is the Haar state of Ao . By (25), h(b*a) = 0 for all b E B and a E C(0). Approaching a* by elements of B (because B is dense in A) and using the continuity of the state h on A, we get h(a*a) = 0 for a E C(0). This contradicts Proposition 29. Thus we have proved that B = A o . Because the matrices u" and uv, v E I, are invertible by Definition 11(ii.2), the class R defined in the prod! of Lemma 41 is contained in Ro) so A is a *-subalgebra of Ao . Repeating the above reasoning, it follows that A is also a CQG algebra such that LA = i6A1- A and A is dense in A by Definition 11(ii.3). As shown in the preceding paragraph, this implies that A = Ao . This completes the proof of Theorem 39. ,

...

11.4.4 Another Definition of CQG C*-Algebras An equivalent definition of CQG C*-algebras is the following Definition 11'. A compact quantum group C*-algebra is a unital C* -algebra A which satisfies the conditions (0 of Definition 11 and (ii)': The linear spaces d(A)(A01) and A(A)(10A) are dense in AA.

11.5 Finite-Dimensional Representations of GL q(N)

435

Lemma 41 above says that Definition 11 implies Definition 11'. From Theorem 1.2 in [Wor7] it follows that Definition 11' implies Definition 11, so that both definitions of a CQG C*-algebra are equivalent. In the proof of Theorem 40 only the conditions (i) and OW are essentially used. Therefore, this proof shows also that each CQG C*-algebra in the sense of Definition 11' admits a unique Haar state. Definition 11 (with condition (ii) rather than (fi)') is better adapted to the construction of examples of CQG C*-algebras, because the standard compact quantum matrix groups (see Subsects. 9.2.4 and 9.3.5) are built from a unitary fundamental corepresentation u = (uii ). But in contrast to that, Definition 11' gives an elegant axiomatic approach to CQG C*-algebras. For the next definition, let M(B) denote the multiplier C*-algebra of a C*-algebra B (see [Mu], 2.1.5). Let £(1 - ) and C(7-0 be the C*-algebras of bounded resp. compact operators on a Hilbert space Definition 13. A unitary corepresentation of a CQG C* -algebra A on a Hilbert space 7-t is a unitary element y of the multiplier algebra M(C(7-0A) such that (id 2i)y = y12v13.

In particular, each unitary element y of the C*-algebra £(7-0 gA - satisfying (id 0 20y= y 12 y13 is a unitary corepresentation of A, because £(7-0 gA - is a C*-subalgebra of the multiplier algebra M(C(7-00A). Clearly, if the Hilbert space 11 is finite-dimensional, then M(C(7-0A) = £(7-0 0 A and Definition 13 reduces to the definition of a unitary corepresentation as used above.

11.5 Finite-Dimensional Representations of

aL q (N)

In classical representation theory (see, for instance, [Zhe], Chap. XVI) all finite-dimensional analytic irreducible representations of the group GL(N, C) can be constructed by means of characters of the subgroup of diagonal matrices and the invariance with respect to the Borel subgroups of triangular matrices. In this section we extend this procedure to the quantum group GL q (N) and show that 0(GL q (N)) is a cosemisimple Hopf algebra when q is not a root of unity. Throughout this section we assume that q is not a root of unity.

11.5.1 Some Quantum Subgroups of GL q (N) Let DN be the diagonal subgroup of CL(N,C). Its coordinate Hopf algebra 0(DN) is the commutative algebra C[ti, tr I , • • • , tN, tA of all Laurent polynomials in N indeterminates t1, t2, • • • , tN with comultiplication 1(t i) = 404 and counit e(ti) = 1. Obviously, there exists a surjective Hopf algebra homomorphism irD : 0(GL q (N)) —> 0(DN) determined by rD(uii) = 8iitil j = 1, 2, • • , N. Therefore, by Definition 9.11, DN is a quantum subgroup of GL q (N), called the diagonal subgroup.

436

11. Corepresentation Theory and Compact Quantum Groups

Next we define the quantum Borel subgroups Bq+ and Bq- of GL q (N). Let 1+ (resp. 1 - ) be the two-sided ideal of the algebra 0(GL q (N)) generated by the elements 4 , j > j (r esp. j < j). It is easily checked that /± is a Hopf ideal of 0(GL q (N)). Hence the quotient (9(13) 0(GL q (N))11 + is again a Hopf algebra. Its structure is described a little more explicitly as follows. As algebras, 0(N) and C9(N) have iN(N + 3) generators z ij , j < j, z 1 and zij , i > j, z 1 , respectively. The defining relations are zii z,V = 1 zii = 1 and the relations (9.17) for 0(Mg (N)) when we set t4 = 0, i > j (resp. t4 = 0, j < j) therein. Under the latter replacement the Hopf algebra structure of O(Bi) (resp. of O(N)) is derived from that of A = O(GLq (N)). There are surjective Hopf algebra homomorphisms 7r B ± : 0(GL q (N)) O(B) such that 7r ± (D-1) = 7 B:

Bq

( 14) = Z ii 7

z NN -1 and
j,

> j,

for

0(B),

7)3,7 (ui ) = 0, i < j,

for

0(N).

7rE: (Uji =

Hence B q+ and B q- are quantum subgroups of the quantum group GL g (N), called the quantum Borel subgroups. From (9.17) it follows in particular that the elements z11, • • • , zNN, zj, • • - , zN -N 1 pairwise commute.

11.5.2 Submodules of Relative Invariant Elements Here we introduce some general notions which will be used for the construction of irreducible representations of GL q (N). Let C.9(G q ) be the coordinate Hopf algebra of a quantum group Gq and let C9(X g ) be the coordinate algebra of a left (resp. right) Gq-quantum space X q . Recall that the latter means that there exists a left coaction 0/, : 0(X q ) -+ C9(Gq )00(X q ) (resp. a right coaction : 0(X q ) 0(X000(G q )) which is an algebra homomorphism.

Definition 14. Let x be an element of C9(Gq ). An element a E C9(X q ) is called left (resp. right) relative Gq-invariant with respect to x if OL(a) = x0 a (resp. OR (a) = a 0 x). The set of all left (resp. right) relative Gq -invariant elements with respect to x is denoted by 0(Gq \X q ;x) (resp. 0(X q lGq ; x)).

In the case x = J. the spaces 0(Gq \X q ;x) and C9(X q lGq ;x) are just the algebras of left- and right-invariant elements of C9(X g ), respectively. Let X q be a left Gq -quantum space with left coaction çL : 0(X, g ) 0(Gq ) 0 0(X g ) and a right Hg-quantum space with right coaction : 0(Xq ) 0 (X q ) 0 O (Hq ). We say that X q is a two-sided (Gq ,I1q )-quantum space if the coactions 0/, and OR commute, that is, if

(95 .L 0 id) OR = (id 0 OR) 0 OL• The following rather simple facts are crucial for what follows.

(41)

11.5 Finite-Dimensional Representations of GL q (N)

437

Proposition 46. Let X g be a two-sided (Gg , Hg )-quantum space and let x E 0(Gg ) and x' E 0(Hq ). Then 0(Gg \X g ;x) is a right 0(Hg )-subcomodule of

0(X g ) and 0(Xg 1Hg ;x') is a left 0(Gg )-subcomodule of 0(X g ). Proof. For a E 0 (G g \X g ; x), we have (q5L id) o Ma) = (id OR ) o L (a) = x 0 R(a) by (41), hence OR(a) E 0(Gg \X g ;x) 0 0(Hg ). The proof for 0(X g IHg ;x') is similar. 0 Now let Kg be a quantum subgroup of Gg with surjective Hopf algebra homomorphism lrKq : 0(Gg ) 0(K g ). Then the formulas Lit-q

(7rK g id) o

and

Ricci := (id ® T-R

-q

) 0 4A

(42)

define a left coaction L Kq and a right coaction RK q of 0(Kg ) on the algebra 0(Gg ) such that Gg becomes a left and right Kg-quantum space. In this way, Gg is a two-sided (Kg , GO-quantum space with coactions OIL = LI{ q = Zi and a two-sided (Gg , Kg )-quantum space with coactions OIL = and OR = • Applying Proposition 46 to these particular cases we obtain Corollary 47. For any element x E 0(Kg ), the subspace 0(Kg \Gg ;x) is a

right 0(Gg )-subcomodule of the right coernodule 0(Gg ) with respect to the comultiplication. Likewise, 0(Gg i Kg ; x) is a left 0(Gg )-subcomodule of 0(Gg ). 11.5.3 Irreducible Representations of GL q (N)

In this subsection we apply the above notions and facts to the quantum group Gg =GLg (N) and its quantum subgroups Kg = DN B. Let P {A = Alf]. + • • • + ANEN I Xi, - • - AN E Z1 be the set of integral weights of the Lie algebra gl(N, C) and let P± = {A = A1€1 + • • • + ANEN E P I A i > - - • > AN} denote the subset of dominant integral weights. For any A = A 1 €1 + • • • + ANEN E P, we define group-like elements

tA =

- • - ekiv E 0q (DN)

and

Clearly, we have t A tm = 0+0 and zA zo = z A+0 for A, 47 we know that the spaces

0(DN\GL q (N);t A )

and

E 0(B q- ).

= 4.'1 • • • z

E P. From Corollary

0(B qT\GL g (N); z A )

are right 0 (G L (n-subcomodules of the comodule 0(G L g (N)). Next we describe some elements of these spaces. For any matrix A -= (aii ) E MN(No) the monomial

UA:= (U1) a 1 I OiD a 12.

(U N1 r 1 n (liD a 21 (11D a22

• • •

(

satisfies the condition L 13, (U A ) = t" (A) UA with a(A) := P, so that uA E O(DN\GLq(N);tcqA)).

Ulkl YINN

Ei (Ej aii ) Ei

E

438

11. Corepresentation Theory and Compact Quantum Groups

Relative invariants for the Borel subgroup Bq- are obtained by means of quantum minor determinants. Let On be the set of n-tuples I = , in} of integers from {1, 2, • • • , N} such that j 1 < i2 C • • • C in . Recall that the quantum n-minor 1.31-.1j for I, J E f2n was defined by formula (9.18). In the case I = {1, 2, • - , n} we write Dj := Dip Since 2A(D j) = > 1 D1 0 DIJ (u: ) = 0 for i < j, we get L B -(DJ) by Proposition 9.7(ii) and E i 7riç (D1 ) 0 DÇ = Dj and so Dj E 0(B,27\GL q (N); zAn), where is an algebra A n = ei + 62 + • • • + En is the fundamental weight. Since homomorphism, it follows that Di, Di, • • • VI E (.9(Bq- /GL q (N); ?),

(43)

for arbitrary mi-tuples Ii E f2m, , j = 1,2, • • • , r, where A = Ami + • • • + A my.. Clearly, an integral weight A E P is of the form A = A mi + Am, + - • - + Arns with N > m 1 > m 2 > • • • > m, > 0 if and only if it can be written as A = AlEi + • - • + AN6N with A 1 > A2 > • • > AN > O. The set (A1, • • • , AN) is obtained from (mi, • • • , m8 ) by transposing the corresponding Young diagram. For such A E P, let T = {Trp I 1 < r < N, 1 < p < Ar } be a tableau of elements from the set {1, 2, - - • , N}. The tableau T is called semistandard of shape A if Tr _ Lp < Trp , < Trp ,

1 < p < s, 1 C r < N,

2 < r < 772p , 2 —s. Then the set {D q-- sDT T E TabAA +s(ei + • • • +EN)» is a basis of the vector space 0(N, \GL q (N,C); ?). (ii) If A E P is not in P+ , then 0(B7q7/GL q (N,C); z A ) = {0}. Proof. The proof is given in [NYM], Theorem 2.5.

By Theorem 9.18, there is a dual pairing (•, -) of the Hopf algebras Ug (ON ) and 0(aL q (N)). By Proposition 1.16, such a pairing turns the algebra 0(GL q (N)) into a Uq (g1N )-bimodule with actions defined by (1.65). Proposition 49. Let A E P. An element a E 0(Dn \GL q (N);t A ) belongs to C3(N\GL q (N);?) if and only if a is annihilated by the right action of all generators Ek E Uq (g1N), k = 1, 2, • • • ,N — 1, that is, a.Ek a(o)a(2) = 0 for k == 1,2, • • • ,N — 1. E(kl Proof. The proof follows from Theorem 2.2 in [NYM]. (Note that our dual pairing of Uq (g1N ) and 0(GL q (N)) differs from that in [NYM].)

11.5 Finite-Dimensional Representations of GL q (N)

439

For A E P± , let TA denote the corepresentation of 0(GL q (N)) on the right 0(GL q (N))-comodule 0(B.7- \GL q (N);?). Then, by Proposition 1.15, the equation i"),(f) := (id® noTA, f E Uq (g1N), defines a representation t of the algebra Uq (g1N ) on the finite-dimensional vector space 0(i3 - \G q (N); ?). In the next proposition and in the proof of Theorem 51 below we shall work with highest weight representations of Uq (g1N ) with respect to the sequence of simple roots -ai , - • — aN (see also Remark 7 in Subsect. 6.2.4). Proposition 50. For any A E P+ TA iS an irreducible corepresentation of 0(G L q (N)) and is an irreducible representation of Uq (g1N ) with highest weight A with respect to the simple roots • , -oe N of glN .

Proof. It is sufficient to prove the assertion for A such that Ai > A2 > • • • > AN > 0 (otherwise we multiply all elements of 0(.13,7\GL q (N); zA) by Dq-8 , s E N, and apply Proposition 48(i)). Let A =-- ri Ai + r2A2 + - - • ± rN AN,• r1, r2,•• , rN E No. Let us consider the standard monomial 1-) TN- 1 a :- D TN Dr2 {1, 2} Dr{ '1 } and set V := t(Uq (g1N ))a. The re• " •

striction of i"), to V is a finite-dimensional representation and hence a weight representation of Uq (g1N ). Since (Kr', = 8kiq 6k1 by the formulas from Sect. 9.4, we obtain 1),(K i-1 )a E a (i) a( 2)) = ea. From the relation A(Fi ) = F 0 1 + 0 Fi (see Subsect. 6.1.2) we get =

n+1

(K71 K i+i )e(n- j+1) 0

Fi 0 1 0(j-1)

Z-d j--=1

Since (Fi , = 0 and (Ki l , ulic) = 0, k 1, by the formulas from Sect. 9.4, we derive that (Fi )a = O. Hence a is a highest weight vector for the representation TA ry with weight A with respect to the simple roots -ai, • • • — CtN• By Proposition 48(i), the representation space of TA has a basis labeled by the standard tableaux of shape A. From classical representation theory it is known that the same is true for the irreducible representation of the Lie algebra gl(N, C) with highest weight A. As noted in Subsect. 7.3.3, the dimensions of irreducible representations of gl(N, C) and Uq (g1N ) with the same highest weight coincide. Since contains a subrepresentation with highest weight A as shown above, it follows that tA is irreducible. The irreducibility of TA obviously implies the irreducibility of TA. LI 11.5.4 Peter-Weyl Decomposition of 0(GL q (N)) The main result of this subsection is the following. Theorem 51. Suppose that q is not a root of unity. Then the coordinate Hopf algebra 0(GL q (N)) decomposes into a direct sum of coefficient coalgebras C(T),) of irreducible corepresentations TA, A E RF, of 0(GL q (N)):

0(GL q (N)) =

ED,E ,+ C(T),).

(44)

440

11. Corepresentation Theory and Compact Quantum Groups

Proof Throughout this proof highest weights always refer to the simple roots - al, • • • - fitN. From the defining relations for Uq (g1N ) we see that there is an algebra anti-homomorphism 0 of Uq (g1N ) such that 19(Ei ) = ,N. We O(Fi) = Ei, j = 1, 2, • • • ,N -1, and 9(Ki ) = Ki , j = 1,2, define a representation p of Uq (g1N ) on 0(aL q (N)) by p(f)a = a.0(f), f E Uq (g1N ), a E 0(GL q (N)). Let a E 0(GL q (N)) be a highest weight vector for p. By definition we have p(Fk)a = a.0(Fk) = a.Ek = 0 for k = 1, • • • ,N - 1. From the definition of O(DN\GL q (N);t") it follows that an element b G 0(GL q (N)) belongs to O(DN\GL q (N); t A ) if and only if = qÀib , j = 1,2, • • • , N. Therefore, a E 0(DN\GL q (N);t A ) for some A E P± and hence a E 0(B q- \GL q (N); zA ) by Proposition 49. Clearly, we have 0(47- \GL q (N);?) C C(L), so that a E C(2-7),). On the other hand, let 0(Mq (N)) 3 be the span of monomials in uji of degree less than or equal to s. The algebra 0(GL q (N)) is the sum of finitedimensional left 0(GL q (N))-subcomodules Dq- TO(M q (N)),, r,s E No , and so of finite-dimensional p-invariant subspaces. Therefore, by Theorem 7.8, p is a direct sum of finite-dimensional irreducible representations. Each irreducible component contains a highest weight vector. Since such vectors belong to some C(T),) as shown in the preceding paragraph and C(T),) is p-invariant, it follows that 0(GL q (N)) =-- EAERE C(TA). If A, p, E P+ and A )2, the rep-

resentations 11A and î are inequivalent (because they have different highest weights). So the corepresentations TA and T are also inequivalent. Hence, by Corollary 9, the sum EA C(TA) is actually a direct sum. Theorem 51 has a number of important corollaries. Recall that, as always in this section, q is not a root of unity. Corollary 52. Any irreducible corepresentation of 0(aLq (N)) is equivalent to one of the corepresentations TA , A E P+ . Proof. If T is an irreducible corepresentation not equivalent to some TA ) then we have C(TA) n C(T) = {O} for all A E P± by Corollary 9. Because of (44), this is impossible.

Corollary 53. The Hopf algebra 0(aL q (N)) is a cosemisimple Hopf algebra and the direct sum (44) is its Peter-Weyl decomposition. Proof. Combine Theorem 51 and the implication (i) -(vi) of Theorem 13. 0

Corollary 54. The dual pairing (•,.) of Uq (g1N) and 0(GL q (N)) from Theorem 9.18 is non-degenerate. Proof. The proof is analogous that of Corollary 23.

Another immediate consequence of Theorem 51 is the existence of a unique Haar functional h on 0(aLq (N)). From formula (11.3) we know that O. h is given by h(1) = 1 and h(a) = 0 for a E C(TA), E P+,

11.5 Finite-Dimensional Representations of

GL q (N)

441

11.5.5 Representations of the Quantum Group U(N) In this subsection we assume that q is a real number such that q O ±1. Recall from Subsect. 9.2.4 that the Hopf *-algebra C9(Uq (N)) is just the Hopf algebra 0(GL q (N)) equipped with the involution defined by (ti i )* = ,

N, and (DV)* -= Dq . Further, as noted in Example S(u3i:), j , j = 1, 2, • 11.7, C9(Uq (N)) is a CMQG algebra. Therefore, all results on the quantum group GL q (N) obtained in the previous subsections and the theory of CQG algebras developed in Sect. 11.3 are valid for the quantum group Uq (N). Most of the corepresentation theory of Uq (N) can be derived by combining these two sources of results. Some of them will be sketched in the following. From Proposition 29, the Haar functional h of the cosemisimple Hopf algebra 0(Glig (N)) is a faithful state of the *-algebra A := 0(Uq (N)). Hence the equation (a, b) := h(a* b), a, b E A, defines a scalar product on A. Formula (36) says that the character f l of A is given by h(-) = (K2 0 , -), where 2p = Ek=i (N - 2k + 1)ck. Now let A e P+ be a dominant integral weight. We choose an orthonormal basis {ei } of the space V), := C9(B\GL q (N), z A ) with respect to the scalar product (-, -) on A such that ei is a weight vector with weight A(i) e P for the representation /IA of Ug (g1N ) on VA. Let Ni l be the matrix elements of the corepresentation TA Of A with respect to the basis fed. Then, w = (u`i ) is an irreducible unitary matrix corepresentation of A. The unique intertwiner FA = ((FA)0) E Mor (0, (0)") such that Tr FA = Tr FA-1 (by Lemma 30) is given by (FA ) ii = q 2 (P ,A (i))6ii (45) We shall prove formula (45). Using Proposition 6.6 and (45) we get

(f, S 2

= (S2 (f), = (1(2pf K2-pl 1 U )q 2("(.1)) = (1{2p f, •) = (f, (FA) in uATii )

for f E Uq (g1N ) and so (FA)in unA i = S 2 (u,L)(F),)5 by Corollary 54. Thus, E Mor (0, (0)). Since Tr FA = Tr IV, the matrix FA defined by (45) is indeed the intertwiner from Lemma 30. Next we combine formula (45) with the Schur orthogonality relation (25) from Subsect. 11.3.2. Then we conclude that the set ui),"i IA E P+, i, j = 1, 2, • • • , dim VA} is an orthogonal vector space basis of A and {

IIu 112 = dilq-2(p,A(i)). The number

q2(p,A(0) Tr 1), = Y.‘ d--d mi)EP(À)

q-2(p,A(i)) L--dx(i)ePoo

(46)

442

11. Corepresentation Theory and Compact Quantum Groups

may be considered as a quantum analog of the dimension of VA • It coincides with the quantum dimension (see Subsect. 7.1.6) of the corresponding representation of Uq (g1N ). Here P(A) is the set of all weights of the representation TA of Uq (g1N ) taken with multiplicities. Finally, let us turn to the character x(A) := Ei 14, of the corepresentation TA, A1 > ••• >AN >O. It is an element of the algebra A = O(Uq (N)). Since t)i" • • • P1;1' for MO = Ale]. + • + ANfiv, IrD(X0,) coincides 7rD(q,) = t A(i) with the corresponding expression in the classical case. The latter is known to be the Schur function S A (t) S A (t 1 ,• • ,tN) (see [Mac]). Thus, we have n-1 1 472-3 7 IrD(X(A)) = SA(t) and d), = S),(q

lq -n-F1) .

11.6 Quantum Homogeneous Spaces 11.6.1 Definition of a Quantum Homogeneous Space Recall that a quantum space for a Hopf algebra may be viewed as a generalization to quantum groups of a space on which a group acts. If the action is transitive, such a space is usually called a homogeneous space. The following definition gives a generalization of the latter notion to quantum groups. Let Gq be a quantum group given by its coordinate Hopf algebra 0(Gq ) and let 0(X q ) be the coordinate algebra of a right Gq-space with coaction ço : 0 (X q ) —+ 0(X q ) 0(G q ). Definition 15. We say that X q is a right quantum homogeneous G q -space if there exist a subalgebra X of 0(G q ) such that A(X) C X 0 0(Gq ) and an algebra isomorphism 0 : 0(X q ) X such that Ao 0 = (0 0 id) o p.

Any subalgebra X of 0(Gq ) such that A(X) C X 0 0(Gq ) is obviously a right quantum Gq -space with right coaction A X. Definition 15 says that up to isomorphisms the right quantum Gq-spaces of this form are precisely the right quantum homogeneous G q-spaces. If X q is such a quantum homogeneous Gq-space, then f := eo0 is a character of the algebra 0(X q), that is, f (xy) = f(x)f(y) for x, y E 0(X q ) and f(1) = 1. This means that any quantum homogeneous G q-space contains at least one "classical point". Left quantum homogeneous Gq-spaces are defined similarly (by requiring that A(X) C 0(Gq ) 0 X and A o 0 = (id 0 0) o ço). Example 10. The quantum vector space CqN is a right and left quantum A homogeneous GL q (N)-space by Proposition 9.11.

Example 11. Let q be not a root of unity. By Proposition 4.31, the quantum 2-sphere 'S Si,, are right quantum homogeneous SL q (*spaces.

A

11.6 Quantum 'Homogeneous Spaces

443

11.6.2 Quantum Homogeneous Spaces Associated with Quantum Subgroups In differential geometry it is well-known that a homogeneous manifold for a Lie group G is isomorphic to the coset space H\G (resp. G I H) of G with respect to the stabilizer subgroup H (see, for example, [Wal, 3.58). Then, omitting technical details, functions on H\G (resp. G I H) can be identified with those functions on G which are invariant under the left (resp. right) action of the subgroup H on G. Thus, it is natural to construct quantum homogeneous G q -spaces in a similar manner. Let _Kg be a quantum subgroup of Gq (see Definition 9.11) with coordinate Hopf algebra 0(1(q ) and surjective Hopf algebra homomorphism 71-K : 0(Gq ) ---+ 0(Kq ). For notational simplicity we abbreviate 0c := 0(Gq ) and OK := 0(K g ). Let Licq and RK g be the left and right coactions of OK on OG, respectively, defined by liK g := (7K 0 id) o z and RK, 2, := (id 0 ir o 6. Definition 16. The elements of the sets

0 K\OG := {a C OG

Lic q (a) = 1 0 a},

OG/OK := {a E OG I RK g (a) = a ® 1} 1 OK\OG/OK

:= OK\OG n OGIOK

are said to be left-invariant, right-invariant and biinvariant, respectively, with respect to the quantum subgroup 1(q . The sets 0 K\OG and OG I OK are called coordinate algebras of the right resp. left quantum homogeneous Gq -spaces .K.q \Gq resp• Gq 1Kq . Since LK, 9, and RK g are algebra homomorphisms, OK\OG, OG/OK and OK\OG/OK are subalgebras of OG Proposition 55(ii) says that Kq \Gq and Gq IKq are indeed right and left quantum homogeneous Gq -spaces. •

Proposition 55. (i) S(OK\OG) g OG/OK, S(OOOK) g OK\OG. ,i(OK\OG) C OK\OG OG, 4(0G/ 401C) C OG OGIOK. (iii) Setting C := {a E OG I 7ric (a) = e(a)1}, we have

OK\OG --= {a E Cl46(a) E C OG}, CC /OK = fa E C I .6(a) C OG C}, OK\OGIOK = {a E C

E C C}.

Proof. (i): Let a E OK\OG, that is, LK .7 (a) = 1 0 a. Since 7 rK is a Hopf algebra homomorphism, 7rKoS=So7rK . Thus, we have RK g (S(a))

=

(id 0 7ric. ) 0 ,i(8(a)) =

E(S = (S

E S(a(2)) 710(am))

S)(a(2) 0 7ric(a(1))) = (S S) O o T(1

TO

0 a) = S(a) 1,

that is, S(a) E Cc/OK. The second inclusion is proved similarly.

LK q (a)

11. Corepresentation Theory and Compact Quantum Groups

444

00 follows from Corollary 47 applied with x = 1. (iii): We verify the first equality. Let a E OK\OG. Applying id. E to (ric 0 id) o 46(a) = 1 0 a yields T-K(a) = E(a)1. Hence, OK \OG C C. By (ii), 46(a) E OK\OG 0 OG • Conversely, suppose that a E C and L(a) E CO 0c• Then (IrK 0 id) 0 L(a) = (a(1))1 a(2) = I 0 a. Thus, a E OK\OG. We illustrate the above notions by restating some results from Chap. 4. Example 12 (Gq = SLq (2)). Throughout this example we assume that q is not a root of unity. Recall that the group Kg of diagonal matrices in SL(2, C) is a quantum subgroup of Gg = SLg (2). From Subsect. 4.2.2 we know that

OK\OG =

A[0, n],

A[n, O] OK\OGIOK = C[bd.

0G/0K =

,

That is, the algebra OK\OG is generated by the elements bc, ac, db, the algebra OG/OK is generated by bc, ab, cd and the algebra OK\OG/OK possesses the single generator bc. Hence 0 K \°c coincides with the algebra generated by the elements = —1,0, 1, from (4.80). Therefore, by Proposition 4.31, the right quantum homogeneous G q-space OK\OG is just the quantum 2sphere 4,0 . It is not difficult to show that the other quantum 2-spheres Q(, are not of the form OH\OG for some quantum subgroup Hg of Gg . Let us note that .6(bc) is neither contained in OK\OG/OK 0 Oc nor in OG OK\OGIOK, so the subalgebra OK\OG/OK of OG is neither a left nor a right quantum homogeneous Gq -space.

4),

Proposition 56. Let 0( 1(g ) be a cosemisimple Hopf algebra with Haar functional hK . Then the linear mappings P1 := (h.K 0 7ri- ® id) o 46 and Pr := (id hK o IrK ) o 46 are projections of OG onto OK\OG and OGIOK, respectively. The projections Pi and Pr commute and PIP, is a projection of OG onto OK\OGIOK• Proof. Since IrK : OG —÷ OK is a Hopf algebra homomorphism, we obtain PIP!

=

071- K 0 id) o o (hK 0 -TrK 0 id) 0 21

= (hKorK0kKoirifOid)0(460id)0.6 =

It

(hK hK

id)(6 g id) o (rK 0 id) o =

is obvious that Pi (a) = a if a E OK\OG. For a E (.9G 7 we have

(7ric

id)

0

.6(pi (a)) =

(7rK

id) 046

OK

7rK

(hic 0 IrK 0 id) o .6(a)

id)(rK 0 id 0 id) 0(z 0 id) 046(a)

OK 0 id 0 id)(46 0 id) 0 (rK 0 id) o 46(a) 1 0 OK 0 71-K 0 id) o 46(a) = 1 0 Pi (a),

so Pi (a) E OK\OG. Thus, we have shown that Pi is a projection of OG onto OK\OG. Similar reasoning proves the assertions concerning P. One verifies that PiP, and Pf .Pi are both equal to (hic 0 7rK 0 id 0 hK 07rK)(id 2,) o Hence PIP, = Pr.Pi. The remaining assertions follow easily.

11.6 Quantum Homogeneous Spaces

445

11.6.3 Quantum Gel'fand Pairs Let us first recall the corresponding classical notion. A pair (G K) of a locally compact group G and a compact subgroup K is called a Gel'fand pair if the algebra L'(K\G I K) of integrable K-biinvariant functions with respect to the convolution product is commutative. If G is compact, this is equivalent to the requirement that the dimension of the subspace of K-biinvariant elements in the coefficient space of each irreducible representation of G is at most one. In the following we shall generalize this concept to quantum groups. Throughout this subsection we assume that Kg is a quantum subgroup of a quantum group Gg such that the Hopf algebras 0(Kg ) and 0(Gg ) are cosemisimple. As we have seen in Example 12, the comultiplication .6 of OG does not map OK\OG/OK to OK\OG/OK OK\OG/OK. For this reason we introduce a new comultiplication 46b i on OK\OG/OK by 4613i :=

(id

hic

id)

71‘

0 (.6

id)

0 = (id 0

Pi ) o = (Pr 0 id) 0,6.

Proposition 57. The vector space Cbi := OK\OG/OK is a coalgebra with comultiplication 2ibi and counit e[Cbi. The restriction to Cbi of a left- (resp. right-) invariant linear functional on OG is also left- (resp. right-) invariant with respect to the comultiplication 2ibi. On CIA we have the identity 2ibi 0 S To

(S S) 0 2ibi•

Proof. We sketch the proof of the first assertion. Let a c Chi . By Proposition 56, PrP/ (a) = a . From the relations ,6 o Pr = (id 0 Pr ) 0 .6 and 2, o P1 = (Pi 0 id) 0,6 and the fact that Pr and P1 are commuting projections we get Lb(a) = (Pr 0 id) o o PrP/ (a) = (13,r .Pi 0 Pr) 0 2ibi (a)

= (id 013/) 0 .6 o PrPi (a) = ( PI ®P1 13,-.) o

These identities imply that 46bi(a) E Cbi Cbi. Some computations show that both expressions (.6b i 0 id) 0 i6bi and (id 0 46bi ) 0 i6bi are equal to (id hK o 71-K 0 id 0 hK 0 7rK 0 id) 0,6(4 ), so 2ibi is coassociative. We have (E.

0 id)

=

( E. 0 id)(id (1/K ric

=

hK o 7:-K id) o (.6 id) ,i(a)

id)(E 0 id 0 id) o (.6 0 id) o

( hK 0K id)

0

.6(a) =

id)(1 0 a) = a

for a E OK\OG/OK. Similarly, (id 0 e) o46b i (a) = a. Hence for CIA. This proves that Cbi is a coalgebra. The comultiplication

2ib1

Efebi

is a counit D

is not an algebra homomorphism in general.

Proposition 58. The following conditions are equivalent: (j ) The coalgebra OK\OG I 0 K with comultiplication 2ib1 is cocominutaave. (ii) dim (OK\OG/OK nC((p)) < 1 for all irreducible corepresentations of OG .

446

11. Corepresentation Theory and Compact Quantum Groups

In the proof of Proposition 58 we will use the following Lemma 59. Let be a corepresentation of0(Gq ) on a finite-dimensional vector space V such that 0 K \OG 10K n c () {0}. Then there exists a basis of V such that the corresponding matrix y = (vii) of (ig satisfies the condition = 1,2, • • r} for some r 5_ dim V. OK\OGIOK nC(W) = Lin {v I Proof. Clearly, (pK := (id 0 -FK ) o(p is a corepresentation of OW on V. Put := {v E V I cp K (v) = v 1 } . Since 0(1Q is cosemisimple by assumption, (plc is a direct sum of irreducible corepresentations. Hence there exists a (PK invariant subspace V2 of V such that V = V1 EDV2 and (pK [V2 does not contain the trivial corepresentation. Take a basis {el, • • • , e ri } of V such that e i E 1717 i < r, and ei E V2 i > r, and let (vij ) be the matrix corepresentation of (p with respect to this basis. Since (pK(ei) = E j ei T- K(v ii ) and V1 and V2 are WK-invariant, we get -n-K(vii ) = 0 if j < r, i > r and if j > r, i < r. Moreover, 7K(vii) = 6ii -1 if i, j < r, because the elements of V1 are (pK-invariant. From these facts it follows that vii E OK\OG/OK for all i, j < r. Since 1172 does not contain the trivial corepresentation, P/ Pr (v ii ) = Em h ic (n-K (v im ))y mn hic (T -K (v ni )) = 0 for all i,j > r by (3). Since PLPT is a Projection of OG onto O K \O G /O K by Proposition 56, we conclude that OK\OG/OK fl Lin {v j, j > r} = {0}. {0}. Proof of Proposition 58. (i) —.(ii): Assume that OK\OG/OK fl C ( ) We then choose a matrix (v in Lemma 59. For i,j < r we then have E OK\OG/OK and -n-K (vii ) = Sii •l, so we obtain )

=

Ern,n

hK(71-K(Vrri n))Vim Vnj

=

E n=1

yin

Vnj-

Since the matrix elements vni,n are linearly independent and r o bj = Zbi by (i ), the latter is only possible if r = 1. (ii)—÷(i): Since the Hopf algebra Oc is cosemisimple, OK\OG/OK is the sum of subspaces OK\OG/OK nC((p) for all irreducible corepresentations (p of Oc. By (ii) and Lemma 59, oK\oGioic nc(v) = C - vu and hence = Em,„ virnhic(irK(vmn))0vni-=vilOvii. Hence r 0 Aibi zbi. I=1 Definition 17. The pair (Gq ,Kq ) is called a quantum Gerfand pair if the equivalent conditions in Proposition 58 are satisfied. If in addition the algebra C9(Ifq )\0(Gq )10(1‘q ) is commutative, then the quantum Gelfand pair is called strict.

A condition for (Gq ,Ifq ) being a quantum Gel'fand pair is contained in Proposition 60. Suppose that there exist a bijective coalgebra anti-homomorphism bc : 0(G q ) 0(Gq ) and an injective coalgebra anti-homomorphism 6K 0( 1(q ) —÷ 0(Ifq ) such that 7r-K 0 6G = (5K 0 7rK and 5G(a) = a for all a E 0 K\OG 10K. Then (Gq , Ifq ) is a quantum Gerfand pair. Proof. The proof is given in [F1].

11.6 Quantum Homogeneous Spaces

447

(2)). Let .9 be the algebra automorphism of 0(S L q (2)) determined by 0(a) = a, 0(b) = e, 0(e) = b, 0(d) = d (see Proposition 4.5(i)). Then the mappings 6G := 0 and 5 K := id have the properties required in Proposition 60, where Kg denotes the diagonal subgroup of Gq . If q is not a root of unity, then the Hopf algebra 0(SL q (2)) is cosemisimple by Theorem 4.17. Therefore, since OK\OG/OK = C[bc] (see Example 12) is commutative, (G q , K g ) is a strict quantum Gel'fand pair if q is not a root of unity. Example 13 (G q = S

11.6.4 The Quantum Homogeneous Space Uq (N-1)\/q (N) Let us briefly recall the corresponding classical situation. It is well-known that the unitary group U(N) acts transitively on the unit sphere Scicv-1 of the complex vector space CN and that the stabilizer of the point eN = (0, • • • , 0, 1) c 41-1 in U(N) is just the subgroup U(N-1). Hence the right coset space U(N-1)\U(N) can be identified with the unit sphere SV 1 . In the following we study a quantum analog of the homogeneous space U(N-1)\U(N). In the course of this we use the representation theory of the quantum group GL q (N) developed in Sect. 11.5. Throughout this sub0, ±1. Whenever we speak about section we suppose that q is real and q highest weights for representations of Uq (gln ), we refer to the simple roots • • • , -an (see also Remark 6.7 and Proposition 50). Let us abbreviate G q := Uq (N) and K g := Uq (N-1). We shall denote the generators of 0(.1Cq ) by yi3:, 1 < j j < N -1, and its quantum determinant by Dq . There exists a unique surjective Hopf algebra homomorphism 7-K : 0(G q ) 0(K q ) such that ,

7K( 1 L ii ) = yip i i ; N -1, 7K(4) r -K(u lkv ) --= 6kN1, k = 1,2, • • • , N.

(Indeed, it suffices to check that the corresponding elements satisfy the defining relations of Uq (N).) Thus, _Kg is a quantum subgroup of Gq . From Sect. 11.5 we recall that for any A E P+ there exists a unique irreducible corepresentation TA of 0(G q ). Let t be the matrix elements of TA with respect to some basis of the carrier space. We fix an index i and set VR (A) = Lin {t I j = 1,2, • • • ,d),} and VL(A) = Lin {t I j =1, 2, • • , }, 4 = dim TA. Then VR(A) and Vi(A) are right and left 0(Kq )-comodules with coactions RA g = (id 0 K ) o zl and Vic = (n-K 0 id) 0 4 respectively. Proposition 61. The right O(ICq )-comodule VR(A) and the left 0(K q )comodule Vi(A) decompose into a direct sum of irreducible subcomodules as

VR (A) = sED t, VR(A,

Vi

= 01, Vii(A71 1 ),

(47)

where in both cases the summations are over all dominant integral weights = Pifi + • - • + PN-ieN-1 such that .À1 >

> A2 >U2 > * • > AN-1 >

AN-1

> AN.

(48)

448

11. Corepresentation Theory and Compact Quantum Groups

17R (A, it) and VL (A,,u) are right and left irreducible 0(Kg )-subcomodules of VR (A) and VL(A), respectively, with highest weights p. Outline of Proof. Clearly, 7rK(Dg ) = V the one-dimensional 0(K g )comodule C • Dql is equivalent to C • (Dq ) 1 , 1 E No. Therefore, by Proposition 48 (i), we may assume that AN > O. For each dominant integral weight ,u satisfying condition (48) we introduce the standard monomials ... -/-02 na2 nbi nai vA = D {1,-..,N} AN DbN -1 {1,—,N-1 {1,-..,N-2,N } } Da' '{1,2}'{1,N } '{1}'{N}I

(49)

where ak = Ak — Ak and bk = Ak — Ak+i, k

1,2, - • • , N — 1. One verifies that vi, is a highest weight vector of weight ,u + (E kN-11 ak)eN for the corresponding representation of Ug (gliv_ i ) on VR(A). Thus, there exists a unique irreducible 0(Kg )-subcomodule VR(A, p,) in VR(A) containing vii, as a highest weight vector of weight p,. This proves that Epi, vR (A, p,) c VR(A). Since both sides have the same dimensions, the first equality in (47) follows. The second equality is proved similarly. D We shall say that the right (resp. left) 0(G g )-comodule VR (A) (resp. VL(A)) is of class 1 with respect to the quantum subgroup Kg if it contains a nonzero Kg -fixed vector, that is, 4 , (v) = v 0 1 (resp. Vi(v) = 1 0 v). Corollary 62. The left 0(Gg )-comodule V L (A) is of class 1 with respect to Kg if and only if A = lei —P€N, 1,1' E No. In this case, the standard monomial VO

= (Dq ) -1 ( D { 1,...,N-1}) e (u iv ) 1 -a SOLliv ) 1 (u iv )1 '

/

is up to a complex multiple the unique Kg -fixed vector in VL(A). Proof. The proof follows from Proposition 61 and formula (49).

0

Proposition 63. The algebra 0(Kg \Gg ) of left K g -invariant elements of 0(Gg ) decomposes into a direct sum of irreducible right 0(Gg )-comodules as

0(Kg \Gg ) =

e 0

Ci(Tici-vcN)•

Here Ci(n ,Ei-PEN) is the irreducible right 0(Gg )-subcomodule of the coefficient coalgebraC(TiEl _ I I EN ) which contains S(u ) I' (unt as its highest weight vector of weight lei — PEN. Proof. Since the Hopf algebra 0(Gg ) is cosemisimple by Corollary 53, the right 0(Gg )-comodule 0(Kg \Gg ) decomposes into a direct sum of irreducible components. In order to determine this decomposition explicitly, we first note that 0(Kg \Gg ) C erit=0C(Tlei-veN) by Corollary 62. For A = /e l — l'EN we choose a basis e l , • • - ,ek of the carrier space of TA such that the vector e l g (N,1)-invariant and the space Lin { e2, • • - , ek} is Ug (N-1)-invariant. isU Then one easily verifies that a matrix coefficient Oi of TA is left Uq (N —1)invariant if and only if i = 1. Moreover, each left Uq (N-1)-invariant element in C(T),) is a linear combination of ti'j , j = 1, 2, . • • , k. Let C1 (TA),

11.6 Quantum Homogeneous Spaces

449

A = 1E 1 - l'eN, be the set of left Uq (N-1)-invariant elements in C(L). Then 0(1-Cq \Gq ) = eli'7, =0 C1 (71/„_/, , N ). The comultiplication /1 realizes the irreof 0(Gq ) on Ci (Thi _ ii, N ). By Corollary ducible coreprese'ntation El 62, S(4,) 1'(ur)1 is the unique highest weight vector in Ci(Tui-vEN)All results developed so far are still valid for the quantum groups Gq = GL q (N) and Kq = GLq (N-1) if q is not a root of unity. From now on we shall use the *-structures of 0(Uq (N)) and 0(Uq (N-1)) too. Since 7rK : 0(Gq ) C9(Kq ) is a *-algebra homomorphism, 0(I-Cq \Gq ) is a *-subalgebra of 0(Gq ) and a right *-quantum space for the Hopf *algebra 0(Uq (N)). By construction, 0(Kg \Gq ) is a right quantum homogeneous Uq (N)-space. Definition 18. The *-algebra 0(Kg \Gq ) is called the coordinate algebra of the quantum sphere related to Uq (N). Our next aim is to describe the *-algebra 0(Kq \Gq ) in terms of generators j = 1,2, • • • , N. Since LiK q (zi) and relations. Let us abbreviate ; = u 7K(u) = 1 zi , the elements ; and z: = ( 41 )* = S(u) belong to 0(Ifq \Gq ). The right coaction (pR of C.9(Gq ) on Zi E 0(Ifq \Gq ) is given by the comultiplication, that is, we have ,

/4, j = 1,2, • • • , N.

(PR(z3) =

Proposition 64. (i) 0(K g \Gq ) is the *-subalgebra of 0(Gq.) generated by the elements z i , j = 1,2, • • • , N. (ii) The generators z1, Z2 7 • • ' 1 zN satisfy the relations z i zi = qzi z i , 2

Zjrc Zk ZkZi: + (1 — q )

E jcic

i < j; 3

j

z;z i = qzj z, i j; k = 1, • • • , N;

E k=1 zkzz =

(50) 1. (51)

As a *-algebra, 0(Ifq \Gq ) is generated by N elements z 1 ,z 2 , • • • , zN with defining relations (50) and (51). (iii) The set {4 1 • (znml • • (z k4 ) 7" I s i ,m i E No} and the set {(4) 7n 1 • • • (z1v )nN z1 1 • • • Z N S N Si Mi G No } are bases of the vector space

0(Kg \Gq ). Proof. (0: The elements (z k 4 ) l'zi = S(4) / ' (un i are highest weight vectors for the 0(Gq )-comodules C1(Ti f ,-1 , , N ). Hence these comodules and so 0(1-Cq \Gq ) by Proposition 63 are contained in the *-subalgebra generated by

zi , = 1,2, • • • ,N• (ii): The relations (50) and (51) follow easily from the commutation relations for the elements u in the *-algebra 0(Uq (N)). That the relations (50) and (51) are already the defining relations of the *-algebra 0(Kg \Gq ) can be shown by using appropriate *-representat ions of 0(Uq (N)) (see [VS2]). (iii) can be proved by means of the diamond Lemma 4.8.

11. Corepresentation Theory and Compact Quantum Groups

450

Note that because of (50) the relations (51) can be rewritten as Z je ZZ = 4.4 — ( 1

q2 )

j,

if

>

where (N- 1 -= EiN VZZ = 1 - zN4, and pi are the little q-Jacobi polynomials.

LI

Proof. The proof is given in [N11114].

Proposition 67 is a quantum analog of classical results on zonal spherical functions for the unitary group U(N) (see [VK1], Chap. 11).

11.6.5 Quantum Homogeneous Spaces of Infinitesimally Invariant Elements

The quantum homogeneous spaces treated in the preceding subsections were constructed by means of quantum subgroups. However, this method does not yield quantum deformations of all compact Riemannian symmetric spaces, since quantum groups do not have as many quantum subgroups as the corresponding Lie groups have Lie subgroups (see [114]). In particular, quantum analogs of the classical symmetric spaces SU(N)ISO(N) and SU(N)IUSp(N) cannot be obtained by means of quantum subgroups. Moreover, as we know from the study of quantum 2-spheres (see Example 12 above and Sect. 4.5), even if there exists a corresponding quantum subgroup, we do not get all interesting deformations of the classical space in this manner. In this brief subsection we sketch the "infinitesimal" method for the construction of quantum homogeneous spaces. Let us first explain the underlying idea in the classical situation. Functions on a homogeneous space I K are usually considered as functions on the Lie group G which are right-invariant with respect to shifts by elements from K. But they can also be defined as those functions on G which are annihilated by the right action of elements from UM, where t is the Lie algebra of K. We assume that LI and A are Hopf algebras equipped with a dual pairing (.,.):11xA-+ C. By Proposition 1.16, the algebra A becomes then a 14bimodule with left and right actions of 14 on A given by X.a = (id 0 X)A(a) and a.X = (X ® id)A(a), X E 11, a E A. Recall also that a coideal of U is a a linear subspace I of 14 such that A(I) CI 0 U + U 01- and E(.1)-= {0}. Definition 19. Let I be a coideal of U. The elements of the sets

2-A := fa E A

X.a = 0, X E 1A1

:= {a E A I a.X = 0, X E

:= 1A n A 1

are called infinitesimally left-invariant, right-invariant and biinvariant with respect to I, respectively.

452

11. Corepresentation Theory and Compact Quantum Groups

Proposition 68. Let I be a coideal of U. Then the sets 1A, A1 and 1A1 are

subalge bras of A and we have ZA(1A) C A® IA, z(A1) C AI 0 A.

Moreover, u.1.4 = 1A and .41.0 = Al. Proof. The fact that A1 is a subalgebra follows immediately from the relation ab.X = E(a.X (1) )(b.X (2) ), a,b E A, X E I, combined with the assumption z(I) = / 0 /I + 11 0 /. For a E A1 and X E I we have 0 = Z1(a.X) -= E a( i ).X 0 a( 2). Hence ZA(a) E A1 0 A. The equation a.XY = (a.X).Y implies the equality Ai. u = Al. The assertions for IA and 1A1 are derived 111 similarly. If A is a coordinate Hopf algebra 0(Gq ), then by Proposition 68 the algebras 1A and A1 are left resp. right quantum homogeneous Gq-spaces. Example 14 (Gq = SL q (2)). Let A = 0(SLq (2)) and 11 = tIq (512 ). Let X p , P E C Wool, be the element of Ûq (S12) defined by (4.84) resp. (4.85). Since z1(X) -= K -1 0 Xp + Xp 0 K, the one-dimensional space ip := C.Xp is a coideal of U. If q is not a root of unity, then Proposition 4.31 says that the right A-quantum space AT, is isomorphic to the coordinate algebra O(S). Let us emphasize once more (see Example 12) that only the quantum 2-sphere A Sq2 „3 can be obtained by means of a quantum subgroup of SL q (2).

11.6.6 Quantum Projective Spaces In this subsection we outline how a family of quantum homogeneous Uq (N)spaces can be constructed by using the coideal approach. These spaces are quantum analogs of the complex projective space (U(N -1) x U(1))\U(N). Throughout this subsection q is a real number such that q 0, ±1. Let A be the Hopf *-algebra 0(Uq (N)) and let ti be the Hopf algebra 1,1(GL q (N)) of L-functionals on the coquasitriangular Hopf algebra A (see Subsect. 10.1.3 for details). Since q is real, II(GL q (N)) becomes a Hopf *algebra, denoted /1(Uq (N)), with involution determined by (i +1)* = S (j ±?) (see (10.48)). By construction, II is a Hopf subalgebra of the dual Hopf algebra A° . Hence the evaluation of functionals f e Li at elements a E A gives a dual pairing of Hopf *-algebras 11 = ti(Uq (N)) and A = O(Uq (N)). Suppose that r and s are nonnegative real numbers such that (r, s) (O, 0). Let er,8) denote the subalgebra of II generated by the elements Oil - l -NN ,

ril - / NN )+ 17-si+N1 ± -1:77s riN - (r - s)(/1 - ril ),

( 53)

1 +1; + Arsrik , 2 < k < N - 1,

(54)

1+ : rkk 1 -1-r i +ki ± -V7-5 1 -kN 1, i,

i1

2< j 0 and s > O. We also set o's) is also denoted by F. For arbitrary a, a = —co if r = 0. The coideal —co < a < +co, we define

er

13`q,'

:= {a E 0(Uq (N)) a.X = 0 for all X E tu l.

By property (1.41) of the dual pairing of the Hopf *-algebras 11 and A we have a* .f = (a.S(f)*)* , a E A, f E U. Therefore, since t' is invariant under the mapping * o S by Lemma 59, the algebra Bq is a *-subalgebra of 0(Uq (N)). From Proposition 68 it follows that B q' === Apa is a right quantum homogeneous Uq (N)-space. The harmonic analysis on the quantum homogeneous space BZ has been studied extensively in the paper [DN], where the proofs of the following two propositions may be found. Proposition 70. (i) The elements

s(u)*u; of 0(Uq (N)) belong to B qr's . They generate the algebra Bqr , s and satisfy the relation (x ii )* = xii. (ii) The right 0(Uq (N))-comodule B q' decomposes as a direct sum =

1=0

v( iEi _IEN)7

where V(lE i lEN) is the irreducible right 0(Uq (N))-subcomodule generated by the highest weight vector x im . The irreducible decomposition of the comodule B°q- is analogous to the irreducible decomposition of the space of functions on the complex projective space U(N)I(U(N —1) x U(1)) (see EVKli, Chap. 11). For this reason we call Bq' the coordinate algebra of the quantum projective space CP 4 . We now consider biinvariant elements of 0(Uq (N)) with respect to the coideal t a and define

11Z := fa E 0(Uq (N)) I X.a = a.X = 0 for all X E

1.

Then, 'HZ is a *-subalgebra of O(Uq (N)). An element of is called a quantum zonal spherical function of Bq' if it belongs to the coefficient coalgebra C(L) for some A E P+ . Proposition Tl. (i ) If we set 7-(7(A) := 7-C7 = M

(TA), then we have

00

.•1***. 1=0

The algebra 7-tZ is commutative dimensional.

n

'1-tt"(ifi — ifN). .1

and the spaces 11Z(lf i — lE N ) are one-

11. Corepresentation Theory and Compact Quantum Groups

454

(ii) If cr E 111, then the algebra 7-(7 is generated by the element s (cr) := X1N + where

±- N1 q7+1±-11

(q2cr+1 q-2cr-1)11

= qu(u)*u.; + q - c r(u li )*u.7 + (41 )* + (u)*u7 E O(Uq (N)).

The assertion of Proposition 71(i) can be expressed by saying that the pair (11, Fr) is an infinitesimal quantum GelIand pair. The preceding exposition and the treatment of the quantum 2-spheres in Subsect. 4.5.4 have shown how one-parameter families of quantum symmetric spaces with the same classical counterpart can be obtained by the infinitesimal approach. This method makes it possible to construct quantum analogs for most of the classical compact Riemannian symmetric spaces (see [DO] and the references therein). The L-functionals and the reflection equation are essentially used in these constructions.

11.7 Notes Cosemisimple Hopf algebras occurred in [Sly], Chaps. 11 and 14, see also [La]. The important Theorem 22 on the cosemisimplicity of the coordinate Hopf algebras 0 (G q ) is due to T. Hayashi [H3]. The theory of compact quantum (matrix) groups and their representations was developed by S. L. Woronowicz in his pioneering papers [Won]] and [Wor3]. In Sect. 11.3 we followed the algebraic approach of M. S. Dijkhuizen and T. H. Koornwinder [DK1] thus avoiding the use of C*-algebras. The discrete quantum groups in [ER] and [VD4] are in fact equivalent to the CQG algebras (see [Ks] for a comparison). The universal CQMG algebras A(E) are in [VDW]. The polar decomposition of the antipode is in [BS]. Both approaches to compact quantum groups C*-algebras are due to S. L. Woronowicz [Wor5], [Wor71. Our proof of the existence of the Haar state follows [VD3]. There exists an extensive literature on compact quantum group algebras and C*-algebras, see [An], [Lei, [LS2], [MN], [Nag], [Riel], [Rie2], [Soil], [Wan1], [Wan2], [Y] • Multiplicative unitaries have been studied in [BS] [B] [Wor81. The representation theory of GL q (N) is taken from the paper [NYM]. The first well-studied quantum homogeneous spaces were the quantum spheres of P. Podleg [Pod]] and of L. L. Vaksman and Y. S. Soibelman [VS2]. In our treatment of the quantum spheres in Subsect. 11.6.4 and of the quantum projective spaces in Subsect. 11.6.6 we have followed the papers [NYM] and [DN], respectively. Quantum Gel'fand pairs were introduced in 1K031 and studied in [Vail], [Vai2], [FI]. The comultiplication ,Abi appeared in [Vai2]. As mentioned in the Notes to Chap. 4, the infinitesimal method for the construction of quantum homogeneous spaces was first used by T. H. Koornwinder [Ko4], [DK21 in the case of the quantum 2-spheres. It was considerably generalized in [N] and [DN] (see also [Sag] and [Dij2]).

Part IV

Noncommutative Differential Calculus

12. Covariant Differential Calculus on Quantum Spaces

Nowadays differential forms on manifolds have entered the formulation of a number of physical theories such as Maxwell's theory, mechanics, the theory of relativity and others. There are various physical ideas and considerations (quantum gravity, discrete space-time structures, models of elementary particle physics) that strongly motivate the replacement of the commutative algebra of Coe-functions on a manifold by an appropriate noncommutative algebra and the study of "noncommutative geometry" there. Differential forms also appear to be a proper framework for doing this. The basic concept in this context is that of a "differential calculus" of an algebra. It allows us to introduce differential geometric notions and carries in this sense the geometry of the "noncommutative space" which may be thought to be behind the algebra. The three final chapters of this book deal with covariant differential calculi on coordinate algebras of quantum spaces and quantum groups. In the first two sections of this chapter we develop general notions and facts on covariant differential calculi on quantum spaces, while the final section is concerned with the construction of such calculi. In particular, a, covariant differential calculus on the quantum vector space CqN is treated in detail. In Subsects. 12.1.1, 12.2.1 and 12.2.2, X denotes an arbitrary algebra. In the rest of this chapter we assume that X is a left quantum space (or equivalently, a left A-comodule algebra) for a Hopf algebra A with left coaction ço:X—>A® X. We retain the convention to sum over repeated indices.

12.1 Covariant First Order Differential Calculus 12.1.1 First Order Differential Calculi on Algebras It is well-known that the exterior derivative d of a C"- manifold M satisfies the Leibniz rule and maps the algebra Coe (M) to the 1-forms on M. If M is compact, any 1-form is-a finite sum of 1-forms f.dg-h with f,g,h E C(M). These facts give the motivation for the following general definition.

Definition 1. A first order differential calculus (abbreviated, a FODC) over an algebra X is an X -bimodule F with a linear mapping d : X—>F such that

458

12. Covariant Differential Calculus on Quantum Spaces

(i) d satisfies the Leibniz rule d(xy) = x.dy + dx.y for any x, y E X, (ii) F is the linear span of elements x.dy.z with x,y,z E X.

We shall say that two FODC 1-1 and 112 over X are isomorphic if there exists a bijective linear mapping V) 112 such that 11)(x•d 1 y.z) = x•d2y-z for x, y, z E X. We shall use self-explanatory notation such as X•dX := Lin{x-dy x, y E X}, X•dX•X := Lin{x-dy-z x, y, z E X}, etc. If F is a FODC, then by (i ) we have x•dy•z = x•d(yz) xy•dz for x, y, z E X and hence F = X•dX = dX•X. Sometimes we omit the dot and write xdy for x-dy. Condition (i) means that d is a derivation of the algebra X with values in the bimodule r. Conversely, if d is a derivation of X with values in an X-bimodule rb then F := X -dX-X is obviously a FODC over X. Let us emphasize that in contrast to the case of classical differential forms for a general FODC the 1-forms dx-y and y-dx are not necessarily equal, even when the underlying algebra X is commutative. As noted above, the 1-forms on a compact Ce()-manifold .A4 constitute a FODC over the algebra C'(,A4). Having this classical picture in mind, we consider the bimodule F in Definition 1 as a variant of the space of 1-forms over the algebra X. We refer to elements of F simply as 1-forms and to the mapping d as the differentiation. We illustrate the generality of the notion of a FODC by a simple but instructive example.

Example 1 (X = C[x]). Let X be the algebra C[x]' of all polynomials in one variable x. Fix a polynomial p E X. Let F be the free right X-module with a single basis element denoted dx. That is, the elements of F are expressions dx. f with f E X and the right action of X is given by the multiplication of X. There is a unique X-bimodule structure on F such that xdx = dx•p. We denote this X-bimodule by fp. It is easily seen that fp becomes a FODC over X with differentiation d defined by

d (E anxn) =

EE Ti

Ti

cEnsi-dx.xi

i±j=n-1

Two such FODC fp and rfi are isomorphic only if p = p. Since dx is a right X-module basis of F, for any f E X there is a unique element a(f) E X such that df = dx a( f). Because f (x)dx = dx. f (p(x)) by definition, we obtain

a(xn)=

E

p(x) 1 x3

i+j=rz— 1

Let us consider the special case p(x) = qx, where q E C, q 1. Then the

X-bimociule fp is characterized by the equation f(x)dx = dxf (qx) and a(f) is just the q-derivative Dq (f) = (f (qs) — f (x)) I (qx — x) (see Subsect. 2.2.1). Another interesting situation is obtained when p(x) c, where c E C[x],

12.1 Covariant First Order Differential Calculus

459

C O. Then we have f (x)dx = dx- f (x c) and 0(f) = (f (x f (x)). Note that in both cases 0(f) becomes the ordinary derivative of f when q 1 and c 0, respectively. Take a fixed 1-form 77 0 of and define dn f = f77 - rj f, f E X. Then Xp ,71 : X•dn X•X is a FODC over X with differentiation d n . If i = rds with r, s E X, then we have dn f =-- dx.r(p(x))0(s)(f (p(x)) - f (x)). A Definition 2. A FODC F over a *-algebra X is called a *-calculus if there exists an involution p p* of the vector space r such that (x.dy.z)* = z*.d(y*).x* for x,y,z E X. Proposition 1. A FODC F over a *-algebra X is a *-calculus if and only if Ei si .dyi = 0 with x i , yi E X always implies that Ei d(yn.x: = O. Proof. The only if part is trivial. Conversely, if the latter condition is fulfilled, then (E i si dyi )* := Ei d( y )-x, x i , yi E X, gives a well-defined(!) antilinear mapping of F which has the desired properties. 111 12.1.2 Covariant First Order Calculi on Quantum Spaces

If the algebra X is a quantum space for a Hopf algebra A, it is natural to look for covariant FODC. The precise definition of this notion is given as follows. Definition 3. A first order differential calculus F over a left quantum space X with left coaction : A® X is called left-covariant with respect to A if there exists a left coaction q : A 0 F of A on F such that

(i) Oxpy) = v(x)0(p)v(y) for all x, y c X and p e F, (ii) 0(dx) = (id 0 d)c,o(x) for all x c X. Conditions (i) and (II) mean that the coa.ction 0 of A on F is compatible with the coaction of A on X and with the differentiation d, respectively. Condition (ii) says that the mapping d intertwines the coactions (p, and 0, that is, d E Mor(vI 0), or equivalently that the following diagram is commutative: X

d

ça A 0X

id 0 d >

14)

A®F

For instance, if x i , • • • , x n are elements of X such that v(x i ) = t4 0 xi , then 0(dx i ) = uii 0dxi by (ii). That is, (ii) implies that the differentials dsi transform under 0 just as the elements si do under v. However, the differentials ds i need not be linearly independent if the s i are.

Example 2 (X = C[s] - continued). The algebra C[x] can be interpreted as the enveloping algebra U(g) with g = R. Hence, by Example 1.6, X is

12. Covariant Differential Calculus on Quantum Spaces

460

a Hopf algebra with comultiplication given by A(x) = xe1 ± 10x. If we consider X as a left quantum space for the Hopf algebra X itself with respect to the comultiplication, then the FODC fp from Example 1 is left-covariant if and only if p(s) = s. (In order to prove this, apply to the equation scis = cis • p(s) and use conditions (i) and (ii).) Let A = CZ be the group Hopf algebra of Z (see Example 1.7). We write A as the algebra C[z, of Laurent polynomials in z with comukiplication A(z) = zgz. Clearly, X is a left quantum space of A with coaction determined by (,o(x) = zgx. Then the FODC fp is left-covariant if and only if p(x) = qx for some q E C. Thus we see that in both cases the covariance requirement is an essential restriction of the wealth of possible FODC. Note that given a FODC over X there is at most one linear mapping as in Definition 3. Indeed, if such a q exists, then by (i ) and :f A (ii), we have . xi dyi )

=

.w(xi)(id

(1)(y) .

(1)

The right-hand side of (1) and so q is uniquely determined by ço and d. An intrinsic characterization of the left-covariance of a FODC which avoids the use of the coaction is given by Proposition 2. For any FODC f over X the following assertions are equivalent: (i) I' is left-covariant with respect to A. (ii) There is a linear mapping ç : such that for all s,y G X .4 0 we have 0(xdy) = v(x)(id00,o(y). Ei si dyi = 0 in implies that Ei (p(xi )(id c) w (y ) = O in A0 F. Proof.

is true by definition and is trivial. (i ): By (Hi), equation (1) defines unambiguously(!) a linear mapping qS : F A f which obviously satisfies condition (ii) of Definition 3. By the properties of the coaction w, we have (A

(e

id)0(xdy) = (A id)ko(x)(id 0,o(y)] = [(A id)(p(x)1Rid id d)(A id)w(y)] = [(id (p)(p(x)i Rid id 0 d)(id 0 ;0)(i, (y)] = (id 0)[(,o(x)(id 0,o(y)] = (id 0)0(sdy), id)0(xdy) = (e id)Ro(x)(id d)(y)] = (6 id)(,a(x)•(e d)w(y) -=

that is, 4 is a left coaction of A on F. To prove condition (0 of Definition 3, let p. = Ei xi dyi . Using the Leibniz rule for d and the fact that ço is an algebra tomomorphism (because X is a quantum space), we obtain

12.2 Covariant Higher Order Differential Calculus

461

E.0(xxid(yiy)) E. 0(xxiyi dy)

0(xpy)

E ,o(xxi )(id

d)cp(yiy) -

E.,(xxiyi)(id

d)(p(y)

= ço(x) (E i 49 (x2)(id d)40 (Yi)) ço(y) =

12.2 Covariant Higher Order Differential Calculus 12.2.1 Differential Calculi on Algebras The de Rham complex of a C"-manifold is formed by extending the exterior derivative to the algebra of differential forms. A differential calculus may be viewed as a substitute of the de Rham complex for arbitrary algebras. Definition 4. A differential calculus (abbreviated, a DC) over X is a graded algebra pA = 6)7.7 o rAn (that is, rA is a direct sum of vector spaces rAn and the product A of f" maps rAn x rAm into r^(n+m)) with a linear mapping rA(n+i),) such that: rA of degree one (that is, d : rAn d : r^

(i) d2 = 0, (ii) d(p A()

dp A ( (-1)np A d( for p (iii) rAo — x and l'An = Lin {x 0 A ds i A n E N. =

and ( E _rA , A dxn I xo , • • •,x n E X} for

E FAn •-•

If condition (iii) of Definition 4 is dropped, then we obtain the definition of a differential graded algebra. The notation p A e reminds us of the cap product of forms in differential geometry. For the product by elements x E rAo = X we shall write simply x p and px, p E rA. Condition (ii) is called the graded Leibniz rule. By induction on n one proves that condition (hi) in Definition 4 can be replaced by (iii)' rA0 = X and _T'An = X•drA(n -1 ) for ri E N. By a differential ideal of a DC rA over X we mean a two-sided ideal {0} and J is invariant under J of the algebra rA such that J n PA0 the differentiation d. Suppose that J is a differential ideal of a DC PA over X. Let 7r : rA ij denote the canonical map of rA to the quotient algebra := rA 1J. We define El(r(p)) 7r(dp), p E rA. The condition n rA0 = folj ensures that PA° = rA0 = X. Then Îv is again a DC over X with differentiation d , called the quotient of PA by the differential ideal J. For any DC over the algebra X the following identities hold:

r^

d(xodxi A (xo dx i A

••• A

dxn ) = dxo A

dx1 A • • - A

dxn ,

(2)

• • • A dxn ) A (xn +1dxn +2 A •• • A dXn+k)

= — 1) 11 SoSi dX2A • - - AdXn-Fic

— 1) n—r SodXi A. - •Ad(sr xr +i)A• • • AdXn±k r=1

(3)

462

12. Covariant Differential Calculus on Quantum Spaces

Indeed, (2) and (3) are easily verified by induction using the Leibniz rule and the equation d2 = O.

Definition 2'. A DC l'^ over a *-algebra X is a *-calculus if there exists an involution p p* of the vector space l'A which coincides with the involution of the *-algebra X on rA0 = iv and satisfies the relations (pi, A pk )* = (_onk pz A pn* and d(p*) = (dp)* for any pn c rA n e roc an d p e rA . Pk Note that the involution of a *-calculus FA (if it exists) is always uniquely determined by the involution of the *-algebra X, because FA is generated as an algebra by rAo = X and dX.

12.2.2 The Differential Envelope of an Algebra The aim of this subsection is to define a DC 12(X) over X which is universal in the sense that any other DC over X is isomorphic to some quotient of f2(X). The calculus f2(X) itself is very large and so is not of interest. We shall need it only as a technical tool in order to construct differential calculi as its quotients (just as free algebras are used to construct algebras in terms of generators and relations). Roughly speaking, the main idea behind the definition of the DC f2(X) can be described as follows: f2(X) is the free algebra generated by elements x c X and symbols dx, x E X, subject to the Leibniz rule dx•y =-- d(xy) x•dy. By the latter relation, any element of f2(X) can be written as a sum of monomials s0dx1 A • • • A dx, with xo, x 1 , • • • , x n E X, where A denotes the product of the algebra (2(X). Thus it suffices to define the product and the differentiation of f2(X) for such monomials. Since we want (2(X) to be a DC, the formulas (2) and (3) must hold. They will give us the corresponding definitions for monomials. The above idea is realized by the following precise mathematical construction. Let X := X/C•1 be the quotient vector space of X by C • 1 and let "î denote the equivalence class x + C • 1 of x E X. We set CX)

m en := $ fr(X),

where Q° (X) := X,

(271 (X) X et- cDn, n c N.

n--=0

We write xodsi A • • • A dxn for xo ±n . It can be shown (see, 0•• for instance, [Con], [CQ] or [Ka]) that the formulas (2) and (3) (with the obvious interpretations for n = O and k = 1) define a differentiation d and a product A, respectively, such that fl(X) becomes a differential calculus over X. Clearly, Q(X) is a *-calculus if X is a *-algebra. Define a linear map Let TA be another DC over X with differentiation Q(X) FA by 'Or(x) x and Cxodsi A • • • A dxn) := xoasi A • • A axn, n E N. (The definition of f2n (X) as a tensor product ensures that IP is welldefined!) From the identities (2) and (3) it follows that g := ker //) is a differential ideal of Q(X) and that re^ is isomorphic to the quotient DC

a.

ir2(X)/Ar.

12.2 Covariant Higher Order Differential Calculus

463

The first order part Q' (X) of f2(X) is obviously a FODC over X and an arbitrary FODC over X is isomorphic to a quotient of ,0 1 (X). The DC 0(X) is called the differential envelope or the universal differential calculus of the algebra X. Likewise, the FODC Q' (X) is called the universal first order differential calculus of X. 12.2.3 Covariant Differential Calculi on Quantum Spaces First we extend Definition 3 and Proposition 2 to higher order calculi.

Definition 3'. A differential calculus F A over the left quantum space X is called left-covariant with respect to A if there exists an algebra homomorphism AOF A which is a left coaction of A on FA such that OA (x) = (p(x) for x E 1 A0 =-- X and O A (dp) = (id 0 d)q5A (p) for p E 1 A • Proposition 2'. For any DC FA over X the following statements are equivalent: (0 FA is left-covariant with respect to A. (ii) There exists a linear mapping O A T A -3 A® F A such that the restriction of O A to X is equal to cp and for any xo , x i , • • , xn E X we have (sodxi A • • • A din ) = (p(x0)(id d)(p(xi) - • - (id 0 cl)(p(xn).

(iii) Ei xt,dit

A --• A

dsin = 0 in FA always implies that d)(p(x1)• ..(id

d)(itn ) = 0 in A® I'.

2

Using formulas (2) and (3) with k = 1, Proposition 2' is proved in a similar manner as Proposition 2. For the proof of the main implication (iii) —3 (i ) the coaction OA is defined as in (ii). From the latter we see in particular that the mapping OA in Definition 3' is uniquely determined by the coaction (p of A on X and the differentiation d. Since (id0d)cp(s) = (id0d)(p(1) for x E X, conditions (iii) of Propositions 2 and 2' are satisfied for the DC f2(X) and the FODC 01 (X). Thus, we obtain Corollary 3'. The universal DC f2(X) and the universal FODC f21 (X) over X are left-covariant.

Now suppose that F is a FODC over X with differentiation dl . In general there are many higher order calculi with first order part F. (A trivial one is obtained by setting _Finn = {O} for n > 2.) We want to define the "most free" differential calculus over X whose first order calculus is the given F. For this purpose we set N{>xdy1 E 01 (X) I Exd 1y1 = o in F}. From the Leibniz rule for d1 it follows easily that At is a X-subbimodule of Q 1 (X) and the FODC F is isomorphic to the FODC Q 1 (X)/N. Let Jr := f2(X)AN .A.0(X)+f2(X)AciATAf2(X) be the differential ideal of the DC f2(X) generated by N. Then 1- 11,\ := 12(X)1,71- is a DC over X whose first order pa rt

464

12. Covariant Differential Calculus on Quantum Spaces

is f2'(X)I.Ar and so is isomorphic to F. It is clear from the construction that the following universal property: If TA =erirAn is another the DC 1' DC over X such that FA' is isomorphic to T, then 1-1^ is isomorphic to a quotient of ru^. Because of the latter property, fu^ = f2(X)1,7r is called the universal differential calculus of the first order differential calculus F. Proposition 4. (i) If r is left-covariant, then so is F. (ii) If X is a *-algebra and F is a *-calculus, then Fu^ is also a *-calculus. Proof. (i): Let 0^ denote the coaction of A for the left-covariant DC (1 (X).

The characterization of left-covariance of r given in Proposition 2(iii) means precisely that 0^ (g) C A0 . By Definition 3' applied to the left-covariant DC r2(X) we have 0"4:1 = (id 0 Or and so 0^ (cW) C A 0 dAr . Since 0" is an algebra homomorphism, it follows from the preceding that O A (Jr ) C A 0 Jr . Therefore, 0^ passes to the quotient ru^ = r2(X)/Jr and defines a left coaction of A on /7,1,,' . Hence ru^ is left-covariant. (ii): Using the Leibniz rule and Definition 2 for the *-calculi 12 1 (X) and I', we conclude that IV is *-invariant. This implies that the differential ideal Jr is *-invariant. Therefore, the involution of the *-calculus P(X) passes to and hence F is also a *-calculus. 111

ruA

Remark. All notions and results concerning left-covariant FODC and DC on

a left quantum space established above have their counterparts for rightcovariant FODC and DC on a right quantum space. They are obtained if we replace "left" by "right", "id 0 d" by "d 0 id", "A 0 r" by "I' 0 A" and "A0 rA" by "FA 0 A" in Definitions 3 and 3', Propositions 2, 2', and 4, and Corollary 3'.

12.3 Construction of Covariant Differential Calculi

on Quantum Spaces 12.3.1 General Method

The purpose of this subsection is to develop and discuss a general result (Proposition 5) concerning the existence of left-covariant FODC on quantum spaces. Let Ixd iE/ be a family of linearly independent generators of the algebra X and let {4} i ,j ,ku be a set of elements of X. Proposition 5. Under the assumptions (5) and (7) stated below, there exists a unique left-covariant FODC r over X such that {dxi}j E I is a free right X module basis of F and the X -bimodule structure off is given by the relations Xi.dXj = dXk•X lici , i,j E

I.

(4)

12.3 Construction of Covariant Differential Calculi on Quantum Spaces

465

Assumption (5) expresses the covariance of the defining relations (4) of the FODC F and (7) requires two consistency conditions, one for the X-bimodule structure of F and one for the differentiation d. Note that the structure of the FODC F is completely described by the equations (4) and the property that {dxi}i c i- is a free right X-module basis of F.

Proof. Let X0 = Lin{x i i E I} and (po (preY o . Let T: X0 0 X0 —) X0 0 X denote the linear mapping defined by T(x i 0 x i ) = xk E I. We assume that Xo and T E Mor ( Po (Po, (Po

(P(Xo) C A

(P)•

(5)

To begin the construction of F, let y be the free algebra with generators xi , i E I. Let dX0 denote a vector space with a basis indexed by the set I. The basis elements are denoted by ax i , j E I. The vector space F := yodxo oy is a Y-bimodule with the obvious right and left actions of y on P. For simplicity we write y•Elx i •z for y 0 z. Since y is the free algebra generated by there are a unique algebra homomorphism b 0 : A 0 Y which extends the linear map w 0 : X0 —> A0 X0 and a well-defined linear mapping P such that a(1) = 0, a(x i) ax i and :Y 71

11(xi 1

"Xi n

= E

X»

,,•axini _xim+

,.

•

ain

m=1

for i, i i , • • , i n E I. Then Y is a left quantum space of A with coaction 00 . Define a coaction qo : clX0 —> AOdX0 by 00 (ax) := (id0a)Vo(zi). It is easily seen that a satisfies the Leibniz rule and 5 := /Po 00 P is a > left coaction of A on P such that c Mor (00 , V)). Hence i; is a left-covariant FODC over 3) with differentiation Now we suppose that J is a two-sided ideal of Y such that the algebra X is (isomorphic to) the quotient algebra y j and 00 (j) c A 0 3. Note that such an J always exists, since we assumed that the algebra X is generated by the set lx i l ic i. Let it : Y —› X be the corresponding quotient map and take elements y i/ci C Y such that 71- (y) = xt. Using once again that y is the free algebra with generators fzi lie j, it follows that there exist linear mappings :y y i i E I, such that pii (yz) =--- p(y)p(z) for y, z E y , pii (1) = kJ ]. and (xi) = y. Then N := Lin{z-ax i •y — dXk.p(z)yz, y E y,i E I} is a Y-subbimodule of P. Hence F1 := P/J14 is a Y-bimodule. That is, as a vector space F1 is isomorphic to dX 0 0 y and the Y-bimodule structure of is determined by the relations

a

—

a.

,

xi •axi

j E I.

(6)

Let N0 := dX0•,7 Linfax i .z IzEJ,iE II and let 30 be a set of generators of the two-sided ideal J. We assume that zo•axi E dX0.„7 and azo E dX0 •,7 for z0 E

go and

i E I.

(7)

466

12. Covariant Differential Calculus on Quantum Spaces

Clearly, .M0 is a y-subbimodule of f1 . The first condition of (7) implies that becomes C M. It is obvious that riJ c No. Hence F :--= F be the a bimodule for the quotient algebra X = yij. Let r i : quotient map. The second condition of (7) and the Leibniz rule for imply that d(r(y)) :=- ri(a(y)), y E y, defines unambiguously a linear mapping d :X f. Then F is a FODC over X with differentiation d. We can consider Mo also as a subset of .T;- rather than of fi, since MinAro = and ?,b0 (J) C NI. From assumption (5) and the facts that ir(y) = .4) C A 0 (go + MO. Since the A 0 3. we conclude that i,b(x i -ax i j E I, Y-subbimodule Ari is generated by the elements xi -jlxi and Mo is a Y-subbimodule of fi we obtain 7,b(Mi) C A 0 (Aro + Mi). Since also O(Aro) C A Aro , we have ip(ri + Aro ) c A 0 (N-0 + MO. Hence the coaction //) of A on f passes to the quotient f/(Ari + Aro) r to define a coaction clo there. The map q satisfies the conditions of Definition 3 for f, since 1,b does for P. Thus we have shown that F is a left-covariant FODC over X. From the preceding construction it is clear that fclxilifi is a free right X-module basis of f and the relations (4) hold.

rum

a

axk

4

Let F be the left-covariant FODC from Proposition 5. Then, for any x E X there are uniquely determined elements ai(x) E X such that

dx =

dxi •ai (x).

(8)

The mappings ai : X X are called the partial derivatives of the FODC F. Let us turn to the universal higher order DC f of r. Recall from Subsect. 12.2.3 that FiLA is the quotient of the universal DC (1(X) over X by the two-sided ideal Jr generated by the defining elements x i -dxj — dxk for the bimodule F and their differentials d(xi •dxi dxk•x) = dxi A dxi +dxk A d4i , i, j E I. That is, the algebra f has the generators x i and dxi , i E I, and the defining relations of Fii^ are those of the algebra X, the relations (4) and

dxi A dxi -= —dxk A dx1Fi , i , j E I.

( 9)

Because of the applications given in the next subsection, we specialize the above procedure. Suppose that the quantum space X is an algebra with generators x i , j = 1, - N, and defining relations written in matrix form as

Buxi x 2 = 0,

(10)

where B is a complex N2 x N2 matrix and x denotes the column vector (x i , • • •, x N )t (see Subsect. 9.1.2 for the corresponding notation). Further, we are linear in the generators x i , that is, we have assume that the elements =X = AZxi for some complex matrix A = (A). Then (4) reads as x i •dxj = Atil dxkai or in matrix notation as

4

j .

12.3 Construction of Covariant Differential Calculi on Quantum Spaces

467

X•dX2 = Al2 dX1 . X2

Now we set Jo := Prikx i xi j n, k = 1,2, • • •,N} and reformulate the key assumption (7) in this case. From the equations (6) and (11) we obtain (B12X1X2) J.X3 = B12A23 xrdX2 'X3 7-- B12 A23Al2aX1 'X2X3, a(Bi2X1X2) = B12(aXi•X2

xrax2)= (B12 +./312Al2)axi-x21

respectively. Therefore, by (10), the assumption (7) is certainly fulfilled if there exists a complex N 3 x N3 matrix T such that (12)

B12A23Al2= T123B23) B12 + B12Al2 =

(13)

0.

12.3.2 Covariant Differential Calculi on Quantum Vector Spaces In this subsection we shall apply the method of the preceding subsection to the following situation: A is the Hopf algebra 0(G q ) with G q q (N),SL q (N),0q (N),Spq (N) and X is the left quantum space with definGL ing relations P_x 1 x2 = O. Here P is the spectral projector of k to the eigenvalue -q -1 (see Sects. 9.2 and 9.3). We suppose that q 2 + 1 4 0 for GL q (N) and SL q (N), (q2 + 1)(1 q 2-N ) 0 for 0q (N) and (1 2 + 1) ( 1 _ c 2N) f(k) for some polynomial for Spq (N). These assumptions ensure that f so X= XL(f; R) is indeed a left quantum space of A by Proposition 9.4. In fact, we have X = O(C) if G q = GL q (N),SL q (N) and X = 0(0,7) if Gq =0q (N). By (9.10), the left coaction o pL of A on X is given by (p(x) = i Oxi, where u = (ui ) is the fundamental matrix of A and x = (x1, • • •, xpi) t is the vector of coordinate functions of X. Then we have (p(X0) C A ® X0 and (idOT) ( (Po O(P0) (xi Oxi) = (idOT)(uluil ®Xk OX/ =U2:k U./ Ankim (San 0Xml ( (Po

(P)T(Xi

xj) = (CPO

(10 )(Xk

AX/)= A ikit unk uL X n,

Xm .

Hence the assumption (5) is fulfilled if A = A+ := q A and if A = A_ . For A -= A+ and A -= A_ both equations (12) and (13) are satisfied with T123 := A23Al2. The relation (13) follows from the equation P_A± 1 --= -q71 P_, while (12) is an immediate consequence of the braid relation for R. Therefore, by the previous considerations, we proved the first assertion of Theorem 6. There exist two. left-covariant FODC 1-1± and F_ over X with respect to A such that the set fclxili=1,-..,N is a free right X-module basis of r± and xi .dx i = q±' (A± 1 )dx k .si . (14) Moreover, we have the following formulas of commutation relations for F±:

468

▪

12. Covariant

Differential Calculus

= 6i3 +

on Quantum Spaces (k ±i)iii,x ka

(15)

(p_)143k a, = o,

(16)

dxi A dxj = —q ±l (k ±1 )Icii dxk A dxi.

(17)

Proof. It remains to verify the formulas (15)-(17). Using (8) and (4) we get

d(si y) = dsi -y -dy--= dxj -y + x -cisi -51(y) = dry q ±1 (k±i)ii.k1 dxi .sk at (y ) .

dzi -ai, (xi y) -=

is a free right X-module basis of F± , the coeffiSince the set cients of dsi on both sides must coincide. This gives (15). Next we turn to (16). Let y E X be such that (P_) ipka/(y) = 0 for all i, j=1, N. By induction it suffices to prove that (p_)ak a,(spy ) = 0 for p = 1, 2, • • , N. The braid relation for implies that fRNÉR1(P--)12 = (P) 2321 I Applying first equation (15) twice, then the identity fRP__ -q -1 P__ and finally the preceding equality we obtain .

P _ )si 3lc ak (x p y) = P _ ) 173k ak( y ) q ±2(p - )/i.ig±1)f 0 mn(k±1.)nr ss ar an (y)

= (p_)icak (y)

. 2.an (y) 7±1.(k±lp4f37 1

= (p_) ipk (y)

q ± 1(_el p _ )

T:L an(y

17±2(fR 21. fR 31. ip )

=q ±2(p_vcr o±Tnk(0±1) .73.si xsar an (y)

( )12)7:Xsaran(Y) q±2 (( p_ )23 mi R23 )ip37-sL r xsaran(y)

= 0.

Finally, note that (17) follows by differentiation of (14).

12.3.3 Covariant Differential Calculus on CN and the Quantum Weyl Algebra In this subsection we treat the covariant FODC F± on X --= O(C) in detail. First we insert the values (9.13) for the numbers Wicji into the formulas (14)-(17) in Theorem 6. Then the commutation relations for the FODC and the defining relations of the algebra O(C) take the following explicit form: (18) x.dx = qdxj•xi + (q 2 1)dxj -xj , j < j,

(19)

xi -dxi = q2 dxj •xi, xj .dxi = q cisi -xi,

j,

(20)

dxi A dx = —tr i dxi A dxi , i < j,

(21)

dzi A dxi = 0,

(22)

sixi =

j,

(23)

12.3 Construction of Covariant Differential Calculi on Quantum Spaces

i < j, aixj = q x -

i

469

(24)

j,

(25)

q2x iai =l+ (q2 -1)E xiai.

(26)

j>i The corresponding relations for the second FODC ./.1_ over X = O(C) are obtained if we replace q by q -1 and the inequality i < j by j < i in the formulas (18)-(26). Of course, the equations (21)-(25) remain unchanged in 1 the FODC 11+ and 11_ are this manner. In particular, we see that for q not isomorphic and that for q = 1 both FODC give the "ordinary" differential calculus on the commutative polynomial algebra C[xi, • • xNI• The above equations (18)-(20) completely describe the bimodule structure of /1+ . The algebra (r+ ).^ of the universal DC associated with r+ admits the 2N generators x i and ds i , j = 1, • • • , N with defining relations (18)-(23). In the case q = 1 the formulas (23)-(26) reduce to the defining relations of the Weyl algebra. This motivates the following ,

Definition 5. The quantum Weyl algebra A q (N) is the unital algebra with

2N generators x l , • • -

v ,a,,. • - ,a,„ and defining relations (23)-(26).

We mention a few algebraic properties of this algebra A q (N).

1. The Weyl algebra A q (N) with q = 1 is simple, that is, it has no nontrivial two-sided ideals. However, the algebra A q (N) is no longer simple if q2 1. Indeed, the kernel of the one dimensional representation given by 1, (1 _ q2 ) _1, xi xi -> 0, a, —> 0, i > 2, is a nontrivial ideal. 2. Let D be the unital algebra with N generators ai,a2,- ,aN and defining relations (24) and let X := O(C). Then the linear map of X 0 O to x•y is a vector space isomorphism. (A proof can A q (N) defined by x y be given by using the diamond Lemma 4.8.) Therefore, we can consider X and D as subalgebras of A q (N) and the set of monomials • -

r _ki

An].

alkiv I rbit. 1

kN n - ••, nN E NO

}

is a vector space basis of Aq (N).

3. The element D := E ixi ai of A q (N) is called the Euler derivation. If fn G X and gn G D are homogeneous elements of degree n, then we have the identities Dfn = [[nlio fn ± fn D where we have set [[n]1 q2 : = ( 172

and gn D = 1 ) 1 ( 72n -

pi llogn ±

Dgn,

1).

4. The partial derivatives ai of the FODC r+ act on the algebra X by

ai (fN(xN)• - -fi(xi))

f N(qxN)• • • f i+i(qx j+i)Dq 2 fi(x i)

• • fl(11)

where fi, - • .1fN are polynomials in one variable and De is the q2-derivative (see (2.43)). It suffices to prove this formula for monomials f(x) =

12. Covariant Differential Calculus on Quantum Spaces

470

First we shift x j:i to the right by using (23) and ai in front of 4i by applying 1 back. [Pci flos iFi -1 and shift x (25). Then we compute ai (xlci) It is well-known that the "ordinary" canonical commutation relations are invariant under symplectic linear transformations. The next proposition contains a quantum analog of this fact. Proposition 7. Let É = (ki ) be the R-matrix and C = (Ci) the matrix of O. the metric for the quantum group Spq (2N) (see Sect. 9.3). Let a E C, a Set yi a q i aN+i-i and YN+i xi for j = 1, 27 • • • , N. Then the generators x i , • • • , x N, satisfy the defining relations (23)-(26) of the algebra • • • , A g (N) if and only if

a,

aN

2N k3lYkYl

qyiyi - a q

i

=0

for

j = 1,2,- • • ,2N.

(27)

k,i=i

The quantum Weyl algebra A q (N) is a left quantum space for the Hopf algebra 0(Sp q (2N)) with coaction ço given by (,n(y0 = uii y i , j = 1, 2, • • • , 2N. Proof. For i, 74 j' we have Cii =-- 0 and (27) reduces to the equations (23)(25). Now we show that the equations (27) for i j' are equivalent to the equations (26). We first rewrite the relations (26) recursively as

=-- 1,2, • • • ,N

= aj x -

aNxN - q 2xN aN

1,

= 1.

These equations are easily verified to be equivalent to yi, yi

2 yi yit = _ q -i a ± (q -2

1)E q k_i ykyk ,, j = 1,2, •••, N.

(28)

kci

On the other hand, inserting the values of C!, and kit' from (9.30) into (27), we see that the equations (27) for i = f are also equivalent to the relations (28). For the second assertion we proceed as in the proof of Proposition 9.4. There exists a unique algebra homomorphism (09 : C(yi )-0(Spq (2N))0C(yi ) such that ço(yi ) = uij Oyi . Let J be the two-sided ideal of the free algebra C(y) which is generated by the elements Pi from the left hand side of (27). Using the relations uCut C and rZu 1 u2 = uiu2rZ (see (9.34) and (9.3)) we passes to the quotient algebra obtain that (p(Jii) = 744 grki. Therefore, D A q (N) = C(yi )/,7 to define a left coaction of 0(Sp q (2N)) on A q (N).

There are two important special cases, where the quantum Weyl algebra A q (N) admits a natural involution. q E R. Then there is a unique algebra involution on Aq (N) such that := x i , j = 1,2, • • • , N. Indeed, since q is real, the defining relations (23)(26) are obviously invariant under this involution. Hence Aq (N) becomes a

Case

1:

12.3 Construction of Covariant Differential Calculi on Quantum Spaces

471

*-algebra. The generators of A(N) can be interpreted as q-boson creation operators a7 = x i and q-boson annihilation operators a i = ai in this way. Case 2: WI = 1. In this case the algebra 0(CqN ) becomes a *-algebra O(R) with involution defined by x7 = xi , i = 1,2, • • • , N, and the FODC 1 -1+ a *-calculus for the *-algebra O(R). The involution of F+ induces theis involution for the generators ai given by ar = —072(N+1-)19i, i = 1,2, • • • , N. Using the assumption WI = 1 one verifies that the relations (23)—(26) are preserved under this involution. Therefore, the algebra A q (N) becomes a * algebra, denoted A q (N; R), such that x1 :- --- x i and ai* = — q 2(N+103. The hermitian elements x i and pi := — 1/--i_ q N+1-i ai of this *-algebra can be viewed as q-analogs of the position and momentum operators, respectively. If one adopts this point of view, it is natural to consider the *-algebra A q (N; R) as a q-deformation of the real quantum mechanical phase space. Let us set a = --I,/-1 - in Proposition 7. Then we have yi = PN+1-i and YN-Ei = Xi therein and the last assertion of Propostion 7 means that the quantum group Spq (2N) acts on the q-deformation A q (N; IR.) of the phase space. -

i

We close this subsection by stating a uniqueness result for the two covariant FODC F± on O(C). Recall from Theorem 6 that the set Idx i l i= i ,...,N is a free right module basis of r+ (and also a free left module basis as follows from the formulas (18)—(20)). The next proposition shows that for N > 3 and for q not a root of unity the latter property characterizes F+ and F_ among all possible left-covariant FODC on O(C). Proposition 8. Suppose that N > 3 and q' 1 for all n E N, n > 2. If F is a left-covariant FODC over X = O(C) with respect to the Hopf algebra A = 0(S L q (N)) such that the set {dxili=1,-.,N is a free right X -module basis of F, then F is isomorphic to F+ or to F_.

Proof. The proof can be found in [PuW]. (In the paper {PuW1, covariant FODC with respect to SU q (N) , q E (-1, 1), are considered, but the proof therein works for SL q (N) if q is not a nontrivial root of unity.) D In the case N = 2 the assertion of Proposition 8 is not true. In this case there are two one-parameter families of such FODC (see [SS1] for details). 12.3.4 Covariant Differential Calculi on the Quantum Hyperboloid

For a nonzero complex number 7, let Xq ,..), denote the algebra with generators x l , x2 and defining relation x1x2 —qx2xi == 7 1-

It is not difficult to check that there is an algebra homomorphism coo : eVy 0(SLq (2))0Xvy defined by ço(x i ) = u.ij eaj , i = 1, 2, such that eVq ,..y becomes a left quantum space for the quantum group SLq (2) with coaction (p. The

472

12. Covariant Differential Calculus on Quantum Spaces

algebra Xq, ..y is called the coordinate algebra of the quantum hyperboloid. Obviously, Xq 2 , 1 is just the quantum Weyl algebra Aq (1). The following proposition says that there exist two distinguished covariant FODC on the quantum space Proposition 9. Suppose that 0 and qn I for n E N, n > 2. Then there are precisely two nonisomorphic left-covariant FODC F over X4,7 with respect to the Hopf algebra 0(S4(2)) such that fclx i ,dx 2 1 is a free right Xq ,..y -rnodule basis of F. The bimodule structures for these two calculi are described by the formulas

-p-l q -1 )dxrx2 + p1

=

+ (1 -p-

1

nv „

p -1 dx i .x 2 + (p - qp-1 )dx 2 •x + (1 - p-1 )7 -1 winv x 2 x i , = pdxj ai + (1- p -1 )7 --i winv(x i ‘2 ) for j = 1, 2, where p := ±(q1 / 2 + q- " 2 ) -1 and w inv

dx1•x 2 qdx2.x1.

This proposition is stated in [SS1]. We shall not carry out its proof. The uniqueness assertion of Proposition 9 is no longer true if q3 = 1 and q 1. Indeed, let q be either 1(-1 + -V7-3) or (-1 Then it can be shown that for any nonzero complex number a there exists a left-covariant FODC Fa over 4,7 with respect to 0(Sli q (2)) such that the set {ds i , dx 2 } is a free right module basis of Fa . The corresponding formulas for the bimodule Fa are

z 1 .dy2 = X2 .11X' =

xi .dxi =

(a -

1- )dxi-x2 + cdx2xi + d Winv X1X2 + (a - q a -1 )dx2 ex i + a winv x2x1l a dx f xi + a Winv (xi ) 2 for j = 1,2,

where de- := -(a + a -1 + 1)7 -1 and winv := dx•.T2 - q dx2-xi

12.4 Notes Differential forms on noncommutative algebras and the differential envelope of an algebra appeared in the work of A. Connes, M. Karoubi, D. Kastler and others (see the references in [Con], [CQ] and [Kaj). The covariant differential calculus on the quantum vector space CqN was obtained independently in [WZ] and [PuW]. These papers were very influential for the study of noncommutative differential calculi, see (for instance) ICSW], ISchj, [0g1, [Z), [3K0), ISc11, [GZ]. First steps towards a q-deformed quantum mechanics have been made in [SW] and [1-1SSWVV]. Apart from quantum vector spaces and quantum groups themselves, covariant differential calculi have been investigated on quantum 2 spheres (see [Pod2], EPod31, [AS ]) . A theory of such calculi on general quantum spaces (if possible) is still at the very beginning. -

13. Hopf Birnodules and Exterior Algebras

This chapter is devoted to some concepts (left-covariant bimodules, bicovariant bimodules) and constructions (tensor algebras, exterior algebras) which are basic tools for the covariant differential calculus on quantum groups in the next chapter. They are also of interest in themselves. The structure theory of covariant bimodules is developed coordinate free in abstract Hopf algebra language. This allows us elegant statements and short proofs of the results. At the end of the subsections the results are reformulated in coordinate form which is more convenient for computations. Throughout this chapter A is a Hopf algebra with invertible antipode S.

13.1 Covariant Bimodules Covariant bimodules can be considered as quantum group analogs of vector fiber bundles over Lie groups which are endowed with left or/and right actions of the groups.

13.1.1 Left-Covariant Bimodules Definition 1. A left-covariant bimodule over A is an A-bimodule 1 -1 which A® 1-1, such that is a left comodule of A, with coaction 2q, : T

p E F.

(1)

Ea(i) p( _ i) b(i) 0 a( 2)p( o)b( 2).

(2)

2q,(apb) = 2i(a),AL(p),A(b)

for

a, b E A

and

In Sweedler's notation, the last condition can be written as

E(apb) (_ 1) 0 (apb)(0)

=

An element p of a left-covariant bimodule f is called left-invariant if 2iL(p) = 1 0 p. The vector space of left-invariant elements of f is denoted by inv T. Lemma 1. Let I' be a left-covariant bimodule over A. There exists a unique linear projection PI, : 1-1 iiiv r such that

PL(aP) = e(a)PL(p),

a E A, p E F.

(3)

474

13. Hopf Bimodules and Exterior Algebras

The map PL is given by PL(P) =

E s(p(_1))P(0)•

For p E F and a E A, we

have (4)

= PL(Pa) = E S(a(i))PL(P)a(2)

adR(a)(PL(P)),

(5)

where adR is the right adjoint action of A on the bimodule F (see Remark 2 in Subsect. 1 .3.4). Proof. Using the formulas (2), (1.30), (1.27), and (1.20) we have ZAL(PL(P))

zAL(s(p(_1))p(0))

=

E s(p ( _2 ) )p ( _i ) s(p(_3 ) )p( 0 ) = EE(p( _ 1) )1

s(p(_2))p(o)

= 1 ® PL(p),

so that PL(p) E inv F. Using the formulas (2), (1.28), and (1.27) we get PL(aP) =

E S((ap)(_1))(ap)( 13) = E S(a(i)P(-1))a(2)P(o) E s(m _ios(a ( oa(2) p(0) = E(a)PL(P).

The relations (4) and (5) follow in a similar way. The second equality in (5) is just the definition of adR(a). If p E inv r, then ,AL(p) = 1 p and hence PL(P) = S(1)P = p, that is, PL is a projection of r ontO P . If P is another projection P : F —■ 1 ,F satisfying (3), then by (4) and (3), P(p)

= E E(p(,)P(PL(p(0))) = E PL(E(p(_1))p(0)) =

PL(P),

which proves the uniqueness assertion. We now begin to describe the structure of left covariant bimodules. Let ro be an arbitrary right A-module, that is, there exists a bilinear map ro X A 3 (w, a) —)w.lac Po such that w a (ab) = (ci.) a a) a b and w a 1 = w for a,b E A and w E ro . It is easy to check that the equations -

a(b w)c = E ahem 0 w ZAL(b

cd) =

c(2))

(6)

cd

( 7)

b(i) ® 1) (2)

define A-bimodule and left A-comodule structures on the vector space rid Aoro such that L becomes a left-covariant bimodule over A. As an example, by computing we verify the right A-module property of (a

cd)bc =

Ea(bc) (i) 0 (4) .1 ((bc)(2)) = E ab(i)c(i) 0 (w< b(2))11 c(2)

= 2 (ab(i)

(.4.) 112 which intertwines the left and the right coactions (that is, T(apb) = aT(p)b for a, b E A, p E F; (idOT) = LL 0 T and (T 0 id) 0 LR = 4iFt 0 T). If T is also bijective, it is called an isomorphism of / 1 and r2. —

Definition 4. The linear mapping o

a(P1 OA

P2 )

: f 1 OA 112 —›

r2 OA

ri

given by

:= E pt,PR(4 ) ) ® A PL,(pto ) )pi,

= E pt_2) ,90) s(pi) )

(24)

® A s(Pt--1) )Pto) 2) P

for pl E .rfi and p2 E 112 is called the braiding of the bicovariant birnodules Fi and f2. This terminology stems from the facts that the bicovariant bimodules over A form a braided tensor category (see Subsect. 10.3.4 for this notion) and a is the braiding in this category. We shall not carry out the details of the proof, because we will not need this in the following. One verifies that the linear mapping a-1 : 112 A ri OA -r2 given by 0.-1

(P2 OA

Pl )

:=

E ,q2,40) s- 1(pt_1 ) ) ® A s-4(41) )4,pt_2)

is the inverse of a. Thus, in particular, a is bijective. We give another form of a and o--1 which is more convenient for applications. For a E A, w E 1 f 1 and w2 E invr2 ) we have

akaw 1 OA (2) =

f 1 2 aw(2o) „0, .'6'A k w < W( 1 )),

(au,2 ®A w 1) = E a(w '

i,„ s_1(41)))

®A

(25)

4)

where a is the right adjoint action of A on invrkl k = 1 1 2 1 (see (8)). We verify the formula (25) and set p1 = aw1 and p2 = w2 in (24). Then we have

13. Hopf Bimodules and Exterior Algebras

482

pt.)

=

E

a(2)®a(3)w',

Epi( _2)

Pi )

®A P2 )

--= Ea ( 1)40) 51(41 ) )

so that 0-(pi

OA S(a(2))a(3)w 1 42)

Ea ( i) 40) OA S (41))6(a(2))W 1 42) E a40) OA (W 1 < 4)Similar reasoning shows that for a E A, 77 1 E (roi n, and 7/2 E (1'2)1nvl 2

C( 7/ 1 OA 77 a)

11 2

»4-1)

(26)

77 ) OA NCO)

where bi> 772 := PR (bn2 ) = ad/J (0(i 2 ) for b E A (see Subsect. 13.1.2).

Proposition 5. The mapping a defined by (24) is an isomorphism of the bicovariant bimodules 111 0A 112 and 112 OA ri over A. It is the unique linear map o- : 1-1 OA 112 —0 r2 OA F1 such that a(ace OA n)

an OA W for a E A, w E invfl and

(27)

E ( 1'2)inv

The map a satisfies the braid relation on i OA 112 OA 113 that is,

(o- id)(id c)(o- id) = (id 0 c)(cr

(28)

id)(id 0 a) .

Proof From Subsect. 13.1.3 it is clear that 1 OA 112 is linearly spanned by each of the sets -taw 0A 77 21 , { aw l 0A w2} and tri l ® A 772 b1 with a, b G A, wk E in v f k and Tik E (fk)inv for k = 1, 2. Hence a is uniquely determined by (27) and it suffices to verify all required relations on f1 OA r2 for such elements. We conclude from (25) that (id0a)oZ1L (p) = ZA L oa (p) and a (a = aa (p) for a E A and p E OA 1 12 • Formula (26) in turn implies that (0' 0 id) o R(P) = Z1 R a(P) and a(pa) = o(p)a. We have already noted that a is bijective. Thus, o- is an isomorphism of bicovariant bimodules. For w E 4,1-' 1 and 7/ E (r2)inv, we have E o.;(_ 2) ®W( - i) w(0) = 1 01 (.4) and (0 ® 77(1) ® 77(2) = 1 ® 1 0 1. Inserting this into (24) we obtain (27). OA ® A 7/3 with w 1 inv- 11 wSince the elements p 2 - inv F2 and 173 E (f3) 1 nv generate ri OA T2 OA 1 13 as a left A-module, it suffices to prove the equality (28) for such elements p. Using (25) and (27), we get

En )

-

(0"

id)(id

o-)(o-

id)(p) = E(a

id)(id (7) (4) ®A W 141 41) OA n3 )

E(aOid)(4 ) 0ATI30Auil 2. Then 23 n, is the group with n - 1 generators s l , s2 , • • • , sn_ i and defining relations

sksk-Fisk = sk-Fisksk+i, S •S si si • =

k = 1, • • • n - 2,

S •S' si si ,

,

2.

The group On plays an important role in knot theory (see, for instance, [Ka], Chap. X). It is the group of equivalence classes of braids on n strands with composition of braids as group multiplication. For k =1, 2, • • • , n - 1, we define an automorphism ak of the bicovariant bimodule Fn by crk = id®A• • •®Ao -OA- • -0Aid, where the braiding a from Proposition 5 stands in the place (k,k+1). Since akak Fiak = cfk+1akak+1 by (28), al, 0. 2, • • •, an-1 satisfy the defining relations of the braid group On . SO there exists a unique group homomorphism r(s) of g3n into the group of automorphisms of the bicovariant bimodule F®n such that r(e) = id and r(sk) = 7k, k = 1,2,•• - n - 1. We give a more explicit description of the homomorphism r. Let tk denote the permutation from the symmetric group 7),2 which interchanges k and k +1 and leaves all other numbers fixed. Each permutation p E Pn can be written as et product p = t ic • •44,) , where 1(p) denotes the length of the permutation p. Then we set r(p) := air ,: • •cri i(p) . It can be shown that the latter is independent of the particular representation p =t il • • •tii(p) of p. Moreover, we have -

7r(PiP2)

= 7 030702) if /(PiP2) = 1030+ l(p 2 ) and P17P2 E Pn

(39)

For n E N and k = 0,1, • • •n we define Bk :=

E

(±1) 1(P) r(p)

(40)

p- lEP,Lk where Pn k is the set of (k, n-k)-shuffles, that is, the set of permutations p E Pn, such that p(1) < • • • < p(k) and p(k + 1) < - • • < p(n).

13.2 Tensor Algebras and Exterior Algebras of Bicovariant Bimodules

487

Let fc.di l ier be a fixed basis of the vector space in„F. Let rh and v.:, j j E ,

be as in Theorem 3. Recall that 2iR(w) = wi v.1 and ph := wi s(v)l iE , is a vector space basis of rinv. For i = (j 11. • •, i) E In we abbreviate the In a similar way we use the product wi , • • •win in the algebra TA (F) by multi-index notation ni and i,j E . Recall from Subsect. 13.1.4 that fc.di liE in is a basis of the vector space inv (F®n ) and that the braiding a leaves iriv (T °2 ) invariant. Thus any operator 7r(p), p E Pn , leaves ii,v (F®n ) invariant. Hence for each B E Lin tir(p) I p E Pn l there exists a pointwise finite matrix (131)i,jEin of complex entries such that B(wi ) = Blwi . For i = (j 17 . • •,ik) E / k and j = (ill • 1jin) E i m , we put T := (ik, - • • 3 il) and (i7i) (ill • ") i, • • •):771Z). Now we are able to give generalizations of the formulas in Examples 1.8 and 1.13. Proposition 8.

For n E N and E I , we have

2iT(Wi) = E (B± n k )("") L4), 0 k=0

(-1) n rir,

(41)

54,(coi ) = (-1) n-2' 1 )71r .

(42)

L ,

ST (w)

ri 46

vT,w j „ ,

4-, Pi) = k=0

Note that there are multi-index summations over j', r E 1k and j" E in - k in (41) and (42). For k = 0 and k = n the summands should be interpreted as 1 0 wi and wr respectively. Proof. We carry out the proof of (42). In order to prove the formula for 24,, we proceed by induction on n. For n = 1 this equation is obviously true. Assume . Put = (i1, i n ). that it holds for all i E In . Let i = (j 1,• •.7 i+1) E n By (42) and Theorem 3(i), 2c4,(win+1 ) = 1 0 win+1 + wm, vrn+ , . Therefore, by the induction hypothesis, equation (42) and the definition (1.42) of the No -graded product., we obtain

i4(wi)

=z

(w)

• 2i1, (win.+1) yi

E(Bri --k)P''''')w r

Carn ®Vimn+i

"W.

E E(-1)/(P)7(ki',in)

(u)

rOvLai,„ win+

lr —kuiruirk+i OVir,C.dittv,rk+i

k=0

(43) where the second sum is over all permutations p E Pn such that p-1 E 'Pnn.• The first summands of the interior sum in (43) correspond to summands (-1) 1 ( 3)7(3) of Bn—±iik withi3-1 _ Pn +1,k, where (i) = p(i) for i = 1, • • • ,n and 23(n + 1) = n + 1. In order to treat the second summands of the interior sum in (43), we observe that wv = v;f4(vnc.4.4 n = v,rolpwin by (29) and hence c.drvirk +' = k+ 18 "-1 crin - 'F28 k 4' • • • cr":". s n' c.d.., with - 3k+inik+2 3k4-2 111 k+3 3n2n+1

13. Hopf Bimodules and Exterior Algebras

488

j " = (ik-1-11• •

',in) and

s"

= (sk + i, • • sa ). Putting

:= (j17 • "IjklMk+i)

and = (Ti, • • rk, rk+1), it follows that the second summands are equal gi s it, to (-1) 1(P)+k (cork+i• • vi.t w if Let /3 pip' G Pn+ i, where \- ' )1 5 • 23 1 := tk+i• • •tn and the restriction to {1, 2, • - n} of p' c Pn+ 1 is p. It is easy to check that 13-1(i) = p--1( i ) for i = 1,2,• • • , k, 23 -1 (k +1) = n + 1 and 1 -1 + 1 ) , 23-1 (i) for i = k + 1, • • , n. Hence 23' is a shuffle permutation 5 (i of Pn+1,k+1 and we have 1 3) = n - k + l(p) = 4p1 ) + 101). Therefore, by (42), (-1) 1( P )+n-k ork+1. • •Grn 7r(P) = ( - 1) 1(73) 7(j3) and the latter is a summand of B:. The preceding shows that there is a one-to-one correspondence (

between summands of (43) and summands of (42) for E in+ 1 . By induction, this completes the proof of the first formula of (42). Formula (38) yields S(w) = - Th for i E I. Since the antipode of the Nograded super Hopf algebra TAM satisfies SI,(xky rn ) = ( -1 ) km514-(Ym)Sli(xk) for xft E ./1 ®k and y, E ./1®m, the second formula of (42) follows. 0

13.2.2 The Exterior Algebra of a Bicovariant Bimodule In this subsection we study quantum analogs of the antisymmetrizer and the exterior algebra for bicovariant bimodules. The basic idea is to replace the flip operator in the corresponding classical constructions by the braiding map

cr • For n E N and k = 0,1, •,n, we define

An := E (-1)/(07(p)

and

Ank := E

pEP

(-1)1(P)7(p).

pEP„k

The map A n is called the quantum antisymmetrizer of f®n. Let k E {1,2, • •, n — 1 } . Each permutation p E Pn admits unique representations p = rpip2 and P P3P4s, where r,s E Pnk and Pi, P3 C Pn (resp. p2, P4 E Pn ) leave the numbers k+1, • - • , n (resp. 1, • • -, k) fixed. Then we have l(p) -= l(r) + l(pi) l(p2) --= - l(p3) + 1(p4) + l(s) and hence (r) 7 (P1) 7 (P2) = 7r(P3)7(p4)7r(s) by (39). From these facts we con1T(P) = clude that Ari = A n k(Ak 0

An-k) (Ak

A n - k) -137;:k •

(44)

We shall use (44) also for k = 0 and k = n, where it is interpreted as An = AnoAn = An Bno and An = Ann An = A n BT. respectively. Let 6 := n = 2 en , where en := ker A. The elements of en can be considered as quantum analogs of symmetric tensors of degree n. From the first equality in (44) we see that 8 is a two-sided ideal of the tensor algebra

TA (r). The quotient algebra re^ := T.,(r)/e Tit (T)/e is called the exterior algebra of the bicovariant bimodule r over A. Clearly, r is again a graded algebra, that is, '

13.2 Tensor Algebras and Exterior Algebras of Bicovariant Bimodules

reA =

co

FAn,

FAO := A,

with

489

F, FArz := rgnierz , n > 2.

n=o The quantum antisymmetrizer A n is a homomorphism of the bicovariant bimodule F®n to itself as a is of F° 2 by Proposition 5. Therefore, 6 is a bicovariant subbimodule of TACT). Hence, the quotient F = TA(T)/6 is also a bicovariant bimodule over A. Taking Proposition 10 stated below for granted, 6 is a Hopf ideal of the No-graded super Hopf algebra T.:49t (f) defined in Proposition 7. Therefore, the quotient TeA = 71:4 9 (nie is also an N o -graded super Hopf algebra with structures induced from 71,34(F). Let (8 2 ) denote the two-sided ideal of the algebra TA(F) generated by 62 = ker (a - I). Since (6 2 ) is a bicovariant subbimodule of TA ( T') and a Hopf ideal of T(T) by the proof of Proposition 10 below, the quotient TA(F)/(6 2 ) is also a bicovariant bimodule over A and an No-graded super Hopf algebra. The following result is needed later in Subsect. 14.4.1.

I"

be a bicovariant bimodule over A and let F be a bicovariant subbimodule of ft (that is, F is an A-subbimodule of r and the left and right coactions of A on F are restrictions of the corresponding coactions on r). Then there exists an embedding F,A C Î , is the identity map on rAo = rAo =- A and the inclusion F c P on FA1 = F, such that 1-1A is an N o -graded super Hopf subalgebra and a bicovariant subbimodule of F.

Proposition 9. Let

Proof. By the universal property of the tensor algebra (see Subsect. 13.2.1), TA (1 )- such there exists an injective algebra homomorphism I : TA(F) that ga) = a and I(p) = p for a E A and p E F. We identify /(C) and for E TA(F), so TA(F) becomes a subalgebra of TACIT Since F is a bicovariant 2 and hence An, = A T, ['ion. (The tilde subbimodule of D , we have a = 6-[F° always refers to the corresponding objects for P.) Thus, 6 = 6 n TA (F) and the assertion follows immediately from the constructions of F,A and Proposition 10. 6 is a Hopf ideal of 774 (11), that is, 6 is a two-sided ideal of the algebra T(F) such that E (6) = {0 } , d1(6) g 60TACT)+TA(T)06 and 514,(e) C e.

Proof. Suppose that C c

= ker A. Since

{wi}iEin is a free left A-module basis of F® n, C can be written as a finite sum = aiwi with ai E A and AC = 0 implies that (An )lai = 0 for all j E in. Let ki denote the k-th summand of the sum for 21.44w1 ) in (42). Since a E Mor (y y) by (30), it follows that (Ak),v1; = v:,(Ak)1:. Using these facts and the second equality of (44) we obtain en

(Ak An_k)Zi(a i )Gi =

Zi(ai )(Bn-k ) i(il 'i") (Ak) , Ws 0 VI: (An-k)1::wr" A(ai )(0k An -OBn-k )fri ' r") w. 0 v:,wrn

= Zi (( An,)fri e ) ai ) we O v t' lwr = 0, n

490

13. Hopf Bimodules and Exterior Algebras

ker A n _ k C so Zgat gki E ker (Ak 0 A n _k) = ker A k 0 ./"° (n-k) f(8) k e k TA(F) TA (r) e n-k . Since Al,(() = L(a)L(w) EL 0 A(aigki by (42), this shows that Al-,(6) C 6 0 TA(T) + TA(f) 0 6. Let po c Pn denote the permutation with P0 (i) = n i 1. Since o- erbni ) = o-rins rh. by (32), we obtain that o-k(m) = (o-n _ k )174 for any i E and k = 1,2, • n - 1. Since o-k = 7r(tk) and an -k = 7r(tn-k) = 7r(PotkPo 1 ) by definition, it follows that (p) (T)) = ir(popp')r j for all p c Pn and hence A(i) = (An ) rti- for i E I. From the latter and (42) we get AS(() = An S(arwi ) = A n (S(w i )S(a1 )) = (-1)(" 42-1 ) A n (rtf)S( )

= ( - 1) ("2-1) T4S((An)la i ) =0, SO SI, (C) C ker An .=

en .

Thus, S.4,(6) C S. Since 6 is a two-sided ideal of TA(F), as noted above, we have proved that 6 is a Hopf ideal of TA (f ). D The construction of the exterior algebra and the above results have their 7r(p) and 2in := counterparts for the symmetric algebra over F. Let Sn := ker Sn . By similar reasoning as above, 2t IT177_2 2in is a Hopf ideal of TA(r). Hence the quotient TA(F)/2t is a Hopf algebra. This Hopf algebra TA(T)01 is called the symmetric algebra of the bicovariant bimodule F. In the case A --= C it is just the symmetric algebra of the vector space F. However, in the important case when f comes from a bicovariant differential calculus as in Sect. 14.6 we have 212 = ker (a + I) {0} if q is not a root of unity (see Subsect. 14.6.2), so the algebra TA(F)Mt is not of interest in this particular situation.

Ep

13.3 Notes The main reference for this chapter is [Wor4]. Yet ter -Drinfeld modules were introduced in [Yt] where they are called crossed bimodules. Their connections to bicovariant bimodules were first observed in [S2]. The relation to quantum double representations (in different forms) appeared in [PoW], [Maj4] and [Tk4]. Propositions 8 and 10 were found in [Dtl] and independently by one of the authors (K.S.).

14. Covariant Differential Calculus on Quantum Groups

This chapter contains the main concepts and results of the general theory of covariant differential calculi on quantum groups. The underlying Hopf algebra structure allows us to develop a rich theory of such calculi which is suggested by ideas from classical Lie theory. In the first two sections left-covariant and bicovariant first order calculi are studied. To each left-covariant FODC F we associate a right ideal 'TZT of ker E and a quantum tangent space Tri . If r is bicovariant, the quantum tangent space Tr carries an analog of the classical Lie bracket and is called the quantum Lie algebra of V. Sections 14.3 and 14.4 deal with higher order left-covariant and bicovariant calculi, respectively. One of the main results is that any bicovariant FODC admits a unique extension to a bicovariant DC on the exterior algebra which becomes then a differential Hopf algebra. Section 14.5 develops a general method for the construction of bicovariant DC on coquasitriangular Hopf algebras. In Sect. 14.6 this method is elaborated for the coordinate algebras of the quantum groups GL q (N), S L q (N), O q (N) and

Spq (N). Throughout this chapter A is a Hopf algebra with invertible antipode.

14.1 Left-Covariant First Order Differential Calculi 14.1.1 Left-Covariant First Order Calculi and Their Right Ideals

Definition 12.3 can also be applied to the Hopf algebra A considered as a left quantum space with respect to the comultiplication. Definition 1. A FODC r over the Hopf algebra A is called left-covariant if

is a left-covariant FODC r over the left quantum space X = A with left coaction = A according to Definition 12.3.

By Proposition 12.2, a FODC r over A is left-covariant if and only if there is a linear mapping AL F —) A TT such that AL(adb) = A(a)(id 0 46(b)

for all

a, b E A.

(1)

If such a mapping AL exists, then, by the proof of Proposition 12.2, AL is a left coaction of A on r which satisfies AL(apb) = A(a)AL(p)A(b) for

492

14. Covariant Differential Calculus on Quantum Groups

a, b E A and p E F. Thus, according to Definition 13.1, F is a left-covariant bimodule over A. Let F be a left-covariant FODC over A. First we restate some results obtained in Subsect. 13.1.1 in the present context. We define a linear mapping inv F by setting wr(a) = PL (da). If no confusion can arise, we omit wr A the subscript F and write simply ci.)(a). From Lemma 13.1 we get w(A) = 11F. Asshown in Subsect. 13.1.1, F = Aw(A) = w(A) A and any basis of the vector space w(A) is a left A-module basis and a right A-module basis of F. Since ,6L(da) = a( 1 ) da(2) by (1), it follows immediately from the definition of PL and (13.4) that

w(a) =

E S(a(o)da (2) and da =

amw(a (2) ) for a c A.

(2)

From the formulas (13.8), (2), and (13.3) and the Leibniz rule we obtain

w(a)

b = PL (E S(a(0)da( 2).b) =

E E(S(a( 1)))PL (da(2)•b) = PL (da•b)

= PL(d(ab)) — PL(adb) = w(ab) E(a)w(b) = w(ab). Recall the notation

a :=

a — E(a)1. Hence, by (13.5) and (13.6), we have

adR(b)(w(a)) ----z w(a) i b = LON, bw(a)c = >

( 3)

bc( l)w(ãc(2) ).

(4)

In order to characterize left-covariant FODC's as quotients of the universal FODC, we use another realization of the universal FODC over A which is better adapted to the Hopf algebra structure of A.

Example 1 (The universal first order differential calculus over A revised). Let f-1 1 (A) := A 0 ker E. Let us write aw(b) instead of a 0 6, where a, b E A. In particular, w(b) denotes the element 1 0 b for b E A. Taking into account the formulas (2) and (4), we introduce an A-bimodule structure of fl l (A) and a linear mapping d : A (21 (A) by c(bw(a)) := cbw(a), (bw(a))c bc(i) w(etc (2) ) and da := E a( i)w(a( 2)) for a, b, c E A. From the relations -

a•db + da•b =

E abmw(b( 2)) + E a(ow(a(2))b

E ab(ow(b( 2)) + E amb( 1)ci.)(a( 2)b(2)) E a( i)E(a( 2))b( 1)w(b( 2)) E a( 1 )b( i)w(a( 2)b(2)) = d(ab), w(a) = e(a( 1 ))44)(a( 2)) = E S(a(1))a(2)w(a(3)) = E S(a(i))da(2) (5) we see that d. satisfies the Leibniz rule and that f-1 1- (A) =

(2 1 (A) is a FODC over the Hopf algebra

A.

A dA, that is,

14.1 Left-Covariant First Order Differential Calculi

493

We prove that 0 1 (A) is the universal FODC over A. Let 1" be another We first show that the linear mapping FODC over A with differentiation r given by 7,b(a.db) = a • a is well-defined. For this we suppose : 0 1 (A) that Ei ai •dbi = 0 in 0 1 (A) with ai, bi E A. By the definitions of d and ,r-21 (A), we have E E.2, az- bi(1) w(bi(2) ) = E E.2 az- bi(1) Obi(2) = 0 in A 0 ker e. S id)(id 21) and using the Hopf Applying the mapping (m id)(id =0 algebra axioms, we compute Ei ( ai bi — ab i0 1) --- O. Hence Ei in r and 71; is well-defined. It is straightforward to verify that the FODC f is Isomorphic to the quotient FODC 0 1 (A )/ M, where W is the A-subbimodule kerli) of 0 1 (A). Therefore, f2 1 (A) is indeed the universal FODC over A. Of course, the universal FODC over A is unique up to isomorphism. Hence the universal FODC 0 1 (A) defined above and the universal FODC constructed in Subsect. 12.2.2 are isomorphic. By Corollary 12.3', the FODC Q 1 (A) is left-covariant. Moreover, formula (5) shows that w(a) is nothing but the 1-form PL (da) w o i (A ) ( a), a E A, in accordance with the notation A introduced above.

a.

For a left-covariant FODC

r over A

we define

:= {a E ker e w F (a) = O}.

(6)

7Zri gives a bijection The next proposition shows that the mapping 1" between left-covariant FODC over A and right ideals of ker E . Proposition 1. (1) Let R, be a right ideal of ker E . Then Ar := A 0).(-21(.A)(R) is an A-subbimodule of f2 1 (A) and the quotient f := if21 (A)/Ar is a leftcovariant FODC over A such that 1 ' = 1. (ii) We suppose that r is a left-covariant FODC over A. Then R. r, is a right ideal of ker E and the FODC r is isomorphic to the quotient FODC -

(21

(A) IA wf 21 (A) (R r) •

Proof. (i): Throughout the proof of (0 we write w(-) for w ni (A )(.). Since R. is also a right ideal of A, we have (4.) (a)c = Ec(o w(ac( 2) ) E A w(1) by (4) for a E R (C ker e) and c E A. That is, Al. = A w(R) is an A-subbimodule of (2 1 (A). Hence r = oi(A)I N. is a FODC over A. Let d i, : 01 (A) A 0 Q' (A) denote the coaction for the left-covariant FODC 0 1 (A). Since d (aw(b)) = d(a)(1 ®w(b)), we have 24,() g Ao.w. Therefore, di, passes to a left coaction of A on the quotient r = 'WAIN * which obviously has all the properties required in Definition 12.3. This shows that the FODC 1" is left-covariant. By the definition of d on the quotient FODC it is clear that R , C Te r. Conversely, suppose that wr(a) = 0 for some a E kere. Then w(a) E N = A w(R.). By the definition of f2 1 (A), this means that 10a E A0 R. which in turn implies a E Te. Thus, R. = R.1-'. (ii): Since r is left-covariant, (3) holds. Therefore, if a E R. and b E ker E 7 then cair. (a) = 0 and hence wr(ab) = adR(b) (W r (a)) = 0 by (3). Thus, R. is a right ideal of kere. The second assertion of (II) follows from (i). El

494

14. Covariant Differential Calculus on Quantum Groups

Proposition 2. Suppose that A is a Hopf *-algebra and f is a left-covariant FODC over A. Then f is a *-calculus if and only if S(RT)* C 1Zr •

Proof. Let

r be a *-calculus and let a E 'RE . Using the identity (S(a)*)

E S(a( 2))* S(a( 1 ))*, the formulas

=

(2) and (1.39), properties of the involu-

tion, and finally the Leibniz rule, we compute

wr(S(a) * ) = E S(S(a(2)) * )d(S(a(1)) * )

Ea ( 2 ) *( dS(a(0 ))*

= E((dS(a(1)))a(2))* --= E(d(S(a (1) )a(2) ) = (E(a)d1 wr(a)) * —wr(a)* = o, ( 7) so S(a)* E R. Thus, S(R. F)* CR T, . Conversely, suppose that S(R.r)* C R.r and write w for wo i( A ). Recall from Subsect. 12.2.2 that the universal FODC over A is a *-calculus. Therefore, by the calculation (7) applied to the *-calculus (2 1 (A) and by the assumption, we have (AO)* = —w(S(a)*) E (.0 (RT ) for a E IZT, so that w(R E )* C w(R. r). Since R.F is a right ideal, cd..)(7Z E )A C Ack)(7Z. r) by (4). Hence we get (Aw(TZT))* = w(RT)*A* C w(Te r )A C Aw(R.r), so the involution on (2 1 (A) passes to the quotient f2 1 (A)/Aw(1Z r) which therefore becomes a *-calculus. Since this quotient FODC is isomorphic to f by Proposition 1, I' is also a *-calculus. .

14.1.2 The Quantum Tangent Space

If f is the "ordinary" differential calculus on a Lie group G, then we have = (ker = ff E A1 f(e) = (df)(e) = 0} and the tangent space at the unit element can be considered as the vector space of all linear functionals on C°° (G) annihilating R.r, and the constant functions. This motivates R.F

Definition 2. For a left-covariant FODC f over A, the vector space

Tr := {X E A' I X(1) = 0 and X(a) = O for a 1ZT 1

(8)

is called the quantum tangent space toi. In what follows we will omit the subscripts f in Ti and RT Propositions 3 and 4 below show that T admits properties analogous to those of the classical tangent space at the identity of a Lie group. However, the commutation relations between elements of T are not necessarily quadratic as in the classical case (see Subsect. 14.1.4 for an example). .

Proposition 3. There is a unique bilinear form (.,.) : T x

(X,a.db) = E(a)X(b)

for

r=

a,b E A,

X E T.

C such that (9)

w(A) and T form a nondegenerate dual pairing The vector spaces inv with respect to this bilinear form (that is, (co), X) = 0 for all X E T implies cd.) = 0 and (w, X) = 0 for all cd.) inv implies X = 0), and we have

r

14.1 Left-Covariant First Order Differential Calculi

(w( a), X) = X(a) for a E A, X E T.

495

(10)

Ei

:= X(E i *Obi ). Proof. For ( = ai dbi E r and X E T, we define (X, We have to prove that this value does not depend on the particular representation a idbi of the form (. For this it suffices to check that the right-hand side of the defining equation vanishes if ( = 0. Indeed, by (13.3), = 0 yields 0 --- PL (() = e(ai )PL(dbi ) w(E i e(a)b), so e(a) E R. and hence X(E i e(a)b) = 0 by the definition of T. Thus, the bilinear form (., .) is well-defined. Since r = A dA, the bilinear form (., .) is uniquely determined by (9). Formula (10) follows at once from (9) and (2). The assertions concerning the dual pairing are easily verified.

Ei

Ei

Ei

It is clear from (8) and the fact that inv dual pairing that

r and T

form a nondegenerate

dim inv f = dim T = dim ker OZ.

(11)

r. r

We call this (cardinal) number the dimension of the left-covariant FODC Let us fix the following notation which will always be kept in the following: {Xi}i c i denotes a basis of the vector space T, Iwi l ie/ is the dual basis of 1 (that is, (X i , w) = (5ii for j j E I) and jE/ is the family of functionals from Proposition 13.2. ,

Proposition 4. With the above notation, we have for a and b in

da =

A that (12)

X i (ab) = e(a)Xi (b)+Ei X j (a)fi (b).

(13)

Proof. Since (Xi , w(a)) = Xi (a) by (10), we have w(a) = Ei Xi (a)w, and hence da = a( i )(.4.)(a( 2)) = E i a(i)Xi(a(2))wi = E i (Xi .a)wi which proves (12). Using (12) and the Leibniz rule, we get

E (X i .ab)(.4.4 = d(ab) = a•db + da•b =

.a(Xi .b)w i

E.a Xi.b)w ( i+ 2

213

Equating the coefficients of wi one obtains

X i .ab = a(Xi .b)+Ei (X i .a)(fii .b),

E I.

Applying e to this equation and using the identity e(f.c) = f(c) for f E A' and c E A, we obtain (13). Formula (12) indicates that the linear mapping Lx i: A A defined by Lx i (a) := Xi .a might be regarded as a quantum analog of a left-invariant vector field in classical Lie theory (see Subsect. 14.4.2 for more details).

496

14. Covariant Differential Calculus on Quantum Groups

Suppose now that the dimension of the vector space T is finite. Then we conclude from (13) and (13.14) that the functionals Xi and f; belong to the dual Hopf algebra A' (see Subsect. 1.2.8) and we have

A(Xi ) = 6

Xi + Xk

fik , 46 (fn = fØf:

(14)

in A°. Hence the linear span of functionals E, X i and g, i, r, s c I, is a subcoalgebra of A°. The antipode of Xi in the dual Hopf algebra A° is given by (15) -X k S(fh. Indeed, by (1.26) and (14), we have the relation

mAo(id S)1(X) -=

(60 S(Xi) + Xk S(fi k )) -=- O =

which in turn implies (15). Proposition 5. A finite-dimensional vector space T of linear functionals on a Hopf algebra A is a quantum tangent space of a left-covariant FODC if and only if X(1) = 0 and Zi(X) X E T 0 A° for all X c T

Proof. The necessity of the second condition follows at once from (14). To prove the sufficiency part, let R = fa E ker s X(a) 0 for all X c Tj. From the assumption d(X) --E0XET0 A° for X E T it follows that R. is a right ideal of ker e. By Proposition 1, there exists a left-covariant FODC F over A such that R, = 7Zr. Since T is finite-dimensional, the definition of TZ implies that T = {X E A' I X(1) = 0 and X(a) = 0 for a E 7Z}. That is, T is the quantum tangent space of F.

Proposition 6. Let

r

be a finite-dimensional left-covariant FODC over a Hopf *-algebra A. Then F is a *-calculus if and only if its quantum tangent space T C A' is *-invariant, where A' is endowed with the involution

Proof. By (1.40), the functional f* is defined by f* (.) = f (S(-) * ) for f E A'. If F is a *-calculus and X E T, we have X* (a) = X (S(a)*) = 0 for all a E by Proposition 2, so that X* c T. Conversely, if T is *-invariant and a E R,, then X (S(a)*) = X* (a) ---- 0 for all X E• T. Since T is finite-dimensional, this implies that S(a)* E R., so r is a *-calculus again by Proposition 2. 0 14.1.3 An Example: The 3D Calculus on SLq (2) -

Throughout this and the next subsection, A is the Hopf algebra 0(514(2)) and we assume that q2 1. Let a = n b = t4 c uf, d = u3 denote the entries of the fundamental matrix of A and let E, F, K be the generators of Oq (s12), see Subsect. 3.1.2. Recall from Subsect. 4.4.1 that there is a dual pairing of the Hopf algebras 6,7 (512) and A =-- 0(SL Q (2)) determined on the generators by the equations ,

,

, (E c) = (F, b) = 1 and zero otherwise. (16) (K , a) = q -1/2 , (K, = /q2

14.1 Left-Covariant First Order Differential Calculi

497

We define three linear functionals X 0 , Xi , X2 on A by A x i := (1 _ q-2)-1 ( 6 ____ K4) . .x.0 := q -112FKI x- 2 :____ q 112E7---1 By the definition (3.12) of the comultiplication of

2■Xi = E 0 X 3 ± Xi 0

K 2 for j=0, 2

iTg (s12) we have

and 2■X1 = E ® Xi +X1 0 K4 . (17)

Therefore, by Proposition 5, T := Lin {X0 , X i , X2 } is the quantum tangent space of a (unique up to isomorphism) left-covariant FODC r over A. This is the so-called 3D-calculus of S. L. Woronowicz. Let {w0,(.4.)1,w2 } be a basis of inv r which is dual to the basis {X0 , X1 , X2} of T. Then, by (12), we can write 2

(18) dx = E(Xi.x)wi, x E A. j=0 Since X0(b) = X2(e) = Xi(a) = 1 and X0 (a) = X0(e) = X2(a) = X 2 (b) = X i (b)= X2 (c)= 0 because of (16), we obtain Wci =----

w(b), W2 = w(c), wi = w(a) = -q -2 w(d),

(19)

da = bw2 + aw i , db = aw0 - q2 bw i , dc = av1 -Fdw2 ,dd= -q2 dwi±cwo. (20) Comparing (17) with (14) we see that fl --= K 2 for j = 0,2, na = K4 and f ;7 = 0 otherwise. This leads to the following commutation rules between matrix entries and 1-forms:

(via = q-l awi , wi b z--- qbw i , wi c=q-l avi , wi d=qdwi for

j = 0,2, (21)

(22) cvia =- q -2 aw1, wib = q2 bwi, w i c= q-2 av 1 , wid= q2 dwi. Let R, := B. A be the right ideal of kerE, where B is the vector space generated by the six elements a + q -2 d- (1+ q-2 )1, b2 , c2 , bc,

(a -1)b,

(a -1)c.

(23)

Using (16) one easily verifies that the functionals X0, X2 1 X1 annihilate these elements, hence R, C R, r. Since dim (ker E )/ R, < 3 by the definition of R, and dim (ker E )/ /RE = dim T = 3 by (11), we conclude that R, = R,r, that is, R, is the right ideal of kerE associated with the FODC r. The six generators s from (23) satisfy the condition S(x)* E R, for the Hopf *-algebras 0(SL q (2,R)), iqj =--- 1, 0(SUq (2)) and 0(SUq (1,1)), q E R. Hence, by Proposition 2, r is a *-calculus for all three Hopf *-algebras. From the commutation rules in the algebra Ûq (512) it follows that the basis elements of the quantum tangent space T satisfy the relations

q2 X1 X0 7

C 2X0X 1 = (1 + q2 )X0,

(24)

q2 X2 X1 - q -2 X1X2 = (1 + q 2 )X2 7 (25) qX 2 X0 - q -1 Xo X2 = —q -1 Xi . (26) In the limit q -p 1 all the above formulas give the corresponding formulas for the ordinary first order differential calculus on SL(2).

14. Covariant Differential Calculus on Quantum Groups

498

14.1.4 Another Left-Covariant Differential Calculus on SL q (2) We put

Xo := q -5/ 2 FK 5 , X2 := q5/2 EK5 , Xi := (1 - q -2)- 1( E.

K 4 ).

Note that X1 is the same as in Subsect. 14.1.3. Now we have

LX E

Xi + Xi

K 6 + (q -2 1)X1 X j, j= O,2.

By Proposition 5, there exists a left-covariant FODC F over Sid g (2) with quantum tangent space T := Lin {X0 , Xi , X21. This FODC F can be developed in a similar manner as in Subsect. 14.1.3, so we only mention the necessary modifications. Formulas (18)-(20), (24) and (25) remain valid without change. In (21) we replace q by q3 and q-1 by q-3 , and in (23) the two last generators of R must be replaced by (a q -2 )b and (a - q -2 )c. Instead of (22), we now have wia ==

q-2 aw1 (q-2

1)bo.)2,

wic = q-2 cw i + (q-2 1)dW2 1

= q2dw 1 (q -2 1)cw o . wib = q2 bw 1 + (q-2 - 1)ac.4.)0 , The commutation relation (26) becomes the equation q5 X2X0

q-5 X0X2 = -C 1 X1 2q -2 (q - q -1 )X?

(q - q ) 2 )q.

1 of the FODC F is just the As in Subsect. 14.1.3, the classical limit q ordinary first order calculus on SL(2). Note that in the case q = 1 the commutation relations of X0 , X1 and X2 for both FODC in Subsects. 14.1.3 and 14.1.4 give the relations of the classical Lie algebra sl(2,C).

14.2 Bicovariant First Order Differential Calculi 14.2.1 Right-Covariant First Order Differential Calculi At the end of Subsect. 12.2.3 we explained how Definition 12.3 and Proposition 12.2 have to be changed in the case of a right quantum space X and a right-covariant FODC F on X. We apply' these modified versions in

Definition 3. A FODC F over A is said to be right-covariant if F is a rightcovariant FODC over the right quantum space X = A with right coaction = A or equivalently if there exists a linear mapping AR : F F A such that (27) AR(adb) = A(a)(d id),6(b) for a, b E A. Suppose that F is a right-covariant FODC over A. Then F is in particular a right-covariant bimodule over A. We introduce a linear mapping n PR(da). Clearly, 71(A) = /inv . Since AR(da) = A Fin , by setting n(a) E da( i ) a( 2) by (27), it follows that li(a) =

E da(1).S(a(2)) and da = Eri(a (i) )a(2) for a

E

A.

14.2 Bicovariant First Order Differential Calculi

499

Bicovariant First Order Differential Calculi

For a Hopf algebra it is quite natural to look for FODC which are compatible with the comultiplication from the left and from the right. These are the bicovariant F0 DC's. Definition 4. A first order differential calculus over a Hopf algebra A is called bicovariant if it is both left-covariant and right-covariant. By Subsects. 14.1.1 and 14.2.1, a FODC F over A is bicovariant if and 4 and AR:r- 4 1-'0A only if there exist linear mappings A L : such that the equations (1) and (27) are satisfied. Suppose that r is a bicovariant FODC over A and let A L and AR be the corresponding mappings. From (1) and (27) we obtain (id 0 AR) 0 AL (adb) ==

(A L 0 id) o A R (adb)

E (id 0 AR)(a(i)b(i) 0 a(2)db(2)) E a( l )b( l ) a( 2)db( 2 ) a(3)b( 3), EGLI L id)(amdb (i) Oa(2 )b( 2) )

E a( i )b( i )

a (2 )db(2)

a(3)b( 3)

for a, b G A. Since r = A-clA, the preceding shows that (id 0 AR) 0 AL = (AL 0 id) o 2IR, so condition (iii) in Definition 13.3 is fulfilled. The validity of the two other conditions in Definition 13.3 have been already mentioned in Subsects. 14.1.1 and 14.2.1. Therefore, r is a bicovariant bimodule over A and the whole theory developed in Chap. 13 applies to F. The following formulas describe the right coaction AR on a left-invariant form w(a) and the left coaction AL on a right invariant form n(a) in terms of the right and left adjoint coactions of A on itself, respectively: A R(w (a)) = (w id) (Ad R(a)) , A

L (7) (a))

= (id 077)(AdL (a)),

a E A. (28)

We verify (for instance) the first equality and compute AR(w(a))

E AR(S(a( 1 ))da( 2)) = E A(S(a(1)))(d 0 id)A(a(2)) E S(a(2))da(3) S(a (1) )a (4) = E w(a(2) ) S(a(o)a( 3) id)(AdR(a)).

(c.i)

Next we characterize bicovariant FODC among left-covariant FODC in terms of their right ideals.

Proposition 7. Let r be a left-covariant FODC over A with associated right ideal R.. The FODC F is bicovariant if and only if R. is invariant under the right adjoint coaction (that is, AdR(R) C R.0 A).

Proof. If

r is bicovariant, then (28) holds. Since R. = {a

E kere I w(a) = OE

(28) implies that AdR(R.) C R. 0 A. Conversely, assume that AdR(R) C

500

14. Covariant Differential Calculus on Quantum Groups

R 0 A. By Corollary 12.3', the universal FODC 0 1 (A) is left-covariant. Similarly, 0 1 (A) is right-covariant (see the Remark at the end of Subsect. 12.2.3) and hence bicovariant. From formula (28) applied to the bicovariant FODC 0 1 (A) and from the AdR-invariance of R it follows that ZAR(A wrp(A)(TZ)) Ç Aci.)(21(..4)(7?)0A. Therefore, the right coaction /AR of (A) passes to the quotient 01 (A)/M, where )f :=-- A w pi ( A ) (R.) , the FODC and the quotient FODC 0 1 (AN is right-covariant. Since 1" is isomorphic is also right-covariant. to 0 1 (A)1 AT by Proposition 1(ii),

r

r

Let be a bicovariant FODC over A. We state two useful formulas for which will be needed later. For any a, b E A, we the braiding a of OA have a(w(a) OA w(b)) = (29) w(b ( 2) ) OA w(aS(b(i))b(3)),

r

r

(w OA w)AdR(a) = (id — (7)(a) OA w)Z(a)-

(30)

Formula (29) follows at once from (13.25) combined with (28) and (3). Using the Hopf algebra axioms, (30) is easily derived from (29). 14.2.3 Quantum Lie Algebras of Bicovariant First Order Calculi

r

Unless stated otherwise we suppose in this subsection that is a bicovariant FODC over A with right ideal R and quantum tangent space T such that dim T < a). For X and Y in T, we define a linear functional [X, Y] on A by setting

[X ,171(a) := (X 0 Y)(AdR(a)),

a E A.

(31)

Since the right ideal R is AdR-invariant by Proposition 7 and the elements of T annihilate R because of (8), it follows from (31) that [X ,Y] also annihilates R. Since X(1) --= Y(1) =-- 0, we get [X, r(1) = O. Hence we have [X, Y} E T by (8). Thus, C : X 0 Y --* [X, Y.] defines a linear mapping of T T to T. The assertions of Proposition 8 below show that the mapping C can be viewed as a quantum analog of the classical Lie commutator bracket. First we need some notation. Let (-, -) be the bilinear form on T x from Proposition 3. There exists a unique bilinear form (-, -) 2 on (T T) x such that inv(r

r)

(X

Y, w 1 OA 2)2 = (X, W 1 )(37, w2 ) for w

,

w2

inv r, X, Y E T (32)

Clearly, the bilinear form (-, -) 2 is nondegenerate as (-, -) is. Since the braiding a maps inv (i" OA II) into itself, its transpose at with respect to the nondegenerate bilinear form (-, -) 2 is a well-defined map of T 0 T into itself. Let mA, : A' A' A' denote the multiplication map of the algebra A', that is, mAi(f g)(a) = fg(a) = >f(a( I ))g(a(2)) for f g E A' and a E A. Recall from Subsect. 14.1.2 that T is contained in the dual Hopf algebra A°, because we have assumed that dim T
3; - a one-parameter family r±,z , z E CVO}, for GL q (2); - 2N calculi r±,z, = q-2 1 for SL q (N), N > 3; - two calculi 11F and L.. for 0q (N), Sp q (N) and SL q (2). It can be shown that different FODC of this list are not isomorphic at least if q is not a root of unity. Note that all these FODC have dimension N 2 . We close this subsection with some remarks about these FODC and Ft (see Sect. As noted in the paragraph before last, the FODC 14.5) are isomorphic. Hence their quantum Lie algebras coincide and the functionals Xiiri from (81) give another basis of the quantum Lie algebra of = T±, z . In the above notation, these functionals are written as

r

i +n — 21C• 23 =EtS(1 7n )1

4

i3

7

r = r±,z•

i7 — - 1 27 • • • 7 N.

(85)

Since S(l ni)/±in is equal to the functional Note that Ec = E if z = of defined by (10.39) for the universal r-form r = r zo , the basis elements the quantum Lie algebra of the bicovariant FODC are just the entries of the matrix E(L - El. This shows the importance of the matrix L = S(L - )L+ introduced in Subsect. 10.1.3 for the study of bicovariant differential calculi. Further, by Proposition 10.16(ii), .

r±,z

c:= Ec Ti LD -1 =

(86)

is a central element of A'. From formula (85) we see that c - c(1)E belongs to the quantum Lie algebra of .r+ , z and

tl(c) = Ej ak 0 Yk71, 1 where Ykn, := E(S(l k2 )/ ±in (D -1 ) i

(87)

If q is not a root of unity, it can be shown that the functionals Yk n,, k, n = 1,- - -,N, are linearly independent. Therefore, by (87) and the first paragraph of the proof of Proposition 11, the vector space T(c) defined by (36) coincides with the linear span of functionals X = Ec /ii . That is, the quantum Lie algebra of the bicovariant FODC F+ ,, is of the form TN, where c is the central element of A' given by (86). An analogous result holds also for the FODC and the universal r-form (r, ) 2 1 , because the passage from r to F21 interchanges the functionals 1+1 and Finally, we turn to the question of *-calculi. Suppose that the parameters q, s and y are real and A is one of the Hopf *-algebras in Proposition 10.10(i). Then the universal r-forms r and s are real and hence, as discussed in Sect. 14.5, the corresponding FODC r±,z , z = sy, are *-calculi.

r_,,

524

14. Covariant Differential Calculus on Quantum Groups

Next we assume that 141 = izi = Izol = 1, z 2 and A denotes one of the Hopf *-algebras from Proposition 10.10(ii). Then the universal r-forms r zo and (,.)21 on A are inverse real. Since z2 = zg, we have C = and hence C c* = ec. Therefore, it follows from the formulas (10.48) and (85) that all functionals (X 'i )* = (e l)* - 6iie = knm (D -1 )7X k/ r,, again belong to the quantum Lie algebra of Therefore, by Proposition 6, F± ,, is a *calculus. In particular, r±,z4 is a *-calculus by applying the preceding with z = 4. Setting z = +1 and zo = +1 we conclude that both FODC Ti+ and are *-calculi for 0q (n, 71), O q (71 1 n + 1) and Sp q (N; 14.6.2 Braiding and Structure Constants of the FODC

r±,,

The matrix coefficients (0-±)T of the braiding o-± and the structure constants (C±),nkim ,„ of the quantum Lie algebra of the FODC F± with respect to the bases Oii OA OM} and {X.0} are given by {

Proposition 22. ( , ±) ritir,nrs

(R fi„ rwzn(r? ±1 )172 (r? ±i )i9 (k f1 4;1 Dk_1 1)191

(c±) ri klm,r5 = (rRT2) ,yukn (i• ±1) xwrm (ik±1)gD y

LIruknulmurs

Proof. As noted in Example 10.3, D = cD i 6ii for some constant c 0. Since S2 (um n ) = an-l um n Dm by Propositions 9.10 and 9.13, we get zo ( r i) pwikp,-D i Di w ipp = fzo (uti l upk)D w—ip i p k-1D p

=

z o (S2 (un 7

S2 (ukP )) = f zo (u7 ,

=

ZoOk —i\wk )11:7 •

Moreover, for F+, z we have and i?' s y rz. Inserting these facts and equation (75) into the formulas in Proposition 21 we obtain the above formulas in the case of 17+ , z. The proof for F- i z is similar. 111

Let us turn to the spectral decomposition of the braiding map a±. Since RT. = (x/y)± 1 R., the assumption made in Sect. 14.5 is fulfilled and formula (80) gives the spectral decomposition of ± when the spectral decomposition of R known. Recall that a± then has precisely the eigenvalues E I, if fk± 1 possesses the eigenvalues A i , i E I. In the case Gq = GL q (N), S Lq (N) the eigenvalues of k are q and so a := o-± has the eigenvalues 1 , -q2 , --q -2 and satisfies the equation (a - I)(cr + q 2 1)(a + q 2 1)

0.

For the quantum groups Gq = O(N), Sp(N) the matrix Ik has the eigenvalues q,- q -1 ,Eq 6- N , hence a := 0-± has the eigenvalues 1, - q2, _ q - 2 7 N 9 q - N -q€N -2 , -4 - €N+ 2 and we have the equation (0._ 1-)( 0. +112.0(0. ±q -2 1-)(0._e_0( 0._ q -Nn(0. +q 6N-2 .0(0. +q -EN+2 .0 = 0.

14.6 Bicovariant Differential Calculi on Quantized Simple Lie Groups

525

In both cases the spectra of g+ and cr_ coincide and for q not a root of unity the operator g±+I is always invertible in contrast to the classical situation. The last formula shows another peculiarity that appears for the quantum groups 0q (N) and Spq (N). Recall that the space of 2-forms for the DC TeA from Theorem 17 is (T OA r)/ ker(o- - ./). Let T denote one of the FODC /-1± for Op (N) or Spq (N). In the classical limit q ---4 1 the eigenspaces of g± for the eigenvalues q N and q -N belong also to the symmetric part of T OA T. Thus, it seems to be natural in this case to take the quotient T A2 = (F 0A 1 )132 as the space of 2-forms, where 32 := ker(a - I) e ker (g - q N I) eker (a - q -N I). This definition has the advantage that the dimension of the space of leftinvariant 2-forms in ,r 2 is ( N22 ) just as in the case of an N2-dimensional rAn is then no longer an No classical first order calculus. But T A super Hopf algebra, because the comultiplication of TA(T) does not-grade map 32 to 32 0 A + A 0 32 when qN 1. Another disadvantage of this definition is that the dimensions of the spaces of higher order left-invariant forms inv rAn 7 n > 3, are smaller than in the classical case.

14.6.3 A Canonical Basis for the Left-Invariant 1-Forms In the preceding subsections we worked with the artificially chosen basis { aii } of the vector space i n, (f+ z ). A more natural and intrinsic subset of this space is formed by the projections

w(u) ==_ PL(duii-) __=_ S(u ik )du;i of the differentials duii of the generators ttii . The purpose of this subsection is to decide when the set fwiil is a basis of the vector space inv (F± z ) and ' to determine the basis transformation {9i.i } -0 fwiil in this case. Since {0 } is a basis of in, (T ±,z ), there is a complex N2 x N2 matrix = ((A±, z )Tir) such that

= (A±,,Yir 0,-,,,. It is clear that the set -PO is a basis of inv W I z ) if and only if the matrix ' A+, z is invertible. We shall use the notation A := q - C I ,

A + := q + q-1 and

B = (Bni) := (Drib„,,bii ),

where as earlier D i = q -22 for Gq =aL q (N), SL q (N) and Di =q2Pi for G q =0q (N), Sp q (N). Case 1 (Gq .--- aLq (N), 51 Lq (N)). Let us abbreviate z := E iAl i q -22 , -- The transformations A± ,, are s + 1 - q - 2 and s_ : =-- s - (2 2 N 4_ q2N2. -

A+ ,z = 4-1 zi ± (z - 1)B, A,z . _ Aq -2N- l z -l i + (.2.-1 — 1)B. The set ILwii i i, j = 1, 2, - • • , N} is a basis of the vector space inv(r± z ) if and only if z± lz± s. If this is fulfilled, the inverses of A±, z are given by

526

14. Covariant Differential Calculus on Quantum Groups

ATiz = qz -'1A - 1 / 7 • ki + (1 '''.- Z)a-f- , ZB),

where a±,,

Ai

_ q2N+i zA -i (i+ k1--.3 , -1

)ce- , z-B),

(z±ls± - s) -1 .

Case 2 (Gq = 0q (N), Spq (N)). Setting A± :-- A±,1 and kt := Eq N—E _ Eti e—N , we have

A+ = AEq N- €1. - A(C 1 )IC,

A_ = -Acq N-( / - 2B + A(C)IC.

The set fwii ( ti , j = 1, 2, • • • , NI is a vector space basis of in,r+, resp. inv f _, if and only if q 2N— 2 E 1, resp. 2+0+20 -1 O. In these cases the inverses AV are

A)I + (c -i )i k cti _ Aa+B ) ,

A -+1

7((EqN-E

A-1 =

--y((fe - A)I + (C -1 ) ti k ti + (A -

where A --.1 6qc-N 7 a , (2 + AA + 20 -1 ) --1 , 7 := Eqf - N A -1 (ti _ A)-'.

We close this subsection by considering the two FODC over 0(GL q (N)) which are obtained from the family r±,z in the particular case z = 1.

Example 8 (The calculi T +,1 and I' i for Glig (N)). Let Gq = GLq (N) and z =--- 1. In this case (83) holds with Oii replaced by wii , because the above transformation A+, 1 is a complex multiple of I. Therefore, we have dui..ur 3 8

= ui w . ur = ui ur fk±1\nmfrZ±Vci n

nj $

nmi

/ kx k

=,__ (1- ±1.vir un u rnik±10F/ Inm k x k

)38S(ux)d y tti

1j$Wsi Y _ (f?,±1)ir „nd,„m(k±1\ lc/ nm uk ill — Jis

(88)

or in matrix notation dill - U2

=--- q21 U1 - C11-12121 •

Thus for both calculi .r±, 1 we have linear commutation relations between the matrix entries u; and their differentials duii . For any higher order DC of f±,1 it follows from (88) by applying d that dui A du2 =---- —q21 dui A du2M1

.

Recall that by Proposition 9.11 the quantum space O(C) can be considered as a subalgebra of 0(aL q (N)) by identifying the generators x i and 74. Since (k±i)ili = eiski sii, under this identification (88) yields

dxi-xi = q±1- (k ±lAim x„dx. That is, on the subalgebra 0(q) the FODC left-covariant FODC FT from Theorem 12.6.

F±,i

of Glig (N) induces the A

14.6 Bicovariant Differential Calculi on Quantized Simple Lie Groups

527

14.6.4 Classification of Bicavariant First Order Differential Calculi It might be natural to find all bicovariant FODC over A for which the differentials du3i- of the matrix entries uji generate the left module of 1-forms. The "ordinary" differential calculus on the coresponding Lie groups obviously possesses this property. We first give some equivalent formulations of this condition and then we state the classification theorem. Lemma 23. For any left-covariant FODC F over ditions are equivalent: (i) F = Lin fa duii a E A, i, (ii) inv r -----.-- Lin {wii

A the following three con-

1,2, • • • ,N},

j = 1, 2, • • ,NI,

(iii) ker E = Ter ± Lin {14 - 6ii 1 I

j = 1, 2, • • • , N}.

Proof. (i)-)(ii) is clear, since PL(adu i ) = e(a)PL(dui ) e(a)w by (13.3). 00-00: Let a G ker e. By (ii), wr (a) = ci..i jj with a ii e C. Thus we 1)) = 0 and so a - aij (14 - 1 ) have co r (a - aij (ujj which proves (iii). (iii) -(i): Let a E A. By OW, Et is of the form a = aí(4 - 6kil) with a kjE C and x E Te r. Thus, wr(a) = w(ã) = akiwki = akiS(undu jz. and hence I" = A ci.) r (A) = Lin la du i }. EJ Theorem 24. Suppose that q is not a root of unity. Then there exist up to isomorphism precisely the following bicovariant FODC r over A such that dim F > 2 and F = Lin { a duji a E A, i,j = 1,2, • • ,N}:

- A = 0(aLi g (N)),N>3 : F±,z with z± 1-5±74 z,z c CVO}, - A = 0(G.4(2)) : 1-1± with z - 1( 1+ q2) 1+ 0, z E CVO; - A = 0(8L q (N)), N > 3: F± , z with z± 15± 74 s, zN q -2 ; ,,

- A = 0(84(2)) and A = 0(Spq (2)) : f+ and F_; - A = 0(0q (N)) and A = 0(8pq (N)), N > 4 : f+ and F_ with 2+ vA + 20 -1 740.

Two FODC over A of this list with different parameters are not isomorphic. For the quantum group 0q (3) under the assumptions of Theorem 24 there are precisely four such bicovariant FODC. These are the two 9-dimensional FODC f± from Sect. 14.6.1 and two additional 4-dimensional FODC developed in [S S3]. Theorem 24 and the preceding assertion show in particular that under the above assumptions for the quantum groups Sid q (N), °q (N) and Sp q (N) there is no bicovariant FODC whose dimension coincides with the dimension of the corresponding classical Lie group. Further, all calculi occurring in Theorem 24 have dimension N2 and belong to the family of FODC F±, z which was constructed by the general method of Sect. 14.5. The conditions z± 1 5± 5 for GL q (N) and Sli g (N),N>3, and 2 + /2A + 212A-1 74 0 for 0q (N) and Spq (N), N > 4, are only needed to ensure that

528

14. Covariant Differential Calculus on Quantum Groups

the corresponding FODC r±,z satisfy the assumption r =Lin {a du} or equivalently inv f = Lin {} (see Subsect. 14.6.3). If N is fixed, then except for the quantum group aL q (N) there exist only finitely many q for which these conditions apply. For all other values of q the list in Theorem 24 contains exactly 2N FODC for SL q (N), N > 3, and 2 FODC for 0q (N) and Spq (N), N > 4.

14.7 Notes The poineering work on covariant differential calculi on quantum groups is due to S. L. Woronowicz {Wor2J, [Worzli. Many results (but not all) in Sects. 14.1-3 appeared in the fundamental paper [Wor4]. The 3D-calculus in [Wr2] was the first example of a noncommutative differential calculus on quantum groups. The short approach given in Subsect. 14.1.3 follows the paper [SS4]. The results in Subsects. 14.3.2 and 14.3.3 on higher order left-covariant calculi are also from [SS4]. Proposition 11 is taken from [BM1]. The higher order bicovariant calculus 1'e^ was obtained in [Wor4j The fact that it yields a differential Hopf algebra was first observed in [Brz], but the proof therein works only for in. Complete proofs have been provided since then by [Dt1], one of the authors (KS.) and probably others. Note that our proof in the text was based on Proposition 13.10. Theorem 18 seems to be new. Quantum Lie derivatives and contraction operators appeared first in [SWZ11 and [AC], but a complete proof of Proposition 19 seems to be still missing in the literature. The method for the construction of bicovariant differential calculi developed in Sect. 14.5 is new at least in the general setting of coquasitriangular Hopf algebras. In the special situation of Subsect. 14.6.1 it gives the procedure used in [Jur] and [CSWW], where bicovariant calculi for the standard quantized simple Lie groups have been constructed in this manner. The two bicovariant calculi in Example 8 of Subsect. 14.6.2 have been found in [Mani.) and also in [Mal]. Theorem 24 was proved in {SS2J and [SS3]. There is now an extensive literature on differential calculi on quantum groups. In addition to the papers mentioned above, an (incomplete) list includes [Bd], [BMDS], [Ct], [CW], [DSWZ], [HS], [IV], [MH11], [Sud2], [Sud3], ISWZ2], [Ts].

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Index

Subject Index Action - adjoint 34, 181 - left 27,34 - right 27, 34 Algebra 7 - associative 6 Birman-Wenzl-Murakami 293 crossed product 349, 383 Drinfeld-Jimbo 163 free 8 h-adic 26 Hecke 293 homomorphism of 7 Hopf 4,13 - involution of 20 left comodule 33 left module 33 - multiplication of 7 - opposite 8 - right comodule 32 - right module 33 - unit of 7 - universal enveloping 18, 160 *-Algebra 20 - Hopf 20 - left comodule 33 - right comodule 33 Antipode 13 - inverse of 20, 247, 334 - order of 15 - polar decomposition of 426 - square of 15, 164, 247,334 341 Bargmann-Fock realization 151, 153 Basis at q = 0 226 Baxterization 296 Bialgebra 11 - braided 373

- comatched pair of 362 - coquasitriangular 331 cotriangular 331 double crossed coproduct 363 double crossed product 360 dual 22 FRT 304 - homomorphism of 12 - matched pair of 359 - quasitriangular 243 - triangular 243 *-Bialgebra 20 Bicharacter 24,336 Biedenharn-Elliott identity 88 12 Biideal Bimodule 28 - bicovariant 477 - - braiding of 481 homomorphism of 481 — tensor product of 481 - left-covariant 473 - right-covariant 477 Birman-Wenzl-Murakami algebra 293 Bosonizat ion 383 Braid - relation 247, 482 - group 173, 486 Braided - Hopf algebra 373, 376 - line 374 - matrix algebra 368 - product 373, 376 — algebra 378 - tensor category 378 - vector algebra 382 Braiding 373, 378,481 Brauer-Schur-Weyl duality 291

546

Index

Cartan - matrix 159 159 - - symmetrized - sub algebra 157 Category 376 - algebra in 378 - tensor, monoidal 377 - - braided 378 Character 422 Clebsch-Gordan coefficient 74,211 - reduced 219 Coaction - adjoint 34 - left 29,34 - right 29,34 Coalgebra 9 - cocommutative 9 - comultiplication of 9 - coopposite 9 - cosimple 398 - dual 22 - homomorphism of 9 - twisting of 357 *-Coalgebra 20 Coassociativity 9 Coboundary 355 2-Cocycle 354,357 - cohomologous 355 - counital 357 - invertible 357 - left 354 - right 354 Coefficient - Clebsch-Gordan 74, 211 - coalgebra 400 - q-binomial 39 - Racah 83 Coherent state 152 Coideal 9 Comodule 29 - algebra 32 - coalgebra 352 - left 30 - morphism of 30 - right 30 Compact quantum group - algebra 415 - C*-algebra 429,434 - matrix algebra 415 Cornultiplication 9 Contraction of a differential form 516 Convolution product 10

Coordinate algebra 98, 100, 304, 308, 310, 315, 320, 324, 388, 449 Copra duct 9 - smash 350 Corepresentation 29, 395, 429 - character of 406 - conjugate 398 - contragredient 398 - equivalent 30 - fundamental 305 - intertwiner 28, 397 - irreducible 30, 398 - isomorphism of 28 - matrix 30,397 - tensor product of 32, 397 - trivial 29 - unitarizable 402 - unitary 30, 114, 402 Counit 9 Crossed coproduct coalgebra 353, 382 Crossed product algebra 349, 383 Crystal basis 227, 231 - global 230,231

3D-Calculus 497,510 4D±-Ca1culs50 Diamond lemma 103 Differential - calculus 461 - - first order 457 - - left-covariant 463 - quotient 461 - - universal 463, 508 - - universal of a first order 463, 509 - *-calculus 462 - envelope 463, 508 - graded algebra 461 - Hopf algebra 511 - ideal 461 Double crossed - copro duct bialgebra 363 - product bialgebra 360 Drinfeld-Jimbo algebra 163 - center of 192, 194 - h-adic 166 - rational 165 Dual pairing 16, 21, 120,327 - nondegenerate 17, 120, 410, 440 Element - biinvariant

105, 443, 451

Index - canonical 256 - group-like 12 - invariant 30,324, 443, 451 - locally finite 181 - primitive 12 - twisted primitive 129 Exterior algebra - of a bicovariant bimodule 488 - of the quantum Euclidean space

Hecke algebra

293

Heisenberg - double 351 - representation 350 Hexagon relation 90 4, 13 Hopf algebra - braided 373, 376

320

- of the quantum plane 102 - of the quantum symplectic space 324

- of the quantum vector space Euler derivation 469

310

First order differential calculus

457

-

bicovariant 499 - classification of 527 - construction of 517, 521 - quantum Lie algebra of 501 - left-covariant 459, 491 dimension of 495 quantum tangent space of 494 quotient 493 right ideal associated with 493

right-covariant 464,498 - universal 463,492 Flip operator 6 Fock representation 149 Fourier transform on SUg (2)

- coquasitriangular 331 - cosemisimple 402 - deformation of 26 - differential 511 - dual 22 - factorizable 372 - h-adic 26,60 16 - homomorphism of - No-graded super 23 - quasi-cocommutative 248 - quasitriangular 61, 243 - super 24 - triangular 243 Hopf - *-algebra 20 - bimodule 477 16 - ideal

Intertwining operator

28, 30

Jordan-Schwinger realization

138

424 KMS condition Kostant partition funtion 176

117

FRT bialgebra 304

Leibniz rule 458 - graded 461 - twisted 506, 515, 516

Function - basic hypergeometric 41 - coordinate 5 - g-exponential 47

L-functional 276, 284, 342, 345

Linear form

- g-gamma 48 - g-trigonometrical 47 - spherical 129 Fundamental matrix 302

- AdR-invariant 335 - dominant 179 - integral 179

Linear functional - central 422 111, 187 - invariant - left-invariant 403 - right-invariant 128, 403

Gel'fand-Tsetlin basis 214 GNS representation 421 Group algebra 19

Haar functional 112,404 - faithfulness 419 Haar state 114, 420, 431

L-operator 275

Matrix coefficient, element

h-Adic

30, 74,

276, 396 Maurer-Cartan formula 507 Modular automorphism group Module 27

- algebra

26 - Drinfeld-Jimbo algebra 166 - Hopf algebra 26 - topology 25 Harish-Chandra homomorphism

547

- algebra 33,349 193

- coalgebra 33

425

548

Index

- isomorphism of 28 - left 27 - morphism of 28 - right 27 - simple 28 - Verma 178 - Yetter-Drinfeld 478

Multiplicative unitary Multiplicity of weight

427 177, 203

No-Graded super Hopf algebra 489, 511,512

23,

Partial derivative 466 176, 295 Partition function Pentagon equation 427 Peter-Weyl decomposition 110, 404, 410,439 Poincaré-Birkhoff-Witt theorem 160, 176

Polynomials - big q-Jacobi 52 - little q-Jacobi 51 - q-Hermite 50

Product - braided

373, 376 - covariantized 368 - smash 350 q-Binomial coefficients 39 q-Coherent state 152 q-Differentiation 44 q-Factorial 37 q-Integral 46 q-Number 37 q-Oscillator algebra 133 - symmetric 134

Quantum - algebra -

-

53 antisymmetric subspace 273 antisymmetrizer 488 Borel subgroup 436 Casimir element 54, 190, 248 codouble 359,363

- determinant 99,312 - dimension 207, 442 - double 252, 356, 361, 483 - Gerfand pair 446,450 - - infinitesimal 454 Quantum group

- compact 317,326 - GL q (N) 315 - GL q (N;R) 316

- inhomogeneous 387,392 320 - 0q (N) - CMN; R) 326 0q (N; v) 326 - Oq (n, n) 325 - 0q (n, n 1) 325 315 - SL q (N) 316 - SL q (N;R) - S L q (2) 100 102 - SLi g (2,R) 323 - S0q (N) 326 - S0q (N;R) - Sp q (N) 320 - Sp q (N; R) 325 - Sp q (N; v) 326 317 - Stlq (N) SUq (N; v) 317 102 - SUq (1, 1) 102 - SUq (2) C*-algebra 118 326 - USp q (N) - Uq (N) 316

Quantum - Euclidean space 320 - homogeneous space 442,451 - hyperboloid 472 - Killing form 189 - Lie algebra 501 - Lie derivative 515 - matrix space 98, 304, 310 - orthogonal sphere 322 - plane 101 - projective space 453 - space 32,436 125 -- isomorphism of - left 32,438 — right 32,436 - sphere 449 - 2-sphere 126 129 - spherical function

- subgroup 105,323 273 - symmetric subspace 324 - symplectic space - tangent space 494 - trace 206 - unitary group 316 - vector space 308, 310 - Weyl algebra 469 Yang-Baxter equation 61, 245, 247, 297, 333, 337

- zonal spherical function 450,453 space 33 *-Quantm 358 Quasi-Hopf algebra

Index

Racah coefficients 83 Real form 21, 58, 102, 172, 316, 325 - compact 172 Realificat ion 363 Reflection equation 346, 368 Representation 27 adjoint 35 character of 203 28, 200 completely reducible conjugate 35 cyclic 69, 145, 234 Fock 149 intertwiner of 28 - irreducible 28 N-finite 140 of class 1 217 of type 1 62, 178, 213 rational 31, 484 self-dual 271 semicyclic 70, 145 tensor product of 28, 69 weight 64, 177 with highest weight 61, 70, 141, 177, 199 with lowest weight 70, 141 *-Representation 28, 123, 420 R-Matrix 61, 247, 264 - of a vector representation 270 - universal 61, 243, 254, 259, 261 Root 157 - negative 158 positive 158 simple 158 subspace 157 - vector 175 Rosso form 188, 257 Schur orthogonality relations 419 Serre relations 160 Shuffle 19 Skew-copairing 358 Skew-pairing 250 - invertible 250 Subcoalgebra 9 Sweedler notation 10,31 Sweedler's Hopf algebra 19, 244, 337 Symmetric algebra 19 - of a bicovariant bimodule 490 Tensor - algebra

- - of a bicovariant bimodule 485

549

8, 19, 24 - - of a vector space - product 8 algebra coalgebra 9 of bicovariant bimodules 481 of corepresentations 32, 397 28, 69 of representations - operator 93, 219 Theorem 160, 176 - Poincaré-Birkhoff-Witt - Wigner-Eckart 93, 219 295 Transfer matrix 375 Transmutation 83 Triangular condition Universal - enveloping algebra 18, 160 - r-form 331 - - factorizable 372 - inverse real 336, 341 -- real 336,340 -- regular 366 61, 243, 254, 259, 261 - R-matrix Vector - representation 210, 267 64 - weight - - highest 64, 177 *-Vector space 20 Verma module 179 Weight 64, 177 - highest 64, 177 177, 203 - multiplicity of - representation 64, 177 64, 177 - space - vector 64 Weyl group 158 93, 219 Wigner-Eckart theorem Yang-Baxter equation 61, 245 - quantum - rational 297 297 - trigonometric Yetter-Drinfeld condition 478 Yetter-Drinfeld module 478, 484 289 Young diagram

Index

550

Index

of Symbols

An , A n k

AdL, AdR

A' A"P

Jr' 13,"13

Ag A[[h]] A[m 7-1.] A(R) A(R) A g (N) A g (N 2t, 2.1n [a], [a]q [ a] , [ a] q (a; q), (a; q),, a v adL, adR a. f

BWMr (p, q) 93„ Zg Cq

488 34 16 22 9 8 14 134 133 25 105 303 310 469 471 490 37 37 38 27 34, 181 36 486 293 453 486 174 54

c"i j,i121+i

82

Cg (t, j)

74 74 47 5 400 445 271, 317 310 25 25 8 11, 303 47 45

i

C, (11 , 12 , 1; i, m) Cosg z C(T) p) C( v) Cbi

c

(

,

311 99, 312 99 438 254 252

D (A, 8; a) D co (A, B; 0 .)

d id°

ai Ec,

Ear Er n)

252 359 457 511 466 157 175

Er

174 47 169

e(z)

106 47

E (z)

Élq - (Z)

Ern) For

174 175 170 422 106 36 313 5 157 21 157 50 293 420 453 51 8 515

Fr

fz f .a G L (N) gii

lin(x q) Hr (q) 7- ( A) 7--C; h(x q)

i(X r ) id

3,7

164

KA I K2p

K() KG

s)

r

Li,-

Lf

Lin

{

u ji }

L±,

r+ L r(A, 13) £(10 1+ , 1 - , 1; 1 i+i

14

i

,xn]

176 274 6 19 13 452 275 514 11 346 276, 346 285 10 27 342 346 276

Index [t, j)

77 285 412 310 377 397 13 7 38 208

0(S p :1 1 , v)) 0(SUq (2)) 0(SUq (N)) 0(Stlq (N; v)) 0(SUq (1, 1)) 0(Uq (N)) P, P± P(Gq )

ni AA' x [ Tri n] q, [7,1 1

208

p.„(,a , P)(x ;

n + , n_ 0(.13)

159 436

((*

101

O(C)

310 435 313 316 392 13 97 310 318 320 325 326 326 325 325 316 125 97 99 102 314 316 325 318 325 324 326

tzi M (n, r) Mq (N)

Mor (X, Y) Mor (u, y) M N (I C) 771

[nr

]

4

!

O (D N)

0(G L q (N)) 0(GL q (N R)) 0(IGq ) Q(M N (1K)) 0(Mq (2)) 0(Mq (N))

0(0q (N)) 0(0 :1 1 ) 0(0q (N; R)) 0(0q (N; u)) 0(0 :1 1 (6, 14) 0(0q (n,n + 1)) 0(04n, n))

(9(11 :1 1. ) O(S) 0(S L(2)) 0(S L q (2)) C9(SL q (2,1' ) 0(S L q (N)) C9(S L q (N; 0(80q (N; R)) 0(Sp q (N)) 0(Sp q (N; • )) )

0(Sp) 0(Sp q (N; v))

d q)

P + , P_ 39 P(g), Pr (0) Pnk q (0) Pn(x; a, bl q)

(21 Q+ Qq Q(q) qi

R

(7) Rep (G) Rep(_4(0) Rvw vw R q (1112131 112123 7 1) R,

rvw rz ryx

ST sisSill q Z 50q (N)

Spec (so), Spec (3,) SR, 8;1 8(.1) Sing Z TalA

TA

Tn rrV

551 326 102 317 317 102 316 179 410

52 273, 309 274 289 20, 486 210 51 164 319 165 161 89 5 232 246, 266 246 83 269, 309, 318 61, 243, 261 493 244 264 331 331 332 339 372 3, 13 426 370 322 485 47 323 195 509 509 47 68 178 199 179

Index

552

199 61

Twi Li

TA (11 ), Ts84 (11 ) T_ Tr T(c) ti( ti) U(g) Uh(g) Uh (5 12)

(g1 N)

q (S12)

Ûq (s12) • (512(R)) item Uq (sliv) Uq (s1(N,R) C7q (soN)) Uq (su2)) Uq (Sill ,1)) •

(SU

N

q• (511(a

• • •

I

al))

Uq (b +), Uq (b_) Ug,(2)

291 485 504 494 502 106 18, 160 166 60 163 53 57 59 164 173 288 59 59 172 172 168 161

• (13) Ùq (g)

189 164

• (g) • (13)

281

Uq(n+) Uq(1—) Uf ( n+)

UV3 (11.--)

U(R) V[[11]] vii va (v, f) Wa

XL(f, R), XR(f; R) Xq,-y z ?'

hq

A

r+1 r-

286 168 168 169 285 5, 11, 303 100, 303 25, 202 30 27 479 158 307 471 437 192

473 477 517

invr Fjflv rC

rr,

rq (n)

48

AO*

3,9 158 442 10 158 473 477 485 3,9 485 7 498 100 102

A(C)

310

A(0 :1 1 )

320

A(0 :1 ( 6, u))

326

A(8p)

324

A(Sp iliv (6, u))

326 24 462 492 525 488 481 483 250 6 43

bi i

6(n)

i6+ L • R

•

46 T

ET ET

77(.)

A(V) Q(X)

On (X)

w(*) Lou

en

o. a2,7 71 .

201(a, 0; -y; q, z)

er

2cpi (a, b; c; q, z)

ou j

E a( i )

408 40 36, 396 395 463 518

a(2)

10

E v(0 ) v ( 1 )

31

f(x)d q s

46

461 457 467, 504

FA 2 rAn

461

( . 7 ')

11‘`

488

1*,

Rh\

463

HI

26 16 500