HOPF Algebras: An Introduction

HOPF ALGEBRAS An Introduction Sorin DGsc5lescu Constantin Ngst5sescu +rban Raianu HOPF ALGEBRAS PURE AND APPLIED MA...

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HOPF ALGEBRAS An Introduction

Sorin DGsc5lescu Constantin Ngst5sescu +rban Raianu

HOPF ALGEBRAS

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Zuhair Nashed Universityof Delaware Newark, Delaware

Earl J. Taft Rutgers University New Brunswick, New Jersey

EDITORIAL BOARD M. S. Baouendi University of California, Sun Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi Universityof California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Universitat Siegen

W. S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

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W. L. Voxman and R. H. Goetschel, Advanced Calculus (1981) L. J. Corwin and R. H. Szczarba, MultivariableCalculus (1982) V. I.Istrijtescu, lntroduction to Linear Operator Theory (1981) R. D. Jawinen, Finite and Infinite Dimensional Linear Spaces (1981) J. K. Beem and P. E. Ehdich, Global Lorentzian Geometry (1981) D. L. Armacost, The Structure of Locally Compact Abelian Groups (1981) J. W. Brewer and M. K Smith, eds., Emmy Noether: A Tribute (1981) K H. Kim, Boolean Matrix Theory and Applications (1982) T. W. Wieting, The MathematicalTheory of Chromatic Plane Ornaments (1982) D. B.Gauld, Differential Topology (1982) R. L. Faber, Foundationsof Euclidean and Non-Euclidean Geometry (1983) M. Canneli, Statistical Theory and Random Matrices (1983) J. H. Camrth et a/., The Theory of Topological Semigroups (1983) R. L. Faber, Differential Geometry and Relativity Theory (1983) S. Bameff, Polynomials and Linear Control Systems (1983) G. Karpilovsky, Commutative Group Algebras (1983) F. Van Oystaeyen and A. Verschoren, Relative lnvariants of Rings (1983) I.Vaisman, A First Course in Differential Geometry (1984) G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984) T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984) K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) T. Albu and C. Ni3stdsescu, Relative Finiteness in Module Theory (1984) K. Hrbacek and T. Jech, lntroduction to Set Theory: Second Edition (1984) F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings (1984) B. R. McDonald, Linear Algebra Over Commutative Rings (1984) M. Namba, Geometry of Projective Algebraic Curves (1984) G.F. Webb, Theory of NonlinearAge-Dependent Population Dynamics (1985) M. R. Bremner et a/., Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985) A. E. Fekete, Real Linear Algebra (1985) S. B. Chae, Holomorphy and Calculus in Normed Spaces (1985) A. J. Jeni, Introduction to lntearal Equations with ADDlications (1985) G. ~arpilovsky,Projective ~epreseniationsof ~ i n i t Groups e (1985) L. Nariciand E, Beckenstein, Topological Vector Spaces (1985) . . J. Weeks, The Shape of Space (1985) P. R. Gribik and K 0. Kortanek, Extremal Methods of Operations Research (1985) J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis (1986) G. D. Crown et a/., Abstract Algebra (1986) J. H. Carmth et a/., The Theory of Topological Semigroups, Volume 2 (1986) R. S. Doran and V. A. Belfi, Characterizationsof C'-Algebras (1986) M. W. Jeter, Mathematical Programming(1986) M. Altman. A Unified Theorv of Nonlinear O~eratorand Evolution Eauations with ~p~lications (1986) A. Verschoren. Relative Invariants of Sheaves (1987) . , R. A. Usmani, .Applied Linear Algebra (1987) P. Blass and J. Lang, Zariski Surfaces and Differential Equations in Characteristic p > 0 (1987) J. A. Reneke et aL, Structured Hereditary Systems (1987) H.Busemann and B. B. Phadke, Spaces with Distinguished Geodesics (1987) R. Harte, lnvertibility and Singularity for Bounded Linear Operators (1988) G. S. Ladde et a/., Oscillation Theory of Differential Equations with Deviating Arguments (1987) L. Dudkin et a/., Iterative Aggregation Theory (1987) T. Okubo, Differential Geometry (1987)

113. D. L. Stancland M. L. Stancl, Real Analysis with Point-Set Topology (1987) 114. T. C. Gard, lntroduction to Stochastic Differential Equations (1988) 115. S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1988) 1 1 6. H. Strade and R. Famsteiner. Modular Lie Alaebras and Their Representations (1988) 117. J.A. Huckaba, ~ornmutative'~in~s with ~ e r i ~ i v i s o(1988) rs 118. W. D. Wallis. Combinatorial Desians (1988) 1 1 9. W. Wi@aw,.Topological Fields (1988) . 120. G. Karpilovsky, Field Theory (1988) 121. S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rinas 11989) 122. W. ko.howski, Modular Function Spaces (1988) 123. E. Lowen-Colebunders. Function Classes of Cauchy Continuous Maps (1989) 124. M. Pavel. Fundamentals of Pattem Recoonition 11989) 125. V. ~aksh'mikanthamet a/., Stability ~ n a l ~ofi son linear Systems (1989) 126. R. Sivaramakrishnan. The Classical Theorv of Arithmetic Functions (1989) . . 127. N. A. Watson, parabolic Equations on an lilfinite Strip (1989) 128. K. J. Hastinas. introduction to the Mathematics of Operations Research (1989) 129. B. Fine, Algebraic Theory of the Bianchi Groups (19i39) 130. D. N. Dikranjan et a/., Topological Groups (1989) 131. J. C. Moraan 11. Point Set Theorv 11990) 132. P. ~ i l eand r A. '~itkowski,problems in ath he ma tical Analysis (1990) 133. H. J. Sussmann. Nonlinear Controllabilitvand O~timalControl (1 . 990) , J.-P. Florens et a/., Elements of ~ a ~ e s i ~tatisbcs an (1990) N. Shell, Topological Fields and Near Valuations (1990) B. F. Doolin and C. F. Martin, lntroduction to Differential Geometry for Engineers

(1990) S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method (1990)

J. Okninski, Semigroup Algebras (1990) K. Zhu, Operator Theory in Function Spaces (1990) G. B. Price, An lntroduction to Multicomplex Spaces and Functions (1991) R. B. Darst, lntroduction to Linear Programming (1991) P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991) T. Husain, Orthogonal Schauder Bases (1991) J. Foran, Fundamentals of Real Analysis (1991) W. C. Brown, Matrices and Vector Spaces (1991) M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces (1991) J. S. Golan and T. Head, Modules and the Structures of Rings (1991) C. Small, Arithmetic of Finite Fields (1991) K. Yang, Complex Algebraic Geometry (1991) D. G. Hoffman et a/., Coding Theory (1991) M. 0. Gonzdlez. Classical Comolex Analvsis (1992) . , 152. M. 0. ~ o n z d l ecomplex i ~ n a l ~ s(1 i 99i) s 153. L. W. Bagoett. FunctionalAnalysis (1992) 154. M. ~nied&ich, Dynamic programming (1'992) 155. R. P. Aganval, Difference Equations and Inequalities(1992) 156. C. Brezinski. Biorthogonalityand Its Applications to NumericalAnalysis (1992) 157. C. Swarfz, An lntroduction to FunctionalAnalysis (1992) 158. S. B. Nadler, Jr., Continuum Theory (1992) 159. M. A.Al-Gwaiz, Theory of Distributions (1992) 160. E. Perry, Geometry: Axiomatic Developments with Problem Solving (1992) 161. E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering (I992) 162. A. J. Jeni, Integral and Discrete Transforms with Applications and Error Analysis 11 897) \'---I 163. A. Charlier et a/., Tensors and the Clifford Algebra (1992) 164. P. Biler and T. Nadzieja, Problems and Examples in Differential Equations (1992) 165. E. Hansen, Global Optimization Using Interval Analysis (1992) 166. S. Guene-Delabriere. Classical Sequences in Banach Spaces (1992) 167. Y. C. Wong, IntroductoryTheory of Topological Vector Spaces (1992) 168. S. H. Kulkami and B. V. Limaye, Real Function Algebras (1992) 169. W. C. Brown, Matrices Over Commutative Rings (1993) 170. J. Loustau and M. Dillon, Linear Geometry with Computer Graphics (1993) ' 171. W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (I993)

172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195.

E. C. Young, Vector and Tensor Analysis: Second Edition (1993) T. A. Bick. Elementary Boundary Value Problems (1993) M. Pavel, Fundamentalsof Pattern Recognition: Second Edition (1993) S. A. Albeverio et a/., NoncommutativeDistributions (1993) W. Fulks, Complex Variables (1993) M. M. Rao. Conditional Measures and Applications (1993) A. Janicki and A. Wemn, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (1994) P. Neittaanmeki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1994) J. Cmnin, Differential Equations: lntroduction and Qualitative Theory, Second Edition (1994) S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) X. Mao. Ex~onentialStabilitv of Stochastic Differential Eauations 11994) . . 6.S. Thomson, Symmetric properties of Real Functions11994) J. E. Rubio. Ootimization and Nonstandard Analvsis (19941 J. L. ~ u e s et'al., o Compatibility, Stability, and ti eaves (1995) A. N. Micheland K. Wang, Qualitative Theory of Dynamical Systems (1995) M. R. Damel. Theory of Lattice-OrderedGroups (1995) 2. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications (1995) L. J. Corwin and R. H. Szczarba, Calculus in Vector Spaces: Second Edition (1995) L. H. Erbe et ab, Oscillation Theory for Functional Differential Equations (1995) S. Agaian etal., Binary Polynomial Transfomls and Nonlinear Digital Filters (1995) M. I.Gil', Norm Estimations for Operation-ValuedFunctions and Applications (1995) P. A. Grillet, Semigroups: An Introduction to the Structure Theory (1995) S. Kichenassamy, Nonlinear Wave Equations (1996) V. F. Kmtov, Global Methods in Optimal Control Theory (1996)

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HOPF ALGEBRAS An Introduction

Sorin Di!isciilescu Constantin N3sti9sescu Serban Raianu University of Bucharest Bucharest, Romania

M A R C E L

MARCEL DEKKER, INC. D EK K E R

Library of Congress Cataloging-in-PublicationData Dascalescu, Sorin. Hopf algebras : a n introduction I Sorin Dascalescu, Constantin Nastasescu, Serban Raianu. p. cm. - (Monographs and textbooks i n pure and applied mathematics ; 235) Includes index. ISBN 0-8247-0481-9 (alk.paper) 1. Hopf algebras. I. Nastasescu, C. (Constantin). 11.Raianu, Serban. 111. Title. IV.Series

This book is printed on acid-6ee paper.

Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000: fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http:llwww.dekker.com The publisher offen discounts on this book when ordered in bulk quantities. For more information, write to Special SaleslProfessional Marketing at the headquarters address above. Copyright O 2001 by Marcel Dekker, Inc. AU Rights Resewed. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit) 1 0 9 8 7 6 5 4 3 2 1

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Preface A bialgebra is, roughly speaking, an algebra on which there exists a dual structure, called a coalgebra structure, such that the two structures satisfy a compatibility relation. A Hopf algebra is a bialgebra with an endomorphism satisfying a condition which can be expressed using the algebra and coalgebra structures. The first example of such a structure was observed in algebraic topology by H. Hopf in 1941. This was the homology of a connected Lie group, which is even a graded Hopf algebra. Starting with the late 1960s, Hopf algebras became a subject of study from a strictly algebraic point of view, and by the end of the 1980s, research in this field was given a strong boost by the connections with quantum mechanics (the so-called quantum groups are in fact examples of noncommutative noncocommutative Hopf algebras). Perhaps one of the most striking aspect of Hopf algebras is their extraordinary ubiquity in virtually all fields of mathematics: from number theory (formal groups), to algebraic geometry (affine group schemes), Lie theory (the universal enveloping algebra of a Lie algebra is a Hopf algebra), Galois theory and separable field extensions, graded ring theory, operator theory, locally compact group theory, distribution theory, combinatorics, representation theory and quantum mechanics, and the list may go on. This text is mainly addressed to beginners in the field, graduate or even undergraduate students. The prerequisites are the notions usually contained in the first two year courses in algebra: some elements of linear algebra, tensor products, injective and projective modules. Some elementary notions of category theory are also required, such as equivalences of categories, adjoint functors, Morita equivalence, abelian and Grothendieck categories. The style of the exposition is mainly categorical. The main subjects are the notions of a coalgebra and comodule over a coalgebra, together with the corresponding categories, the notion of a bialgebra and Hopf algebra, categories of Hopf modules, integrals, actions and coactions of Hopf algebras, some Hopf-Galois theory and some classification results for finite dimensional Hopf algebras. Special emphasis is

PREFACE put upon special classes of coalgebras, such as semiperfect, co-Frobenius, quasi-co-Frobenius, and cosemisimple or pointed coalgebras. Some torsion theory for coalgebras is also discussed. These classes of coalgebras are then investigated in the particular case of Hopf algebras, and the results are used, for example, in the chapters concerning integrals, actions and Galois extensions. The 'notions of a coalgebra and comodule are dualizations of the usual notions of an algebra and module. Beyond the formal aspect of dualization, it is worth keeping in mind that the introduction of these structures is motivated by natural constructions in classical fields of algebra, for example from representation theory. Thus, the notion of comuItiplication in a coalgebra may be already seen in the definition of the tensor product of representations of groups or Lie algebras, and a comodule is, in the given context, just a linear representation of an affine group scheme. As often happens, dual notions can behave quite differently in given dual situations. Coalgebras (and comodules) differ from their dual notions by a certain finiteness property they have. This can first be seen in the fact that the dual of a coalgebra is always an algebra in a functorial way, but not conversely. Then the same aspect becomes evident in the fundamental theorems for coalgebras and comodules. The practical result is that coalgebras and comodules are suitable for the study of cases involving infinite dimensions. This will be seen mainly in the chapter on actions and coactions. The notion of an action of a Hopf algebra on an algebra unifies situations such as: actions of groups as automorphisms, rings graded by a group, and Lie algebras acting as derivations. The chapter on actions and coactions has as main application the characterization of Hopf-Galois extensions in the case of co-Frobenius Hopf algebras. We do not treat here the dual situation, namely actions and coactions on coalgebras. Among other subjects which are not treated are: generalizations of Hopf modules, such as Doi-Koppinen modules or entwining modules, quasitriangular Hopf algebras and solutions of the quantum Yang-Baxter equation, and braided categories. The last chapter contains some fundamental theorems on finite dimensional Hopf algebras, such as the Nichols-Zoeller theorem, the Taft-Wilson theorem, and the Kac-Zhu theorem. We tried to keep the text as self-contained as possible. In the exposition we have indulged our taste for the language of category theory, and we use this language quite freely. A sort of "phrase-book" for this language is included in an appendix. Exercises are scattered throughout the text, with complete solutions a t the end of each chapter. Some of them are very easy, and some of them not, but the reader is encouraged to try as hard as possible to solve them without looking at the solution. Some of the easier

PREFACE

v

results can also be treated as exercises, and proved independently after a quick glimpse at the solution. We also tried to explain why the names for some notions sound so familar (e.g. convolution, integral, Galois extension, trace). This book is not meant to supplant the existing monographs on the subject, such as the books of M. Sweedler [218], E. Abe [I], or S. Montgomery [149] (which were actually our main source of inspiration), but rather as a first contact with the field. Since references in the text are few, we include a bibliographical note a t the end of each chapter. It is usually difficult to thank people who helped without unwittingly leaving some out, but we shall try. So we thank our friends Nicol6.s Andruskiewitsch, Margaret Beattie, Stefaan Caenepeel, Bill Chin, Miriam Cohen (who sort of founded the Hopf algebra group in Bucharest with her talk in 1989), Yukio Doi, Josk Gomez Torrecillas, Luzius Griinenfelder, Andrei Kelarev, Akira Masuoka, Claudia Menini, Susan Montgomery, Declan Quinn, David Radford, Angel del Rio, Manolo Saorin, Peter Schauenburg, Hans-Jiirgen Schneider, Blas Torrecillas, Fred Van Oystaeyen, Leon Van Wyk, Sara Westreich, Robert Wisbauer, Yinhuo Zhang, our students and colleagues from the University of Bucharest, for the many things that we have learned from them. Florin Nichita and Alexandru StZ.nculescu took course notes for part of the text, and corrected many errors. Special thanks go to the editor of this series, Earl J . Taft, for encouraging us (and making us write this material). Finally, we thank our families, especially our wives, Crina, Petrufa and Andreea, for loving and understanding care during the preparation of the book.

Sorin Dkciilescu, Constantin N k t k e s c u , Serban Raianu

Contents Preface

1 Algebras and coalgebras 1.1 Basic concepts 1.2 The finite topology 1.3 The dual (co)algebra 1.4 Constructions in the category of coalgebras 1.5 The finite dual of an algebra 1.6 The cofree coalgebra 1.7 Solutions to exercises 2

Comodules 1.1 The category of comodules over a coalgebra 2.2 Rational modules 2.3 Bicomodules and the cotensor product 2.4 Simple comodules and injective comodules 2.5 Some topics on torsion theories on M~ 2.6 Solutions to exercises

3

Special classes of coalgebras 3.1 Cosemisimple coalgebras 3.2 Semiperfect coalgebras 3.3 (Quasi)co-Frobeniusand co-Frobenius coalgebras 3.4 Solutions to exercises

4

Bialgebras and Hopf algebras 4.1 Bialgebras 4.2 Hopf algebras 4.3 Examples of Hopf algebras

vii

...

CONTENTS

vlll

4.4 4.5

Hopf modules Solutions to exercises

5 Integrals 5.1 5.2 5.3 5.4 5.5 5.6 5.7

6

7

The definition of integrals for a bialgebra The connection between integrals and the ideal H*"' Finiteness conditions for Hopf algebras with nonzero integrals The uniqueness of integrals and the bijectivity of the antipode Ideals in Hopf algebras with nonzero integrals Hopf algebras constructed by Ore extensions Solutions to exercises

Actions and coactions of Hopf algebras 6.1 Actions of Hopf algebras on algebras 6.2 Coactions of Hopf algebras on algebras 6.3 The Morita context 6.4 Hopf-Galois extensions 6.5 Application to the duality theorems for co-Frobenius Hopf algebras 6.6 Solutions to exercises Finite dimensional Hopf algebras 7.1 The order of the antipode 7.2 The Nichols-Zoeller Theorem 7.3 Matrix subcoalgebras of Hopf algebras 7.4 Cosemisimplicity, semisimplicity, and the square of the antipode 7.5 Character theory for semisimple Hopf algebras 7.6 The Class Equation and applications 7.7 The Taft-Wilson Theorem 7.8 Pointed Hopf algebras of dimension pnwith large coradical 7.9 Pointed Hopf algebras of dimension p3 7.10 Solutions to exercises

181 181 184 189 192 194 200 22 1

233 233 243 25 1 255 267 276

CONTENTS A The category theory language A.l Categories, special objects and special morphisms A.2 Functors and functorial morphisms A.3 Abelian categories A.4 Adjoint functors

B C-groups and C-cogroups B. 1 Definitions B.2 General properties of C-groups B.3 Formal groups and affine groups Bibliography Index

ix

Chapter 1

Algebras and coalgebras 1 .

Basic concepts

Let k be a field. All unadorned tensor products are over k. The following alternative definition for the classical notion of a k-algebra sheds a new light on this concept, the ingredients of the new definition being objects (vector spaces), morphisms (linear maps), tensor products and commutative diagrams.

-

-

Definition 1.1.1 A k-algebra is a triple (A, M, u), where A is a k-vector A and u : k A are mo~phismsof k-vector spaces space, M : A @ A such that the following diagrams are commutative:

C H A P T E R 1. A L G E B R A S A N D C O A L G E B R A S

W e have denoted by I the identity map of A , and the unnamed arrows from the second diagram are the canonical isomorphisms. ( I n general we will denote by I (unadorned, i f there is no danger of confusion, the identity I map of a set, but sometimes also by Id.) Remark 1.1.2 The definition is equivalent to the classical one, requiring A to be a unitary ring, and the existence of a unitary ring morphism 4 : k A, with I m q5 Z ( A ) . Indeed, the multiplication a . b = M ( a @ b) defines o n A a structure of unitary ring, with identity element u(1); the role of 4 is played by u itself. For the converse, we put M ( a 8 b) = a . b and u = 4. Due to the above, the map M is called the multiplication of the algebra A, and u is called its unit. The commutativity of the first diagram in the definition is just the associativity of the multiplication of the algebra. I

-

The importance of the above definition resides in the fact that, due to its categorical nature, it can be dualized. We obtain in this way the notion of a coalgebra.

-

-

Definition 1.1.3 A k-coalgebra is a triple (C, A, E ) , where C is a k-vector C @ C and E : C k are morphisms of k-vector spaces space, A : C such that the following diagrams are commutative:

1.1. 1.1.BASIC BASIC CONCEPTSCONCEPTS

33

C®CC®C The maps The Amaps A and and ~ ~are are called called the the comultiplication, comultiplication, and and the the counit, counit, respec-respectively, tively, of of the the coalgebra C. coalgebra C. The commutativity The commutativity of of the the first first diagramisdiagram iscalledcalled coassociativity. coassociativity. || Example 1.1.4 Example 1.1.4 1) 1) Let Let S Sbe be a anonempty set; nonempty set; kS kS is isthe the k-vector k-vector space space withwith basis basis S. S. Then kS Then kS is isa acoalgebra with coalgebra with comultiplication comultiplication A and A counit and counit ~ ~defineddefined by by A(s) A(s) = s= s® s, ®s,~(s) ~(s) fofo r rany any s E sS.S. EThTh is isshowsshows that athat ny veny a ctorctor vespacespace can can be be endowedwith endowed with a ak-coalgebra k-coalgebra structure.structure. 2) 2)Let Let HH be be a ak-vector k-vector space space with with basis basis {cm {cm I Im m ~ ~N}. N}. Then Then H is H isaa coalgebra with coalgebra with comultiplication comultiplication AA and counit and counit ~ ~defined defined byby

for for any any mm ~ ~N N (5iy (5iy will will denote denote throughout throughout the the Kronecker Kronecker symbol). symbol). ThisThis coalgebra is coalgebra iscalled called the the divided divided power coalgebra, power coalgebra, and we andwill we will come back cometo back to itit later.later. 3) 3) Let Let (S, (S, ~) ~) be be a apartially partially ordered locally orderedlocally finite finite set set (i.e. (i.e. for for any any x,y x,y with with x x~ ~y, y,the the set set {z {z ~~ S S~ ~ x x< z

(Irn@ Ai-l 18I ~ - ~ )-0 ~An-i+l + ~ (by the induction hypothesis) = (17" @ Ai-l @ p - i - m f l ) 0 ( I r n@ A @ In+rn) (by generalized coassociativity) = (I" @ ((Ai-1€4 I ) 0 A) @ In-i-m) 0 A,-i = ( I m@ Ai @ I . - ~ - ~ )0 (using 1)) =

0

Anpi

These formulas allow us to give the following computation rule, which is essential for computations in colagebras, and which will be used throughout in the sequel. 1.1.11 Computation rule. Let ( C ,A, E ) be a coalgebra, i

( i n the preceding tensor product C appearing i

2 1,

+ 1 times) and

-

f:C-c

linear maps such that f o Ai = 7. Then, if n 2 i , V is a k-vector space, and g : C @ ....

+

c-v

(here C appearing n 1 tinzes i n the tensor product) is a k-linear map, for any 1 5 j 5 n 1 and c E C we have

+

This happens because

CHAPTER 1. ALGEBRAS AND COALGEBRAS @Cj+i+l €9.. . @ cn+i+l) = ( ~ j - 18 f 8 ~ n - j + l ) o An+i(~) - g 0 (1j-18 f 8 p - j + l ) 0 ( I H @ A, @ 1n-j+1 ) o An (c) = g o ( 1 j - l 8 ( f 0 A,) @ P - j + l ) 0 An(c) ( ~ j - 1@ 7 ~ n - j + l o An ( c ) -

@

=

~ ~ ( C I @ . . . @ ~ ~ - ~ @ ~ ( C ~ ) B C ~ + ~ @ . . . @ C ~ + ~ )

This rule may be formulated as follows: if we have a formula (*) in which an expression in c l , . . . ,c,+l (from A i ( c ) ) has as result an element in C ( f o Ai = T), then i n a n expression depending o n c l , . . . , cn+i+l (from An+i(c)) in which the expression i n the formula (*) appears for c j , . . . ,cj+i (i+l conpositions), we can replace the expression depending o n c j , . . . ,cj+i secutive by f ( c j ) , leaving unchanged c l , . . . ,cj-1 and transforming cj+i+l,. . . ,cn+i+l I i n cj+l,. . . , cn+l.

Example 1.1.12 If (C,A, E ) is a coalgebra, then for any c E C we have

This is because having in mind the formula ~ E ( c ~=) Cc, ~we can replace i n the left hand side &(c2)c3by c2, leaving cl unchanged. Therefore, C E ( C ~ ) E ( C Z ) C=~ C E ( C ~ ) C Z , and this is exactly c. I We end this section by giving some definitions allowing the introduction of some categories.

Definition 1.1.13 A n algebra ( A ,M , u ) is said to be commutative if the diagram

is commutative, where T : A @ A ---t A 8 A is the twist map, defined by T ( a 8 b) = b 8 a. ii) A coalgebra (C,A, E ) is called cocommutative if the diagram

1.1. BASIC CONCEPTS

is commutative, which may be written as c E C.

C cl 8 c2

=

C cz 8 cl

for any

I

-

Definition 1.1.14 Let ( A , MA,u A ) ,( B ,M B ,u g ) be two k-algebras. The k-linear map f : A B is a morphism of algebras if the following diagrams are commutative

-

zi) Let ( C ,A c , ~ c )(,D ,AD,&,) be two k-coalgebras. The k-linear map D is a morphism of coalgebras if the following diagrams are g : C commutative

The commutativity of the Jirst diagram may be written i n the sigma notation as: a D ( g ( c ) )=

C g(c)l @ g(c)2 = C g ( c d €9 g(c2).

10

CHAPTER 1. ALGEBRAS AND COALGEBRAS

In this way we can define the categories k - Alg and k - Cog, in which the objects are the k-algebras, respectively the k-coalgebras, and the morphisms are the ones previously defined.

Exercise 1.1.15 Show that in the category k - Cog, isomorphisms (i.e. morphisms of coalgebras having an inverse which is also a coalgebra morphism) are precisely the bijective morphisms.

1.2

The finite topology

Let X and Y be non-empty sets and YX the set of all mappings from X to Y. It is clear that we can regard yXas the product of the sets Y, = Y, where x ranges over the index set X. The finite topology of yXis obtained by taking the product space in the category of topological spaces, where each Y, is regarded as a discrete space. A basis for the open sets in this topology is given by the sets of the form ( 9 E yx

I g(xi) = f (xi), 1 5 i 5 n),

where {xi 1 1 5 i 5 n ) is a finite set of elements of X , and f is a fixed element of YX, so that every open set is a union of open sets of this form. Assume now that k is a field, and X and Y are two k-vector spaces. The set Homk(X,Y) of all Ic-homomorphisms from X to Y, which is also a k-vector space, is a subset of y X . Thus we can consider on Homk(X,Y) the topology induced by the finite topology on YX. This topology on Homk(X, Y) is also called the finite topology. If f E Homk(X,Y), the the sets

1). We have S ' n- 1

C kei+ken

n-1

n ef)"+ke, i=l

n

n efne;)' i=l

+ 2 kei = i= 1 n

( S n0 e $ ) l . i=l i=l ii) Since the proof is similar to the one of assertion i), we only sketch the ku 2 ( S n u L ) l . For case n = 1. We put u = u1. We clearly have S' the reverse inclusion, we can assume S n u L # S. Clearly in this case we have u # 0. Since u l = K e r ( u ) , we have that V/u' = V / K e r ( u ) has dimension one. So S / ( S n u l ) E ( S + u " ) / u L I V / u L also has dimension one. There exists e E S , e # S n u L , such that S = ( S n u L ) @ ke. So u ( e ) # 0. If now f E ( S n u L ) l , we put g = f - f(e)u(e)-'u. If x E S , then x = y + Xe, with y E S n u L . But f ( x ) = f ( y ) + X f ( e ) = X f ( e ) , and ( f (e)u(e)-'u)(x) = f (e)u(e)-lu(y)+ X f (e)u(e)-'u(e) = X f ( e ) . So g(x) = X f ( e )- A f ( e ) = 0 , and therefore g E S'. Since f = g+ f (e)u(e)-lu, we obtain f E S L + ku. SL+

=(Sn

=(Sn

=

+

Theorem 1.2.6 i) If S is subspace of V , then (SL)" = S . iz) If S is a subspace of V * , then (SL)" = 3, where 3 is the closure of S in the finite topology. Proof: i) We have clearly that S C ( S L ) " . Assume now that there exists x E ( S L ) I , x @ S . Since S is a subspace, then kx n S = 0. Thus there

1.2. THE FINITE TOPOLOGY

13

exists f E V* such that f ( x ) = 1 and f ( S ) = 0. But f E SL,and since z E (SL)L, we have f ( x ) = 0, a contradiction. Hence (S'-)I = S . ii) S'- is a subspace of V. Hence (5'')'- = n W L , where W ranges over the finite dimensional subspaces of S L . Since WL is an open subspace of V*, it follows that WL is also closed (see Exercise 1.2.2). Hence nW'- is closed, so (SL)' is closed in the finite topology (see also Exercise 1.2.7 below). Since S 5 (SL)'-,it follows that S's (SL)I.Let f E (SL)' and W c V a k-subspace of finite dimension. We show that (f W'-) n S # 0. Clearly i f f E WL then f WL = W'- (because WL is a subspace), and therefore W L ) n S = WL n S # 0 (because it contains the zero morphism). (f Also if W E SL,then (S'-)I C W L , and therefore f E WL. Hence we S L . Thus we can can assume that f $! wL and so it follows that W write W = ( W n S L ) @ W', where W' # 0 and dimk(W1) < oo. Also since f (SL) = 0 and f (W) # 0 it follows that f (W') # 0. Let {el,. . . ,en) (n 2 1) be a basis for W'. We denote by a, = f (e,) (1 5 i 5 n ) , hence not all the a,'s are zero. By Proposition 1.2.5 i), we have

+

+

+

+

Exercise 1.2.4 The set of all f W'-, where W ranges over the finite dimensional subspaces of V , form a basis for the filter of neighbourhoods of f E V * i n the finite topology. Solution: We know that a basis for the filter of neighbourhoods of f E V *

1.7. SOLUTIONS TO EXERCISES in the finite topology is f + O ( 0 ,X I , .. . , x,). Note that O ( 0 ,X I , . . . , x,) = W L , where W is the subspace of V spanned by X I , . . . , x,.

Exercise 1.2.7 If S is a subspace of V * , then prove that (SL)' is closed i n the finite topology by showing that its complement is open. Solution: I f f @ ( S L ) I ,then there is an x E S1 such that f ( x ) # 0. Then (f + X I )n = 0.

(sL)l

Exercise 1.2.10 If V is a k-vector space, we have the canonical k-linear map d v :v (v*)*,d v ( x ) ( f )= f ( x ) , V x E v, f E V * .

-

Then the followzng assertzons hold: a) The map d v zs znjective. b) I m ( 4 " ) zs dense in ( I / * ) * . Solution: a ) Let z E K e r ( 4 v ) . If x # 0, there exists f E V* such that f ( x ) # 0. Since d v ( x ) = 0, we obtain that f ( x ) = 0, tlf E V * , a contradiction. b) We prove that ( ~ m ( + ~=) )( 0l) . Let f E ( I m ( d v ) ) l C V * . Hence d v ( x ) ( f )= 0 for every x E V . Thus f ( x ) = 0 V x E V , and therefore f = 0.

Exercise 1.2.11 Let V = Vl $ V2 be a vector space, and X = X I $ X 2 a subspace of V * ( X , C V,*, i = 1,2). If X is dense in V * , then X , is dense in V,*, i = 1,2. Solution: Let x E If f E X , then f = f l f 2 , with f l E X I and f i E X2. Since V * = V; $ V,*, we have that f2(V1) = 0 , SO f ( x ) = f l ( x ) f 2 ( x ) = 0. Hence x E X L = ( 0 ) .

+

+

Exercise 1.2.14 Let X C V * be a subspace of finite dimension n. Prove that X is closed i n the finite topology of v* by showing that d i m k ( ( X L ) L5 n.

- -

Solution: Let { f l , . . . , f,)

f)Ker( f,), 2=1 kn, so d i m k ( V / X L ) 5 n, and hence

be a basis of X . Then X L =

and therefore 0 V/XL d i m k ( V / X L ) *5 n. On the other hand, from the exact sequence

we have

0

-

-- -

(v/xL)* v*

(xL)* 0 , Hence d i m k ( ~ ' ) ~T"(M), and i : M + T ( M ) , defined by i(m) = m E T1(M) for any m 2 M. On T ( M ) we define a multiplication as follows: if x = m l 8 . . . a m n E T n ( M ) , and y = h 1 8 . . . 8 h, E TT(M),then define the product of the elements x and y by

The multiplication of two arbitrary elements from T ( M ) is obtained by extending the above formula by linearity. In this way, T ( M ) becomes an , the pair (T(M), i) is a tensor algebra, with identity element 1 E T O ( M ) and algebra of M .

Remark 4.3.2 The existence of the tensor algebra shows that the forgetful functor U : k-Alg + k M has a left adjoint, namely the functor associating to a k-vector space its tensor algebra. I We define now a coalgebra structure on T ( M ) . To avoid any possible confusion we introduce the following notation: if a and P are tensor monomials from T ( M ) (i.e. each of them lies in a component Tn(M)), then the tensor monomial T ( M ) 8 T(M) having a on the first tensor position, and ,B on the second tensor position will be denoted by aBj3. Without this notation, for example for a = m 8 m E T ~ ( M )and ,Ll = m E T1(M), the elements a 8 0 and ,O 8 a from T ( M ) 8 T ( M ) would be both written as m m 8 m, causing confusion. In our notation, a 8 p = m 8 m g m , and ,B@a=rngm8m. Consider the linear map f : M -+ T ( M ) €9 T ( M ) defined by f (m) = m g l + l g m for any m E M . Applying the universal property of the tensor algebra, it follows that there exists a morphism of algebras A : T ( M ) + T ( M ) 8 T ( M ) for which Ai = f . Let us show that A is coassociative, i.e. ( A 8 I ) A = ( I 8 A)A. Since both sides of the equality we want to prove are morphisms of algebras, it is enough to check the equality on a system of generators (as an algebra) of T ( M ) , thus on i(M). Indeed, if m E M , then

and

4.3. EXAMPLES OF HOPF ALGEBRAS which shows that A is coassociative. We now define the counit, using the universal property of the tensor algebra for the null morphism 0 : M k. We obtain a morphism of algebras E : T ( M ) 4 k with the property that ~ ( m = ) 0 for any m E i ( M ) . To show that ( E @ I ) A = 4, where 4 : T ( M ) -+ k @ T ( M ) , 4(z) = 1 @ z is the canonical isomorphism, it is enough to check the equality on i ( M ) , and here it is clear that -+

Similarly, one can show that (I@ E)A = $', where 4' : T ( M ) 4 T ( M ) @ k is the canonical isomorphism. So far, we know that T ( M ) is a bialgebra. We construct an antipode. Coni sider the opposite algebra T(M)OP of T ( M ) , and let g : M -+ T(M)OP be the linear map defined by g(m)= -m for any m M. The universal property of the tensor algebra shows that there exists a morphism of algebras S : T ( M ) -+ T(M)OP such that S(m) = -m for any m E i(h/l). For an arbitrary element ml @ . . . @ mn E T n ( M ) we have S ( m l @ . . . @ m,) = (-l)nmn@. . .@ml. We regard now S : T ( M ) T ( M ) as an antimorphism of algebras. We show that for any m E T1(M) we have -+

Indeed, since A(m) = m B l

+l

m we have

and similarly the other equality. Therefore, the property the antipode should satisfy is checked for S on a system of (algebra) generators of T ( M ) . The fact that S verifies the property for any element in T ( M ) will follow from the next lemma.

Lemma 4.3.3 Let H be a bialgebra, and S : H -+ H an antimorphism of algebras. If for a , b E H we have (S * I)(a) = ( I * S)(a) = u&(a) and ( S * I)(b) = ( I * S)(b) = u&(b), then also ( S * I)(ab) = ( I * S)(ab) = u ~ ( a b ) . Proof: We know that

and

CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS

162 Then

( S * I)(ab) = = =

~((ab)l)(ab)2

C s(aibl)az$ S ( b l ) S ( a l ) a 2 b 2 ( S is an antimorphism of algebras)

= E & ( a ) s ( b l ) b 2 ( from the property of a ) =

x ~ ( a ) ~ ( b ) (l from the property of b)

I Similarly, one can prove the second equality, and therefore we know now that T ( M ) is a Hopf algebra with antipode S. We show that T ( M ) is cocommutative, i.e. r A = A, where 7 : T ( M )8 T ( M ) -+ T ( M )8 T ( M ) is defined by ~ ( 8zv ) = v 8 z for any z, v E T ( M ) . Indeed, it is enough to check this on a system of algebra generators of T ( M ) , hence on i ( M ) (because r A and A are both morphisms of algebras), but on i ( M ) the equality is clear. 3) The symmetric algebra. We recall the definition of the symmetric algebra of a vector space.

Definition 4.3.4 Let M be a k-vector space. A symmetric algebra of M is a pair ( X ,i ) , where X is a commutative k-algebra, and i : M -, X is a k-linear map such that the following universal property holds: for any commutative k-algebra A, and any k-linear map f : M 4 A, there exists a unique morphism of algebras 7 : X --+ A such that Ti = f, i.e. the following diagram is commutative.

The symmetric algebra of a k-vector space M exists and is unique up to isomorphism. It is constructed as follows: consider the ideal I of the tensor algebra T ( M ) generated by all elements of the form x @J y - y @ x

4.3. EXAMPLES OF HOPF ALGEBRAS

163

with x, y E M . Then S ( M ) = T ( M ) / I , together with the map pi, where i : M -+ T ( M ) is the canonical inclusion, and p : T ( M ) -+ T ( M ) / I is the canonical projection, is a symmetric algebra of M. Remark 4.3.5 The existence of the symmetric algebra shows that the forgetful functor from the category of commutative k-algebras to the category I of k-vector spaces has a left adjoint. We show that the symmetric algebra M has a Hopf algebra structure. By Proposition 4.2.13, this will follow if we show that I is a Hopf ideal of the Hopf algebra T ( M ) . Since A and E are morphisms of algebras, and S is an antimorphism of algebras, it is enough to show that

~ ( x g y - y @ x ) = O a n d S ( x @ y - y @ x )I~ for any x, y E M . Indeed,

and this is clearly an element of I @ T(M)

+ T ( M ) @ I . Moreover,

and S ( x @ y - y @ x) = S(y)S(x) - S(x)S(y) = = (-9) @ (-x) - (-x) @ (-9) € I.

We obtained that S ( M ) has a Hopf algebra structure, it is a factor Hopf algebra of T ( M ) modulo the Hopf ideal I . It is clear that S ( M ) is a commutative Hopf algebra, and also cocommutative, since it is a factor of a cocommutative Hopf algebra. 4) The enveloping algebra of a Lie algebra. Let L be a Lie kalgebra, with bracket [ , 1. The enveloping algebra of the Lie algebra L is the factor algebra U(L) = T ( L ) / I , where T(L) is the tensor algebra of the k-vector space L, and I is the ideal of T(L) generated by the elements of the form [x,y] - x @ y y @ x with x, y E L. A computation similar to the one performed for the symmetric algebra shows that

+

164

CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS = ([x,y] - x 8 y

which is in I @ T ( M )

+ y 8x)%1 + 1G([x,y] - x 8y + y 82)

+ T ( M ) @ I,

and S([x,y ] - x 63 y 3- y 8 x) = -([x,Y ] - @ Y

+ Y 8x) E I ,

so I is a Hopf ideal in T(L). It follows that U(L) has a Hopf algebra structure, the factor Hopf algebra of T(L) modulo the Hopf ideal I. Since T(L) is cocommutative, U ( L ) is also cocommutative. 5) Divided power Hopf algebras Let H be a k-vector space with basis {cili E N ) on which we consider the coalgebra structure defined in Example 1.1.4 2). Hence

for any m E N . We define on H an algebra structure as follows. We put

for any n , m E N , and then extend it by linearity on H. We note first that co is the identity element, so we will write co = 1. In order to show that the multiplication is associative it is enough to check that (cncm)cp= c, (cmcp) for any m, n , p E N. This is true because

4.3. EXAMPLES OF HOPF ALGEBRAS

165

We show now that H is a bialgebra with the above coalgebra and algebra structures. Since the counit is obviously an algebra map, it is enough to show that A(cncm) = A(cn)A(cm)for any n, m E N. We have

It remains to prove that the bialgebra H has an antipode. Since H is cocommutative, it suffices to show that there exists a linear map S : H -.. H such that C S ( h l ) h a = ~ ( h ) for l any h in a basis of H. We define S(cn) recurrently. For n = 0 we take S(co) = S(l) = 1. We assume that S(co), . . . ,S(cn-1) were defined such that the property of the antipode checks for h = ci with 0 5 i 5 n - 1. Then we define

and it is clear that the property of the antipode is then verified for h = cn too. In conclusion, H is a Hopf algebra, which is clearly commutative and cocommutative. 6. Sweedler's 4-dimensional,Hopf algebra.

Assume that char(k) # 2. Let H be the algebra given by generators and relations as follows: H is generated as a k-algebra by c and x satisfying the relations c2 = 1, x2 =o, x c = -CX Then H has dimension 4 as a k-vector space, with basis { 1,c, x, cx ). The coalgebra structure is induced by


166

In this way, H becomes a bialgebra, which also has an antipode S given by S(c) = c-I, S(x) = -CX. This was the first example of a non-commutative and non-cocommutative Hopf algebra.

7. The Taft algebras. Let n 2 2 be an integer, and X a primitive n-th root of unity. Consider the algebra Hn2 (A) defined by the generators c and x with the relations

On this algebra we can introduce a coalgebra structure induced by

In this way, Hnz (A) becomes a bialgebra of dimension n2, having the basis { cixj I 0 5 i, j 5 n - 1 ). The antipode is defined by S(c) = c-' and S(x) = -c-'x. We note that for n = 2 and X = -1 we obtain Sweedler's 4-dimensional Hopf algebra. 8. On the polynomial algebra k[X] we introduce a coalgebra structure as follows: using the universal property of the polynomial algebra we find a unique morphism of algebras A : k(X] -+ k[X] C3 k[X] for which A(X) = X @I 1 1@ X . It is clear that

+

and then again using the universal property of the polynomial algebra it follows that A is coassociative. Similarly, there is a unique morphism of algebras E : k[X] -+ k with E(X) = 0. It is clear that together with A and E , the algebra k[X] becomes a bialgebra. This is even a Hopf algebra, with antipode S : k[X] -+ k[X] constructed again by the universal property of the polynomial algebra, such that S(X) = -X. This Hopf algebra is in fact isomorphic to the tensor (or symmetric, or universal enveloping) algebra of a one dimensional vector space (or Lie algebra). We take this opportunity to justify the use of the name convolution. The polynomial ring R[X] is a coalgebra as above, and hence its dual,

167

4.3. EXAMPLES O F HOPF ALGEBRAS

U = R[X]* = Hom(R[X],R ) is an algebra with the convolution product. If f is a continuous function with compact support, then f * E U, where f * is given by

and P is the polynomial function associated to P E R[X]. We have that A ( P ) E R [ X ] @3 R[X] R[X, Y],

--

A(P) =

C

PI@P2= P ( X

+ Y).

If g is another continuous function with compact support, the convolution product of f * and g* is given by

where h(t) = f(x)g(t - x)dx is what is usually called the convolution product (see [199]). 9. Let k be a field of characteristic p > 0. On the polynomial algebra k[X] we consider the Hopf algebra structure described in example 8, in which A ( X ) = X @ 1 + 1 €4 X ,E(X) = 0 and S ( X ) = -X. Since A(Xp) = XP @ 1 1 8Xp (we are in characteristic p, and all the binomial coefficients (p) with 1 5 i 5 p - 1 are divisible by p, hence zero), E ( X P )= 0 and S(X" = -XP (remark: if p = 2, then 1 = -I), it follows that the ideal generated by XP is a Hopf ideal, and it makes sense to construct the factor Hopf algebra H = k[X]/(XP). This has dimension p, and denoting by x the coset of X , we have A(x) = x @ 1 + 1@ x and x p = 0. This is the restricted enveloping algebra of the 1-dimensional p L i e algebra.

+

10. The cocommutative cofree coalgebra over a vector space.

Let V be a vector space, and (C, T) a cocommutative cofree coalgebra over V. We show that C has a natural structure of a Hopf algebra. Let ~(c)) p : C @ C -+ V @ Vbe the map defined by p ( c 8 d ) = ( ~ ( c ) ~ ( d ) , ~ ( d )for

168

CHAPTER 4. BlALGEBRAS AND HOPF ALGEBRAS

any c, d E C. Proposition 1.6.14 shows that (C @ C,p) is a cocommutative cofree coalgebra over V $ V . The same result shows that if we denote by y : ( C @ C ) @ C - +( V $ V ) $ V the map defined by

we have that (C @ C @ C, y) is a cocommutative cofree coalgebra over V@V$V. Let m : V @ V V be the map defined by m ( x ,y) = x + y. Then V induces a morphism of coalgebras M : the linear map m : V $ V C €3 C -+ C between the cocommutative cofree coalgebra over these spaces. Also the linear map m $ I : V $ V @ V --, V $ V induces the morphism M @ I : C @ C @ C -t C @ C of coalgebras between the cocommutative cofree coalgebras over the two spaces (this follows from the relation p(M @ I ) = ( m $ I ) y , which checks immediately). By composition it follows that M(M €3 I ) : C €3 C @ C -+ C is the morphism of coalgebras associated to the linear map m(m @ I) : V @ V @ V + V (using the universal property of the cocommutative cofree coalgebra). Consider now the map y' : C €3 C @J C -+ V $ V $ V as in Proposition 1.6.14, for which (C @ (C @I C), 7') is a cocommutative cofree coalgebra over V &, (V @ V ) . Similar to the above procedure, one can show that M(I @ M ) is the morphism of coalgebras associated to the linear map m ( I $ m ) . But it is easy to see that y = y', and that m ( I $ m ) = m ( m @ I ) , hence M(M @J I ) = M(I €4 M ) , i.e. M is associative. Using Proposition 1.6.13, the zero morphism between the null space and V induces a morphism of coalgebras u : k -+ C. Also the linear map s : V -+ V, s(x) = -x, induces a map S : C -+ C , using again the universal property. As in the verification of the associativity of M , one can check that u is a unit for C , which thus becomes a bialgebra, and that S is an antipode for this bialgebra. In conclusion, the cocommutative cofree coalgebra over V has a Hopf algebra structure. -+

-+

Exercise 4.3.6 (i) Let k be a field which contains a primitive n - t h root of 1 (in particular this requires that the characteristic of k does not divide n ) and let Cn be the cyclic group of order n. Show that the Hopf algebra kcn is selfdual, i.e. the dual Hopf algebra (kc,)' is isomorphic to k c n . (iz) Show that for any finite abelian group C of order n and any field k which contains a primitive n - t h root of order n, the Hopf algebra kC is selfdual.

4.4. HOPF MODULES

169

Exercise 4.3.7 Let k be a field. Show that (i) If c h a r ( k ) # 2, then any Hopf algebra of dimenszon 2 is isomorphic to k c 2 , the group algebra of the cyclic group with two elements. (ii) If c h a r ( k ) = 2, then there exist precisely three isomorphism types of Hopf algebras of dimension 2 over k , and these are k c 2 , ( k c 2 ) * , and a certain selfdual Hopf algebra. Exercise 4.3.8 Let H be a Hopf algebra over the field k , such that there k x k x . . . x k (k appears n times). exists an algebra isomorphism H Then H is isomorphic to ( k G ) * , the dual of a group algebra of a group G with n elements.

--

Two more examples of Hopf algebras, the finite dual of a Hopf algebra, and the representative Hopf algebra of a group, are treated in the next exercises.

Exercise 4.3.9 Let H be a bialgebra. Then the finite dual coalgebra H O is a subalgebra of the dual algebra H * , and together with this algebra structure it is a bialgebra. Moreover, if H is a Hopf algebra, then H O is a Hopf algebra. Exercise 4.3.10 Let G be a monoid. Then the representative coalgebra R k ( G ) is a subalgebra of kGJ and even a bialgebra. If G is a group, then R k ( G ) is a Hopf algebra. If G is a topological group, then

R ~ ( G=) { f

E

R R ( G ) I f continuous)

is a Hopf subalgebra of R R ( G ) .

4.4

Hopf modules

Throughout this section H will be a Hopf algebra.

Definition 4.4.1 A k-vector space M is called a right H-Hopf module if H has a right H-module structure (the action of an element h E H on an element m E M will be denoted by m h ) , and a right H-comodule structure, given by the map p : M -+ M 8 H , p ( m ) = C m ( o ) 8 m ( ~ such ) , that for anym€M,h€H

170


Remark 4.4.2 It is easy to check that M B H has a right module structure over H 8 H (with the tensor product of algebras structure) defined by ( m@ h ) ( g B p )= m g @ h p f o r a n y m g h ~M 8 H , g B p E H B H . Considering then the morphism of algebras A : H H B H , we obtain that M 8 H becomes a right H-module by restriction of scalars via A. This structure is given by ( m @I h)g = C mgl 8 hg2 for any m 8 h E M 8 H , g E H . With this structure i n hand, we remark that the compatibility relation from the preceding definition means that p is a morphism of right H-modules. There is a dual interpretation of this relation. Consider H 8 H with the tensor product of coalgebras structure. Then M @ H has a natural structure of a right comodule over H 8 H , defined by m B h H C m ( ~ 8 h) 1 8 m ( l ) 8 h 2 . The multipliction p : H 8 H ---, H of the algebra H is a morphism of coalgebras, and then by corestriction of scalars M 8 H becomes a right H comodule, with m 8 h H E m(o)8 hl 8 m(l)h2. Then the compatibility relation from the preceding definition may be expressed by the fact that the map 4 : M 8 H -+ H , giving the right H-module structure of M , is a I morphism of H-comodules. -+

We can define a category having as objects the right H-Hopf modules, and as morphisms between two such objects all linear maps which are also morphisms of right H-modules and morphisms of right H-comodules. This category is denoted by M S , and will be called the category of right H-Hopf modules. It is clear that in this category a morphism is an isomorphism if and only if it is bijective.

Example 4.4.3 Let V be a k-vector space. Then we define on V 8 H a right H-module structure by ( v 8 h)g = v @hg for any v E V , h , g E H , and a right H-comodule structure given by the map p : V 8 H -+ V B H 8 H , p(v 8 h ) = C v @I hl B h2 for any v E V, h E H . Then V 8 H becomes a right H-Hopf module with these two structures. Indeed

p((v @ h ) g ) = p(v @ hg) =

Cv

@

( h g ) l @(hg)2

=

B hlgi B h2g2

=

C((v@hl)gl)Bh292

=

x ( v 8 h)(o)giB ( v 8 h)(l)gz

proving the compatibility relation.

I

We will show that the examples of H-Hopf modules from the preceding example are (up to isomorphism) all H-Hopf modules. We need first a definition.

4.4. HOPF MODULES

171

Definition 4.4.4 Let M be a right H-comodule, with comodule structure given b y the map p : M -+ M @ H . The set

is a vector subspace of M which is called the subspace of coinvariants of M. I

Example 4.4.5 Let H be given the rzght H-comodule structure induced b y A : H -, H @ H . Then HCoH= k l (where 1 is the identity element of H ) . , A ( h ) = C hl @ h2 = h @ 1. Applying E on the Indeed, if h E H " " ~ then first position we obtain h = ~ ( h )El k1. Conversely, if h = a1 for a scalar I a , then A ( h ) = a1 @ 1 = h @ 1. Theorem 4.4.6 (The fundamental theorem of Hopf modules) Let H be a Hopf algebra, and M a right H-Hopf module. Then the map f : M~~~ @ H -t Mydefined b y f(m @ h ) = m h for any m E MCoH and h E H , is an isomorphism of Hopf modules (on MCoH@ H we consider the H-Hopf module structure defined as in Example 4.4.3 for the vector space MCoH). Proof: W e denote the map giving the comodule structure o f M by p : 8 m ( l ) . Consider the map g : M -+ M , M + M @ H , p(m) = defined by g ( m ) = C m ( o ) S ( m ( l )for ) any m E M . I f m E M , we have

P(Cm ( o ) ~ ( m ( l ) ) ) C ( m ( o , ) ( o ) ( s ( m ( l , )@) l( m ( o ) ) , l ) ( s ( m ( l , ) ) 2 (definitiono f Hopf modules) C ( m ( o ; ) ( o , s ( ( m ( l , ) 28) ( m ( o , ) ( l ) s ( ( m ( l , ) l ) (the antipode is an antimorphism of coalgebras)

C m(o)s(m(s)) m(l)S(m(2)) @

(using the sigma notation for comodules) C m ( o ) s ( m ( 2 )8) ( m ( l ) ) i S ( ( m ( l ) ) n ) (using the sigma notation for comodules) m ( o ) S ( m ( 28 ) )~ ( m ( ~ (definition ))l o f the antipode)

C m ( O ) ~ ( m ( 2 ) & ( m ( l )1) ) @

C m ( o , ~ ( ( m ( l , ) 2 ~ ( ( m ( l )@ ) l 1) ) (using the sigma notation for comodules)

CHAPTER 4. BIALGEBRAS AND HOPF ALGEBRAS m ( o ) ~ ( m c l@ ) )1 (the counit property)

=

= g(m)@l,

which shows that g(m) E M~~~for any m E M . It makes then sense to define the map F : M --+ MCoH@ H by F ( m ) = C ~ ( m ( ~@ )m(l) ) for any m E M . We will show that F is the inverse of f . Indeed, if m E M C O ~ and h E H we have

=

C g(m(o)hl)@ m(l)h2 (definition of Hopf modules)

= = =

g(mh1) €3 h

(since m E M ~ O ~ )

~ ( m h 1 ) ( o ) ~ ( ( m h l )@ ( l h2 )) m ( o ) ( h l ) i ~ ( m(h1)2) ( ~ ) @ h2 (definition of Hopf modules)

=

x m ( h l ) l ~ ( ( h l ) 2@) hz (since m E M " " ~ )

=

x m ~ ( h@~h2) (by the antipode property)

= m @ h (by the counit property)

hence Ff = I d . Conversely, if m E M , then

=

C m(o)s((mil,)l)(m,l))2 (using the sigma notation for comodules)

=

m ( o ) ~ ( m ( l )(by ) the antipode property)

= m (by the counit property)

which shows that f F = I d too. It remains to show that f is a morphism of H-Hopf modules, i.e. it is a morphism of right H-modules and a morphism of right H-comodules. The first assertion is clear, since f ( ( m 8 h)hl) = f ( m 8 hh') = mhh' = f (m 8 h)h'. In order to show that f is a morphism of right H-comodules, we have to prove that the diagram

4.5. SOLUTIONS TO EXERCISES

is commutative. This is immediate, since

( p f ) ( m @ h ) = p(mh) = m h l 8 h2 (since m E M ~ O ~ )

x

which ends the proof.

a

Exercise 4.4.7 Let H be a Hopf algebra. Show that for any right (left) H-comodule M , the injective dimension of M in the category M H is less than or equal the injective dimension of the trivial right H-comodule k . In particular, the global dimension of the category M* is equal to the injective dimension of the trivial right H-comodule k . Exercise 4.4.8 Let H be a Hopf algebra. Show that for any right (left) H-module M , the projective dimension of M i n the categoy M H is less than or equal the projective dimension of the trivial right H-module k (with action defined by a: +- h = ~ ( h ) for a any a: E k and h E H ) . I n particular the global dimension of the category M H is equal to the projective dimension of the right H-module k .

4.5

Solutions to exercises

Exercise 4.1.9 Let k be a field and n 2 2 a positive integer. Show that there is no bialgebra structure on M n ( k ) such that the underlying algebra structure is the matrix algebra. Solution: The argument is similar to the one that was used in Example 1.4.17. Suppose there is a bialgebra structure on M n ( k ) , then the counit E : M n ( k ) -+ k is an algebra morphism. Then the kernel of E is a two-sided ideal of M n ( k ) , so it is either 0 or the whole of M n ( k ) . Since ~ ( 1 =) 1, we have K e r ( ~ = ) 0 and we obtain a contradiction since dim(M,(k)) > d i m ( k ).

174


Exercise 4.2.16 Let H be a finite dimensional Hopf algebra over a field k of characteristic zero. Show that if x E H is a primitive element, i.e. A(x) = x 8 1 + l a x , t h e n x = 0. Solution: If there were a non-zero primitive element x, we prove by induction that the 1,x, . . . ,xn are linearly independent for any positive integer n, and this will provide a contradiction, due to the finite dimension of H. The claim is clear for n = 1, since a1 bx = 0 implies by applying E that a = 0, and then, since x # 0, that b = 0. Assume the assertion true for n - 1 (where n 2), and let Cp=O,napxP = 0 for some scalars ao, . . . ,a p . Then by applying A we find that

+

>

+

Choose some 1 5 i , j 5 n - 1 such that i j = n, and let h;,h; E H* such that hf(xt) = biYt for any 0 5 t 5 n - 1 and hj*(xt) = 6j,t for any 0 5 t 5 n - 1 (this is possible since 1, x , . . . , xnP1 are linearly independent). Then by applying hf 8 h5 t o the above relation we obtain that a,(:) = 0, and since k has characteristic zero we have an = 0. Then again by the induction hypothesis we must have ao, . . . ,an-1 = 0.

Exercise 4.2.17Let H be a Hopf algebra over the field k and let K be a field extension of k. Show that one can define on ?? = K 8 k H a natural structure of a Hopf algebra by taking the extension of scalars algebra structure and the coalgebra structure as in Proposition 1.4.25. Moreover, if 3 is the antipode of X,then for any positive integer n we have that Sn = Id if and only if --n S =Id. Solution: The comultiplication and counit Z of & are given by

for any 6 E K and h E H. It is a straightforward check that 2 is a bialgebra over K , and moreover, the map 3 : -+ defined by S ( 6 ~ ~=h 6@kS(h) ) is an antipode of H. The last part is now obvious.

Exercise 4.3.6 (i) Let k be a field which contains a primitive n-th root of 1 ( i n particular this requires that the characteristic of k does not divide n ) and let Cn be the cyclic group of order n. Show that the Hopf algebra k c n is selfdual, i.e. the dual Hopf algebra (kcn)* is isomorphic to k c n . (ii) Show that for any finite abelian group C of order n and any field k which contains a primitive n-th root of order n, the Hopf algebra kC is

175

4.5. SOLUTIONS T O EXERCISES

selfdual. Solution: (i) Let Cn =< c > and let J be a primitive n-th root of 1 in k. Then k c n has the basis 1,c, c2,. . . ,cn-l, and let pl, p,, . . . ,pcn-l be the dual basis in (kcn)*. We determine G = G((kCn)*). We know that the elements of G are just the algebra morphisms from k c n to k. If f E G, for some 0 i 5 n - 1. Conversely, for then f ( c ) ~= 1, so f (c) = any such i there exists a unique algebra morphism fi : k c n -+ k such that fi(c) = t i . More precisely, fi(cj) = tij for any j (extended linearly). Thus fo, f l , . . . ,fn-1 are distinct grouplike elements of (kc,)*, and then a dimension argument shows that (kcn)* = kG. On the other hand fi = f: for any i, so G is cyclic, i.e. G E Cn. We conclude that (kcn)* = k c n . (ii) We write C as a direct product of finite cyclic groups. The assertion follows now from (i) and the fact that for any groups G and H we have that k(G x H) 2. kG @ kH.

2 and a = (al, . . . ,at) E {O,lIt.

The

is a Hopf ideal if and only if qj = cj*(cj) is a primitive nj-th root of unity for any 1 5 j s t .

5.6. HOPF ALGEBRAS CONSTRUCTED B Y ORE EXTENSIONS 207 Proof: Since c y -1 is a (1, ~7~)-prirnitive, it follows that X,") - a j ( c y 1) is a (1, ~ y ) - ~ r i m i t i vif eand only if so is By (5.10) and Remark 5.6.3, this occurs if and only if = 0 for every 0 < k < n j , i.e. if and only

(7)qi

x?.

if qj is a primitive nj-th root of unity. Moreover, since S(Xj) = - c y l x j , induction on n shows that

Now, since qTi = 1, checking the cases n j even and n j odd, we see that (-l)"lq~n~(n'-1)12 = -1 and hence

for 1 5 j 5 t , so that the ideal J ( a ) is invariant under the antipode S, and is thus a Hopf ideal. 1 By Lemma 5.6.13, H = At/ J ( a ) is a Hopf algebra. However the coradical may be affected by taking this quotient. Since we want H to be a pointed Hopf algebra with coradical kC, some additional restrictions are required. We denote by xi the image of Xi in H and write xp for xY1 . . . x y , p = (pl, . . . ,pt) E N t .

Proposition 5.6.14 Assume J ( a ) as in Lemma 5.6.13 is a Hopf ideal. Then J ( a ) n k C = 0 if and only if for each i either ai = 0 or ( ~ f =) 1.~ ~ If this is the case then {gxplg E C , p E Nt, 0 5 pj 5 n j - 1) is' a basis of AtlJ(a). Proof: By Lemma 5.6.13, we know that J ( a ) is a Hopf ideal if and only if qz = cf (ci) is a primitive ni-th root of unity for 1 5 i 5 t . Now suppose that J ( a ) n k C = 0. Since

is in J ( a ) for every g E C, it follows that

c l ~ and (1-cf (g)-nt)Xr' But then for every g E C , both ~ ~ ( l - c f ( g ) - ~ " ( -1) are in J ( a ) . If ai # 0, which by our convention implies that cli - 1 # 0, then we must have cf (g)ni = 1 for all g, and thus c:ni = 1. Conversely, assume that cfna = 1 whenever ai # 0. By Definition 5.6.8 (iii), XTag = ~ f ( g ) ~ i g X "In~ .particular, Xznig= gXTi if ai # 0.. Also, if

208 i

CHAPTER 5. INTEGRALS

< j then by (5.7),

So, if bij = 0, then XjXzna= cj*( c ~ ) ~ ~ where X ~ ~ c; (ci)"z X ~ ,= cf ( ~ ~ ) = - ~1 i if ai # 0. If bij # 0 then cfcj' = 1, hence cf(ci) is a primitive ni-th root of unity, so that XjXzni = XYiXj. A similar argument works for i > j . Thus, Xzni is a central element of At if ai # 0. It follows that

so that J ( a ) is equal to the left ideal generated by

{x? - aj(c,"'

j _< t), and At is a free left module with basis {XplO

- 1)(15 - 1)

5 pj 5 n j

over the subalgebra B generated by C and X;', . . . ,X r t . We now show that no nonzero linear combination of elements of the form gXP, p E N t , 0 5 p j 5 n j - 1 lies in J(a). Otherwise there exist f j E A,, 1 j t, not all zero, such that

<
= C, a fixed primitive m t h root of I. (i) H H* if and only i f there exist h , .rr as i n Proposition 5.6.38 such that for all 1 5 j 5 t , n,(j) = n j ,

huj

= d,(j)

mod m, u , z ( j )

--

uj

mod m.

In particular a Tuft Hopf algebra is selfdual. (ii) H 2 HcOp i f and only if there exist h,.rr such that for all 1 < j 5 t ,

n,(j) = n j , h u j (iii) H

-u,(~) mod m, d j

hd,(j) mod m.

Hop if and only if there exist h, IT such that for all 1 5 j 5 t , n,(j) = nj, h u j

= u,(j)

mod m, d j

-hd,(j) mod m.

I We study now Hopf algebras of the form H ( C , n, c*, c, 0 , I ) , where b = 1 means that bij = 1 for all i < j. Thus, the skew-primitives xi are all nilpotent and for i # j , xixj - cf ( c j ) x j x iis a nonzero element of k c . It is easy to see that if a = 0 and all bij are nonzero, then a change of variables ensures that all bij equal 1. This class produces many interesting examples. The following two definitions are particular cases of Definition 5.6.15.

Definition 5.6.40 For t = 2, let n 2 2, c = ( c l ,cp) E C 2 , g * E C* with g*(cl) = g*(c2) a primitive n-th root of unity, and clc2 # 1. Denote the pair ( n , n ) by ( n ) , and, if cl = c2 = g, denote (cl,c2) by ( g ) . Then H ( C , ( n ) (, c l ,c 2 ) ,( g * , g*-l), 0 , l ) denotes the Hopf algebra generated by the commuting grouplike elements g 6 C , and the ( 1 ,cj)-primitives xj, j = 1,2, with multiplication relations

23 = 0 , x1g =< g*,g > gx1,

x2g

=< g*-l,g > gxp

Definition 5.6.41 Let t > 2 and let c E Ct,g* E C* such that g*(ci) = -1 for all i and cicj # 1 if i # j . We denote the t-tuple ( 2 , . . . , 2 ) by ( 2 ) , and the t-tuple (g*,. . . , g * ) by (g*). Then H ( C , ( 2 ) ,( c l ,. . . , c t ) , ( g * ) ,0 , l ) is the Hopf algebra generated by the commuting grouplike elements g E C , and the ( 1 ,c j )-primitives xj , with relations

x: = 0 ,

xig = g*(g)gxi, xixj

+ x j x i = cicj - 1 for i # j.

5.6. HOPF ALGEBRAS CONSTRUCTED BY ORE EXTENSIONS 219 Example 5.6.42 (i) Let C, =< g > be cyclic of finite order m 2 2, let n be an integer 2 2, and let cl = gul, c2 = gu2,g* 6 C* be such that g*(g) = X where Am = 1, ul + u2 $ 0 mod m, and Xu' = Xu2, a primitive nth root of 1. Then H = H(C,, (n), c, (g*, g*-'), 0 , l ) is a Hopf algebra of dimension mn2, with coradical kc, and generators g, X I ,x2 such that g is grouplike of order m , xi is a (l,gut)-primitive, and

, =< g >. (ii) Let m L 2 , t > 2 be integers, m even, and let C = C Let u l , . . . , ut be odd integers such that ui u j $ 0 mod m if i # j and let ci = gui,cf = g* where g*(g) = -1. Then the Hopf algebra H(Cm, (2),c, (g*),0 , l ) has dimension 2tm and has generators g, X I , .. . ,xt such that g is grouplike, xi is a (1,gut)-primitive, and

+

(iii) Suppose C =< g > is infinite cyclic, and n 2 2. Let ul,u2 be integers such that ul u2 # 0, and let X E k such that Xu' = Xu2 is a primitive nth root of 1. Let g* E C* with g*(g) = A. Then there is an infinite dimensional pointed Hopf algebra with nonzero integral

+

with generators g, x1,x2 such that g is grouplike of infinite order, xi is a (1, gu"-primitive, and

(iv) Let C =< g > be infinite cyclic, t > 2 and let 211,. . . , ut be odd integers such that u,+uJ # 0 for i # j . Then there is an infinite dimensional pointed Hopf algebra with nonzero integral H(C, (2), c, (g*),0, l ) , where c, = gut and g*(g) = -1. The generators are g , x l , . . . ,xt such that g is grouplike of infinite order, x, is a (1, gua)-primitive,and

By an argument similar to the proof of Theorem 5.6.27, we can classify the Hopf algebras from Definition 5.6.40.

220


Theorem 5.6.43 There is a Hopf algebra isomorphism from

to H' = H ( C 1 ,(n'),c', (g*',(g*')-I),0,1) if and only if C = C ' , n = n' and there is an automorphism f of C such that (i) f ( c l ) = ci ,f (c2)= C: and g* = g*' o f ; or (ii) f ( c l ) = cb, f (c2)= C; and g* = (g*')-' o f.

Proof: I f H H', then exactly as in the proof o f Theorem 5.6.27, there exists an automorphism f o f C and a bijection n o f { 1 , 2 } such that f (ci) = c$) and cf = c S g o f . The conditions ( i ) and (ii) in the statement correspon t o n the identity and n the nonidentity permutation. Conversely, i f ( i ) holds, define an isomorphism from H t o H' by mapping g t o f ( g ) and xi t o xi. I f (ii) holds, define an isomorphism from H t o H' by mapping g t o f ( g ) ,X I t o x i and xz t o -g* ( c l ) x i . I Corollary 5.6.44 If C =< g > is cyclic, then the Hopf algebras H and H' above are isomorphic i f and only i f C = C',n = n', and there is an integer h such that the map taking g to g h is an automorphism of C and either (2) cf = C f ' h and c f = guih = gu: = c: for i = 1,2; or (ii) cf = ( C f ' ) - hand g"lh = g";, guZh = 9U''. 4 For the Hopf algebras o f Definition 5.6.41 there is a similar classification result.

Theorem 5.6.45 There is a Hopf algebra isomorphism from

to H' = H ( C ' , (2), c', (g*'),0 , l ) if and only if C = C 1 , t = t' and there is a permutation n E St and an I automorphism f of C such that f (ci) = c&, and g* = g*' o f .

Corollary 5.6.46 Suppose C =< g > is cyclic. Then H and H' as above are isomorphic if and only if C = C', t = t' and there exists a permutation n E St and an azltomorphism of C taking g to gh, such that cf = gUih = for all i. I


221

In Exercise 5.6.31 we saw that if a # 0, Ore extension Hopf algebras with nonzero derivations may be isomorphic to Ore extension Hopf algebras with zero derivations. The following theorem shows that if a = 0, this is impossible.

Theorem 5.6.47 Hopf algebras of the form H ( C ,n,c, c*) = H ( C ,n, c, c*,0,O) cannot be isomorphic to either the Hopf algebras of Definition 5.6.40 or Definition 5.6.41. Proof: Suppose that f : H ( C 1 ,(n'),c', ( g * ' g*'-l), , 0 , l ) -r H ( C ,n, c, c*)

is an isomorphism of Hopf algebras. Then, as in the proof of Theorem 5.6.27, we see that C = C', f ( x i ) = Cicvixi and f ( x ; ) = Pixi for scalars ai, Pi. But f applied to the relation

xi

yields C , , u i P j ( z j z i - (9"')-'(c;)xixj) = 1 - 1 in H ( C , n, c, c*), where 1 # 1 is a grouplike element. The relations of an Ore extension with zero derivations show that this is impossible. Similarly, H ( C , n, c, c*) cannot be I isomorphic to a Hopf algebra as in Definition 5.6.41.

5.7

Solutions to exercises

Exercise 5.1.6 Let H be a Hopf algebra over the field k, K afield extension H the Hopf algebra over K defined in Exercise 4.2.17. of k, and H = K If T E H* is a left integral of H , show that the map T E p* defined by T ( 6 @ k h ) = S T ( h ) is a left integral of p. Solution: Let S @ k h E H . Then

showing that

T is a left integral of H .


222

Exercise 5.1.7 Let H and H' be two Hopf algebras with nonzero integrals. Then the tensor product Hopf algebra H 8 H' has a nonzero integral. Solution: Assume that H and H' have nonzero integrals. Then we show that t 8 t' is a left integral for H 8 H ' , and this is obviously nonzero. Note that we regard H* 8 H" as a subspace of ( H 8 H1)*,in particulat t 8 t' E ( H 8 HI)* is the element working by ( t 8 t l ) ( h8 h') = t ( h ) t l ( h ' )for any h E H , h' E HI. If h E H , h' E H' we have that

showing that indeed t @ t' is a left integral.

Exercise 5.2.5 Let H be a Hopf algebra. Then the following assertions are equivalent: i) H has a nonzero left integral. ii) There exists a finite dimensional left ideal i n H * . iii) There exists h* E H* such that K e r ( h * ) contains a left coideal of finite codimension i n H . Solution: It follows from Corollary 5.2.4 and the characterization of H*Tat given in Corollary 2.2.16. Exercise 5.2.12 A semisimple Hopf algebra is separable. Solution: Let t E H be a left integral with ~ ( t=)1. We show that

is a separability idempotent. Since compute

C t l S ( t z ) = ~ ( t )=l 1, let x

E H and


Exercise 5.2.13 Let H be a finite dimensional Hopf algebra over the field k , K a field extension of k , and H = K @ k H the Hopf algebra over K defined i n Exercise 4.2.17. If t E H is a left integral i n H , show that T = l K B~ t E H is a left integral i n H . A s a consequence show that H is semisimple over k i f and only if H is sen~isimpleover K . Solution: For any S 81,h E H we have that

which means that ? is a left integral in ?T. The second part follows immediately from Theorem 5.2.10. Exercise 5.3.5 Let H be a finite dimensional Hopf algebra. Show that H is injective as a left (or right) H-module. Solution: Since H* has nonzero integrals, we have that H* is a projective left H*-comodule. By Corollary 2.4.20 we see that H is injective as a right H*-comodule. Since the categories M ~ and * H M are isomorphic, we obtain that H is an injective left H-module.

h

and h E H be such that x o S ( h ) = 1. Then x Exercise 5.4.8 Let x E spans Solution: The existence of x and h as in the statement was proved in 5.4.4, only in that proof we have used the uniqueness. So we show first that this can be done directly. Let J. be the injective envelope of k l , considered as a left H-comodule. Then J is finite dimensional and H = J @I K for a left coideal K of H . Let f : H -+ k be a nonzero linear map such that f (K) = 0 and f ( l H )= 1. Since K K e r ( f ) we have that f E Rat(H;J.) = rat(^-H*). By Theorem 5.2.3 there exist h, E H and t , E such that f = C t, h,, so (Ct, h , ) ( l ) # 0. Therefore, one of the ( t , h , ) ( l ) is not zero, and we can take this element of H as our h and a suitable multiple of the corresponding left integral for X . We will denote the right integral x o S by xS. We show first that for any t E and g E H there exists an 1 E H such that

h.

I

c

-

h

h

-

-


224 where (g

-

t)(x) = t(xg). Let x E H and compute

-

=

=

(9 t)(x) = xS(h)t(xg) (by Lemma 5.4.4 ii))

x

~ S ( x ~ g l h ) t ( x 2 g 2(t) left integral)

X ~ ( x g h l ) t ( ~ ( h 2(xS ) ) right integral)

=

C ghlt(~(h2))

=

xs(x

=

xS(xu) (where u = C ghlt(S(h2)))

= xS(xu)xS(h)

~ S ( x ~ u ~ ) ~ ( x 2 ~ 2(xS S ( right h ) ) integral)

=

xS(h2)x(xuS(hl)) (X left integral) = ~ ( x l )(where 1 = C u S ( h l ) x ( S ( h ~ ) ) ) =

=

(1

-

x)(x)

so (5.35) is proved, and we can choose r E H such that

Now t(x> = xS(h)t(x) = xS(hl)t(xh2) (xS right integral) = ~S(S(~l)~2hl)t(~3h2) = X ~ 2 ( x l ) t ( x 2 h(t) left integral) =

xS2(x1>x(x2r) (by (5.36))

=

xS(r)x(x),

where the last equality follows by reversing the previous five equalities. It follows that t = xS(r)x, i.e. x spans and the proof is complete.

-

Exercise 5.5.6 Prove Corollary 5.5.5 directly. Solution: We know that 4 : H * H, q5(h*) = C t i h * ( t h ) is a bijection. Hence there exists a T E H* such that C t i T ( t h ) = 1. Applying E to this equality we get T(tl) = 1. For h E H we have

5.7. SOLUTIONS T O EXERCISES

In particular, we have S ( t ) = C t ' , T ( t l t L ) = C t i T ( t 1 A ' ( t L ) ) = (t' A1)T(t')= t' A'.

-

-

Exercise 5.5.2 Let f : C -t D be a surjective morphism of coalgebras. Show that if C is pointed, then D is pointed and C o r a d ( D ) = f ( C o r a d ( C ) ) . Solution: It follows from Exercise 3.1.13 and from the fact that for any grouplike element g E G ( C ) , the element f ( g ) is a grouplike element of D . Exercise 5.5.7 Let H be a Hopf algebra such that the coradical Ho is a Hopf subalgebra. Show that the coradzcal filtration Ho C H1 C . . . H , G . . . zs an algebra filtration, 2.e. for any positzve integers m , n we have that HmHn C Hm+n. Solution: We remind from Exercise 3.1.11 that the coradical filtration is a coalgebra filtration, i.e. A(H,) C Cz=o,nHz @ Hn-, for any n. In Ho @ Hn. particular this shows that A ( H n ) 2 Hn 8 Hn-l We first show by induction on m that HmHo = Hm for any nz. For m = 0 this is clear since Ho is a subalgebra of H . Assume that Hm-l Ho = Hm-l. Then

+

where for the second inclusion we used the induction hypothesis. Thus H,,Ho C Ho A Hm-1 = H,,,. Clearly, Hm C HmHo since Ho contains 1. Similarly HoHm = Hm for any m. Now we prove by induction on p that for any m, n with m n = p we have that H, Hn C H,. It is clear for p = 0. Assume this is true for p - 1, where p 2 1, and let m, n with m + n = p. If m = 0 or n = 0, we already proved the desired relation. Assume that m , n > 0. Then we have that

+

which shows that HmHn C Ho A HPp1= H, = Hm+,.

Exercise 5.5.9 Let H be a Hopf algebra. Show that the following are equivalent.

226


(1) H is cosemisimple. (2) k is an injective right (or left) H-comodule. (3) There exists a right (or left) integral t E H* such that t ( 1 ) = 1. Solution: ( 1 ) and ( 2 ) are clearly equivalent from Theorem 3.1.5 and Exercise 4.4.7. To see that ( 2 ) and ( 3 ) are equivalent, we consider the unit map u : k -, H, which is an injective morphism of right H-comodules. Then k is injective if and only if there exists a morphism t : H -+ k of right H-comodules with tu = I d k . But such a t is precisely a right integral with t ( 1 ) = 1. Exercise 5.5.10 Show that i n a cosemisimple Hopf algebra H the spaces of left and right integrals are equal, and i f t is a left integral with t ( 1 ) = 1, we have that t o S = t . Solution: By Exercise 5.5.9 we know that there exist a left integral t and a right integral T such that t ( 1 ) = T ( l )= 1. Then t = T ( 1 ) t = Tt = t ( 1 ) T = T , so = J,. We know that t o S is a right integral. Since (t o S ) ( l ) = 1, we see that t o S = t. Exercise 5.5.12 Let H be a Hopf algebra over the field k , K a field extension of k , and 2 = K 81,H the Hopf algebra over K defined i n Exercise

4.2.17. Show that i f H is cosemisimple over k , then

is cosemisimple over

K . Moreover, in the case where H has a nonzero integral, show that if is cosemisimple over K , then H is cosemisimple over k . Solution: We know from Exercise 5.1.6 that if T is a left integral of H ,

z.

then T E %* defined by T ( 6 8 k h ) = 6 T ( h ) is a left integral of Everything follows now from the characterization of cosemisimplicity given in Exercise 5.5.9. Exercise 5.5.13 Let H be a finite dimensional Hopf algebra. Show that the following are equivalent. (1) H is semisimple. (2) k is a projective left (or right) H-module (with the left H-action o n k defined by h . a = ~ ( h ) a ) . (3) There exists a left (or right) integral t E H such that ~ ( t#)0. Solution: It is clear that ( 1 ) and ( 2 ) are equivalent from Exercise 4.4.8. If the left H-module Ic is projective, since E : H -+ k is a surjective morphism of left H-modules, then there exists a morphism of left H-modules 4 : k -t H such that E o 4 = I d k . Denote t = 4 ( l k )E H . We have that

for any h E H , showing that t is a left integral in H. Clearly ~ ( t=) & ( + ( I ) )= 1. Conversely, if there exists a left (or right) integral t E H such that ~ ( t#) 0, we can obviously assume that ~ ( t=) 1 by multiplying

,


227

with a scalar. Then the map 4 : k -+ H , 4(a) = a t is a morphism of left H-modules and E o 4 = I d , so k is isomorphic to a direct summand in the left H-module H . This shows that k is projective.

Exercise 5.6.12 Give a different proof for the fact that At does not have nonzero integrals, by showing that the injective envelope of the simple right At-comodule kg, g E C , is infinite dimensional. Solution: Let E, be the subspace of At spanned by all

Then by Equation (5.9), E, is a right At-subcomodule of At and kg is Thus the ES7sare essential in E,. On the other hand, At = $SEC&,. injective, and we obtain that &, is the injective envelope of kg.

Exercise 5.6.24 Let A be the algebra generated by an invertible element a and an element b such that 6" = 0 and ab = Xba, where X is a primitive 2n-th root of unity. Show that A is a Hopf algebra with the comultiplication and counit defined by

Also show that A has nonzero integrals and it is not unimodular. Solution: Let C =< a > be an infinite cyclic group and a* E C* such that a*(a) = It is easy to see that A e H(C, n, a2, a*). Everything else follows from the properties of Hopf algebras of the form H(C, n, c , c*,a , b).

a.

Exercise 5.6.25 Let H be the Hopf algebra with generators c, X I , .. . ,xt subject to relations

Show that H is a pointed Hopf algebra of dimension 2t+1 with coradical of dimension 2. Solution: Let C = C2 =< c >, the cyclic group of order 2, c;', . . . ,cf E C* defined by c,*( c ) = -1, and cj = c for all 1 5 j 5 t . Then H = H(C, n, c, c*) and all the requirements follow from the general facts about Hopf algebras defined by Ore extensions. Exercise 5.6.26 Let 4 be an automorphism of k C of the form 4 ( g ) = c*(g)g for g E C , and assume that c*(g) # 1 for any g E C of infinite order. Show that i f 6 is a 4-derivation of k C such that the Ore extension (kC)[Y,4, S ] has a Hopf algebra structure extending that of k C with Y a (1, c)-primitive,


228

then there is a Hopf algebra isomorphism ( k c )[Y, $,6] 2 ( k c )[X,$1. Solution: Let U = {g E Clc*(g) # 1) and V = {g E Clc*(g) = 1). Thus, if g E V then by our assumption g has finite order. In this case, $(gn) = gn for all n , and induction on n 2 1 shows that S(gn) = ngn-lS(g). Then S(1) = mg-lS(g), where m is the order of g, and S(1) = 0 imply that 6(g) = 0. Now let g E U. Applying A to the relation Yg = c*(g)gY + S(g), we find that A(S(g)) = cg @ S(g) + S(g) 8 g. Thus S(g) is a (g, cg)-primitive, and so 6(g) = a,g(c - 1) for some scalar a,. Therefore, for any two elements g and h of U

and similarly

+ a,c*(h))(c - 1)gh Since C is abelian a, + a h < c*,g >= a h + a, < c*, h >, or S(hg) = ( a h

Denote by y the common value of the a,/(l - c*(g)) for g E U. We have a, - y c*(g)y = 0. Let Z = Y - y(c - 1). For any g E U we have that

+

Obviously, Zg = g Z if g E V, so (kC)[Y,4, S] (kC)[Z,41 as algebras. Since 2 is clearly a (1, c)-primitive, this is also a coalgebra morphism, which completes the solution.

Exercise 5.6.31 (i) Let C = C4 =< g > , t = 2 , n = (2,2),c = (g,g), C* = (g*,g*) where g*(g) = -1, b12 = 1, a = (1, I),a' = ( 0 , l ) . Show that there exists a Hopf algebra isomorphism H ( C , n , c, c*, a , b) E H(C,n , c, c*,a', b). (ii) Let C = C4 =< g > , t = 2,n = (2,2),c = (g,g),c* = (g*,g*) where g*(g) = -1, a = ( 1 , l ) and b12 = 2, a' = (0, l ) , bi2 = 0. Show that the Hopf algebras H(C, n , c, c * , a', b') and H(C, n , c, c*, a, b) are isomorphic. Solution: (i) The map f : H(C, n, c, c*, a, b) --+ H(C, n, c, c*, a', b) defined


+

229

+

by f ( g ) = g, f ( x 1 ) = -(p2 P)xi Pxi, f ( x 2 ) = x i , where P E k is a primitive cube root of -1 is a Hopf algebra isomorphism. (ii) The map f from H ( C , n , c, c*,al,b') to H ( C , n, c, c*,a , b) defined by f ( g ) = g, f ( x l ) = 2 2 , f ( x 2 ) = x1 - 2 2 , is a Hopf algebra isomorphism. Note that one of the Hopf algebras is an Ore extension with nontrivial derivation while the other is an Ore extension with trivial derivation.

Exercise 5.6.32 Let C be a finite abelian group, c E C t and c* E C*t such that we can define H ( C , n , c, c*). Show that H ( C , n, c, c*)* H ( C * ,n, c*,c ) , where in considering H ( C *, n, c* , c ) we regard c E C** by identifying C and C**. Solution: Suppose C = C 1 x C2 x . . . x C , =< gl > x . . . x < g, > where Ci is cyclic of order mi. For i = 1, . . . , s, let Ci € k* be a primitive mi-th root of 1. The dual C* =< g: > x . . . x < g,* >, where g,t(gi) = Ci and gf ( g j ) = 1 for i # j,is then isomorphic to C . We identify C and C** using the natural isomorphism C C** where g**(g*)= g* ( g ) . First we determine the grouplikes in H*. Let hf E H* be the algebra map defined by h f ( g j ) = gf ( g j ) and h f ( x j ) = 0 for all i , j . Since the hf are algebra maps from H to k , H* contains a group of grouplikes generated by the g,', and so isomorphic to C * . Now, let yj E H* be defined by y j ( g x j ) = ~ j + - l ( ~and ) , y j ( g x w ) = 0 for xw # x j . We determine the nilpotency degree of yj. Clearly y; is nonzero only on basis elements gx;. Note that by (5.19) and the fact that qj = cj*(cj),

By induction, using the fact that

(T)

'?i

= (1

+ qj + . . . + q;-l),

we see that

for r l j = q:l, 3

Since 9,-, and thus r l j , is a primitive n j - t h root of 1, this expression is 0 if and only if r = nj. Thus the nilpotency degree of yj is nj. Let g* E H* be an element of the group of grouplikes generated by the gb above. We check how the yj multiply with g* and with each other. Clearly,


230

both yjg* and g*yj are nonzero only on basis elements gxj. We compute

so that

g*yj = g*(cj)yjg*,or yjg* = ~

j**-~(~*)~*~~.

Let j < i. Then yjyi and yiyj are both nonzero only on basis elements gxixj = cf ( c j ) g x j x i . We compute

and

Therefore for j < i,

Finally, we confirm that the elements yj are ( e H ,~j*-l)-~rimitives and then we will be done. The maps cj*-' @ yj yj @ E H and m * ( y j )are both only nonzero on elements of H @ H which are sums of elements of the form g @ l x j or gxj @ 1, where m : H @I H -+ H is the multiplication of H and m* : H* -r ( H @ H ) * is regarded as the comultiplication of H * . We check

+

and

m * ( y j ) ( g@I Zxj) = y j ( g l x j )= ~ j * - l ( ~ l ) Similarly,

and

yj(gxj1) = yj(c;(l)glxj) = ~ j * ( l ) c j * - ~=( ~c l; )- ' ( ~ ) Thus the Hopf subalgebra of H* generated by the hr, yj is isomorphic to H(C*,n,c*-',c-') and by a dimension argument it is all of H*. Now we only need note that for any H = H ( C , n, c, c*), the group automorphism of C which maps every element to its inverse induces a Hopf algebra isomorphism from H to H ( C , n,c-', c*-'), and the solution is complete.


Bibliographical notes Again we used the books of M. Sweedler [218], E. Abe [I],and S. Montgomery [149]. Integrals were introduced by M. Sweedler and R. Larson in [120]. The connection with H*Tat was given by M. Sweedler in [219]. Lemma 5.1.4 is also in this paper. In the solution of Exercise 5.2.12 (which we believe was first remarked by Kreimer), we have used a trick shown to us by D. Radford. In [218], M. Sweedler asked whether the dimension of the space of integrals is either 0 or 1 (the uniqueness of integrals). Uniqueness was proved by Sullivan in [217]. The study of integrals from a coalgebraic point of view has proved to be relevant, as shown in the papers by B. Lin [123], Y. Doi [72], or D. Radford [189]. The coalgebraic approach produced short proofs for the uniqueness of integrals, given in D. Stefan [211], M. Beattie, S. Diisciilescu, L. Grunenfelder, C. N5st6sescu, [28], C. Menini, B. Torrecillas, R. Wisbauer [145], S. Dbciilescu, C. N k t b e s c u , B. Torrecillas, [68]. The proof given here is a short version of the one in the last cited paper. The idea of the proof in Exercise 5.4.8 belongs to A. Van Daele [236] (this is actually the method used in the case of Haar measures), and we took it from [198]. The bijectivity of the antipode for co-Frobenius Hopf algebras was proved by D.E. Radford [189], where the structure of the 1-dimensional ideals of H* was also given. The proof given here uses a simplification due to C. Ciilinescu [52]. The method for constructing pointed Hopf algebras by Ore extensions from Section 5.6 was initiated by M. Beattie, S. Dkciilescu, L. Griinenfelder and C. Nkt6sescu in (281, and continued by M. Beattie, S. D6sc6lescu and L. Grunenfelder in [27]. A different approach for constructing these Hopf algebras is due to N. Andruskiewitsch and H.-J. Schneider [13], using a process of bosonization of a quantum linear space, followed by lifting. This class of Hopf algebras is large enough for answering in the negative Kaplansky's conjecture on the finiteness of the isomorphism types of Hopf algebras of a given finite dimension over an algebraically closed field of characteristic zero, as showed by N. Andruskiewitsch and H.-J. Schneider in [13], M. Beattie, S. Dbciilescu and L. Grunenfelder in [25, 271. The conjecture was also answered by S. Gelaki [86] and E. Muller [154]. A more general isomorphism theorem for Hopf algebras constructed by Ore extensions was proved by M. Beattie in [24]

Chapter 6

Actions and coactions of Hopf algebras 6.1 I11

Actions of Hopf algebras on algebras

this chapter k is a field, and H a Hopf k-algebra with comultiplication E . The antipode of H will be denoted by S.

A and counit

Definition 6.1.1 W e say that H acts on the k-algebra A (or that A is a (left) H-module algebra i f the following conditions hold: (MA1) A is a left H-module (with action of h E H on a E A denoted by h . a). (MA2) h . (ab) = C ( h l . a ) ( h a. b), V h E H , a , b E A . (MA3) h . lA= ~ ( h ) l V~h ,E H . Right H-module algebras are defined i n a similar way. I Let A be a lc-algebra which is also a left H-module with structure given by

By the adjunction property of the tensor product, we have the bijective natural correspondence

-

H o m ( H @ A , A)

H o m ( A ,H o m ( H , A ) ) .

If we denote by $ : A H o m ( H , A ) the map corresponding to v by the above bijection, we have the following Proposition 6.1.2 A is an H-module algebra if and only i f $ is a morphism of algebras ( H o m ( H ,A ) is an algebra with convolution: ( f * g ) ( h ) = C f (h1)dhz)).

234

CHAPTER 6. ACTIONS AND COACTIONS

Proof: Since $ corresponds to v , we have that v ( h 8 a ) = $ ( a ) ( h ) ,Qh E H , a E A. Hence ( M A 2 ) holds @ v ( h @ a b ) = ~ v ( h l @ a ) u ( h 2 8 bV) ,~ E Ha , b E A @ $(ab)(h)= C $ ( a ) ( h l ) $ ( b ) ( h a )= ( $ ( a ) * $ ( b ) ) ( h ) V h E H , a,b E A @ $(ab) = $(a) * $(b), Va,b E A @ $ is multiplicative. Moreover, ( M A 3 ) holds I @ v ( h @ 1 ~=)$ ( l ~ ) ( h=) ~ ( h )@ l $~ ( 1 ~ = ) IHO~(H,A). L e m m a 6.1.3 Let A be a k-algebra which is a left H-module such that ( M A 2 ) holds. Then i ) ( h - a)b = C hl . ( a ( S ( h 2 ). b)), Va,b E A, h E H. ii) If S is bijective, then a ( h . b) = C hZ . ( ( S W 1 ( h l.)a)b), Qa,b E A , h E H.

Proof: By ( M A 2 ) we have:

ii) is proved similarly.

-

I

Proposition 6.1.4 Let A be a k-algebra which is also a left H-module. Then A is an H-module algebra if and only if p : A @ A A , p(a@b) = ab, is a morphism of H-modules ( A C3 A is a left H-module with h . ( a C3 b) = [email protected]). Proof: The assertion is clearly equivalent to ( M A 2 ) . To finish the proof it is enough to show that ( M A 3 ) may be deduced from ( M A 2 ) . We do this using Lemma 6.1.3. Indeed, taking in Lemma 6.1.3 a = b = l A ,we have

so ( M A 3 ) holds and the proof is complete.

I

Definition 6.1.5 Let A be an H-module algebra. We will call the algebra of invariants = { a E A I h . a = ~ ( h ) a V, h E H ) .

6.1. ACTIONS O F HOPF A L G E B R A S O N A L G E B R A S

235

A H is indeed a k-subalgebra of A : if a, b E A H ,then for any h E H we have h . (ab) = E ( h l a)(hz - b) =

E &(hi)a&(hz)b

=

Ec(hl)&(hz)ab

=

x

~ ( h l e ( h 2 ) ) a=b &(h)ab.

Another algebra associated to an action of the Hopf algebra H on the algebra A is given by the following

Definition 6.1.6 If A is an H-module algebra, the smash product of A and H , denoted A # H , is, as a vector space, A#H = A @I H , together with the following operation (we will denote the element a @ h by a#h):

-

Proposition 6.1.7 i ) A # H , together with the multiplication defined above, is a k-algebra. ii) The maps a a#lH and h l A # h are injective k-algebra maps from A , respectively H , to A # H . iii) A#H is free as a left A-module, and if {hi)iEl is a Ic-basis of H , then { l A # h i ) i E 1 is an A-basis of A # H as a left A-module. iv) If S is bijective (e.g. when H is finite dimensional, see Proposition 5.2.6, or, more general, when H is co-Frobenius see Proposition 5.4.6), then A # H is free as a right A-module, and for any basis {hi)iel of H over k , { l A # h i ) i E I is an A-basis of A # H as a right A-module.

-

Proof: i) We check associativity:

hence the multiplication is associative. The unit element is l A # l H :


ii) It is clear that (a#lH)(b#lH) = ab#lH, Qa, b E A. We also have (lA#h)(lA#g) = hl 1 ~ # h 2 g= l ~ # & ( h l ) h a= l ~ # h g . The injectivity of the two morphisms follows immediately from the fact that 1~ (resp. l A ) is linearly independent over k. iii) The map a#h a @ h is an isomorphism of left A-modules from A#H to A @ H , where the left A-module structure on A @ H is given by a(b @ h) = ab @I h. iv) will follow from

-

-

Lemma 6.1.8 If A is an H-module algebra, and S is bzjective, we have

Proof:

I We return to the proof of iv) and define

and

0 :H

@I A + A#H,

6(h @I a ) = ( l A # h ) ( a # l ~ ) .

By Lemma 6.1.8 it follows that 0 o 4 = lA#jy. Conversely,

hence also 4 o 0 = lHBA. Since 0 is a morphism of right A-modules, we I deduce that A#H is isomorphic to H @I A as right A-modules. We define now a new algebra, generalizing the smash product.

6.1. ACTIONS O F HOPF A L G E B R A S O N A L G E B R A S

237

Definition 6.1.9 Let H be a Hopf algebra which acts weakly on the algebra A (this means that A and H satisfy all conditions from Definition 6.1.1 with the exception of the associativity of multiplication with scalars from H : hence we do not necessarily have h . ( 1 . a ) = ( h l ) . a for V h ,1 E H , a E A. A s it will soon be seen, this condition will be replaced by a weaker one). Let a :H x H A be a k-bilinear map. W e denote by A#,H the k-vector space A @ H , together with a bilinear operation ( A @ H ) @ ( A @ H-4 ) (ABH), ( a # h ) @ (b#l) H (a#h)(b#l), given by the formula

-

where we denoted a @ h E A @ H by a#h. The object A#,H, introduced above, is called a crossed product if the operation is associative and l A # l ~ I is the unit element (i.e. if it is an algebra). Proposition 6.1.10 The following assertions hold: i ) A#,H is a crossed product if and only if the following conditions hold: The normality condition for a :

The cocycle condition: x ( h l . a(11,m l ) ) o ( h 2 ,hmz) = The twisted module condition:

x

~ ( h lll)a(h212, , m ) , V h . 1, m E H (6.3)

-

For the rest of the assertions we assume that A#,H is a crossed product. (ii) The map a a # l H , from A to A#,H, is an injective morphism of 12-algebras. iii) A#,H 21 A @ H as left A-modules. iv) If c is invertible (with respect to convolution), and S is bijective, then A#,H E H 18 A as right A-modules. In particular, i n this case we deduce that A#,H is free as a left and right A-module. Proof: i) We show that l A # l H is the unit element if and only if (6.2) holds. We compute:

Hence, if u ( 1 , h) = & ( h ) l A V , h E H , it follows that l A # l H is a left unit element. Conversely, if l A # l H is a left unit element, applying I @ E to the equality l A # h = C u ( 1 ,hl)#h2

238


we obtain ~ ( hlA ) = a(1, h). Similarly, one can show that lA#lHis a right unit element if and only if u(h, 1) = &(h)lA. We assume now that (6.2) holds, and that the multiplication defined in 6.1 is associative, and we prove (6.3) and (6.4). Let h, 1, m E H and a E A. From (l#h)((l#l)(l#m)) = ((l#h)(l#l))(l#m) we deduce (6.3) after writing both sides and applying I 8 e. From (l#h)((l#l)(a#m)) = ((l#h)(l#l))(a#m) we deduce (6.4) after writing both sides, using (6.2) and applying I @ E. Conversely, we assume that (6.3) and (6.4) hold. Let a , b, c E A and h, 1, m E H. We have:

where we used, (MA2) for the second equality, (6.3) for h3, 12,ml for the third one, and (6.4) for ha, 11,c for the fourth. On the other hand, ((a#h)(b#l))(c#m) = C a(hl . b)a(h2,11)((h312) + ~ ) ) 4 h 4 1 3m, l ) # h s b m , hence the multiplication of A#,H is associative. ii) and iii) are clear. iv) We define a:H@A-A#,H,

where a-' is the convolution inverse of u, and S-' is the composition inverse of S . We also define

We show that a and p are isomorphisms of right A-modules, inverse one to each other. We prove first the following

Lemma 6.1.11 If u is invertible, the following assertions hold for any h , l , m E H: a) h . 4 , m) = C d h l , ldu(h212, ml)~-'(h3,13m2). b) h , a-'(1, m) = C u(hl, llml)a-1(hz12, m2)u-'(h3, 13). C) C ( h l . o-'(S(h4), h5))o(h2, S(h3)) = ~ ( h ) l ~ .

6.1. ACTIONS OF HOPF ALGEBRAS ON ALGEBRAS

239

Proof: First, if up' is the convolution inverse of o , we have ~ o - ' ( l l , m l ) o ( 1 2 , m 2 ) = ~ o ( l l , m l ) o - ' ( ~ , m 2=) = €(l)€(m)lA, V1,m E

H.

a) We have

where we used (6.3) for the last equality. b) Multiplying by h E H both sides of the equality

we get

- -

h . 0-'(l,m), from H 8 H 8 H to We deduce that the map h 63 163 m A, is the convolution inverse of the map h 8 18 m h . a(1, m). To finish the proof of b), we show that the right hand sides of the equalities in a) and b) are each other's convolution inverse. Indeed,

=

o(h1, ll)o(h212,ml)~1(h313,m2)o-1(h4, 14) = ~ ( h ) ~ ( l ) ~ ( m ) l ~ .

c) The left hand side of the equality becomes, after applying b) for hl, S(h4),h5:


and the proof of the lemma is complete. I We go back to the proof of the proposition and show that a and ,B are each other's inverse. We compute

=

C hs €3 (SP1(h7)- o-'(h2,

S-'(hi)))

(s-l(he) . (h3 . a))o(S-l (h5), h4) (by (MA.2)) =

C hs B (S-'(hi)

. o-' (h2, S-l (hi)))

ff(s-l(h6)' h3)((S-'(hS)h4) . a ) (by (6.4)) =

hr 4 (S-'(he)

= =

x

. o-'(hz,

S-l(hi)))o(S-'(hs), ha)r(hr)a

he €3(S-'(he) . o-l(h2,

S-l

(hl)))o(s-'(h4), h3)a

C h2 4 E(S-' (hl))a (by Lemma 6.1.11, c) for S-' (hl)) =

hz 4€(hl)a = h 4 a ,

hence ,B o a = l , y @ ~Conversely, .

=

1op1(h5, S-'(h4))(h6 =

((SV1(h3). a ) o ( S - ' ( h ~ ) ,hl))#h7

o-'(h5, s-l(h4))(h6 (sW'(h3) a)) (h7. u(S-'(hz), hi))#hs (by (MA211

=

u-l(h8,

S-l

(h7))(hg - (S-l(h6). a))o(hlo, S-' (h5))o(hllS-'(h4), h l )

a-l (hi2,S-I (h3)h2)#h13 (by Lemma 6.1.11, a)) =

o-'(h6, s-l(h5))(h7 . (s-l(h4) . a)) o(h8, S - ' ( h 3 ) ) ~ ( h 9 S - ~ ( h ~hl)#hlo ),

=

o-l(h6, s 1 ( h 5 ) ) o ( h 7 ,S-'(h4))((h~S-'(h3)) . a )

6.1. ACTIONS OF HOPF ALGEBRAS ON ALGEBRAS

=

241

x

a&(h1)#h2 = a#h,

hence also a o p = lA#,H. Finally, we note that

= =

o-l(h2, s - ' ( h l ) ) ( h ~. ( a b ) ) # 4 a ( h 8 ab) = a ( ( h @ a)b),

hence a is a morphism of right A-modules. It follows that phism of right A-modules, and the proof is complete.

-

a!

is an isomor-

I

Remark 6.1.12 In case u : H 8 H A is trivial, i.e. o ( h , l ) = ~ ( h ) ~ ( 1 ) 1A * , is even an H-module algebra, and the crossed product A#,H is the smash product A#H. I We look now at some examples of actions:

Example 6.1.13 (Examples of Hopf algebras acting on algebras) 1) Let G be a finite group acting as automorphisms on the k-algebra A. If we put H = kG, with'^(^) = g 8 g, ~ ( g = ) 1, S(g) = g-l, Vg 6 G, and g . a = g(a) = ag, a E A, g E G, then A is an H-module algebra, as it may be easily seen. The smash product A#H is in this case the skew group ring A * G (we recall that this is the group ring, in which multiplication is altered as follows: (ag)(bh) = (abg)(gh), = is the subalgebra of the elements fixed Qa, b E A, g, h E G), and by G (which explains the name of algebra of the invariants, given to AH in general). The smash product A#H is sometimes called the semidirect product. Here is why. Let K be a group acting as automorphisms on the group H (i.e.


242

+

-

: K Aut(H)). Then K acts there exists a morphism of groups as automorphisms on the group ring kH, which becomes in this way a kK-module algebra. Since in kH#kK we have, by the definition of the multiplication,

we obtain that kH#kK 21 k ( H X + K ) , where H x 4 K is the semidirect product of the groups H and K. 2) Let G be a finite group, and A a graded k-algebra of type G. This means that A = @ A, (direct sum of k-vector spaces), such that AgAh Agh.

c

OEG

If 1 E G-is the unit element, Al is a subalgebra of A. Each element a E A writes uniquely as a = C a,. The elements a, E A, are called the sEG

homogeneous components of a. Let H = kG* = Homk(kG,k), with dual basis {p, I g E G, p,(h) = 6,,h). The elements p, are a family of orthogonal idempotents, whose sum is l H . We recall that H is a Hopf algebra with A(pg) = C ~ ~ h @ph, - 1 E(P,) = 6,,1, S(pg) = pg-1. For a E A we put hEG

p, . a = a,, the homogeneous component of degree g of a. In this way, A becomes an H-module algebra, since p, . (ab) = (ab), = agh-1bh = hEG

C (pgh-1 . a)(ph.b ) . The smash product A#kG* is the free left A-module ~ E G with basis { p , I g E G), in which multiplication is given by The subalgebra of the invariants is in this case Al, the homogeneous component of degree 1 of A. 3) Let L be a Lie algebra over k, and A a k-algebra such that L acts on A Derk(A) a morphism as derivations (this means that there exists cr : L of Lie algebras). For x E L and a E A, we denote by x . a = cr(x)(a). Let H = U(L), be the universal enveloping algebra of L (for x E L A(x) = x 63 1 163 x, E(X) = 0, S(x) = -x). Since H is generated by monomials of the form X I . . .a,, xi E L, we put

-

+

XI..

.x, . a

= XI

. (xz . (. . . (x, . a ) . . .),

a E A.

In this way, A becomes an H-module algebra, and AH = {a E A ( x . a 0, Vx E L). 4) Any Hopf algebra H acts on itself by the adjoint action, defined by h . I = (ad h)1 =

hlls(h2).

=

6.2. COACTIONS O F HOPF ALGEBRAS ON ALGEBRAS

243

This action extends the usual ones from the case H = kG, where (ad x)y = xYx-l, x , y E G, or from the case H = U ( L ) , where (ad x)h = xh - hx, x E L, h E H (the second case shows the origin of the name of this action). We have then HH = Z ( H ) (center of H ) . Indeed, if g E HH, then Qh E H

The reverse inclusion is obvious. 5) If H is a Hopf algebra, then H* is a left (and right) H-module algebra h*)(g) = h*(gh) (and (h* h)(g) = h*(hg)) with actions defined by (h I for all h , g H, ~ h* E H*.

-

6.2

-

Coactions of Hopf algebras on algebras

We have seen in Example 6.1.13 2) that a grading by an finite group G on an algebra is an example of an action of a Hopf algebra. To study the case when G is infinite requires the notion of a coaction of a Hopf algebra on an algebra. Definition 6.2.1 Let H be a Hopf algebra, and A a k-algebra. We say

that H coacts to the right on A (or that A is a right H-comodule algebra) if the following coditions are fulfilled: (CAI) A is a right H-comodule, with structure map

(CA2) C(ab)o @ (ab)l = Caobo 8 albl, Va, b E A. (CA3) p(1) = 1~@ 1 ~ . The notion of a left H-comodule algebra is defined similarly. If no mention of the contrary is made, we will understand by an H-comodule algebra a right H-comodule algebra. I The following result shows that, unlike condition (MA2) from the definition of H-module algebras, conditions (CA2) and (CA3) may be interpreted in both possibile ways.

-

Proposition 6.2.2 Let H be a Hopf algebra, and A a k-algebra which is a

right H-comodule with structural morphism p : A assertions are equivalent: i) A is an H-comodule algebra.

A@H. The following


244

-

ii) p is a morphism of algebras. iii) The multiplication of A is a morphism of comodules (the right comodule structure o n A @ A is given by a @ b C a0 @ bo @ albl), and the unit of A, u : k A is a morphism of comodules.

-

Proof: Obvious. I As in the case of actions, we can define a subalgebra of an H-comodule algebra using the coaction. Definition 6.2.3 Let A be an H-comodule algebra. The following subalgebra of A ~ c o H = { a € A I p(a) = a @ 1). is called the algebra of the coinvariants of A .

I

In case H is finite dimensional, we have the following natural connection between actions and coactions. Proposition 6.2.4 Let H be afinite dimensional Hopf algebra, and A a k-

algebra. Then A is a (right) H-comodule algebra if and only if A is a (left) H*-module algebra. Moreover, i n this case we also have that A ~ =* AcoH. Proof: Let n = d i m k ( H ) , and { e l , . . . , e n ) C H , {e;, . . . , e;) C H* be dual bases, i.e. e * ( e j ) = b i j . Assume that A is an H-comodule algebra. Then A becomes an H*-module algebra with

f -a= xaof(al), Vf

E H * , a € A.

Indeed, we know already that A is a left H*-module, and

Conversely, if A is a left H*-module algebra, A is a right H-comodule with

6.2. COACTIONS OF HOPF ALGEBRAS ON ALGEBRAS We have, for any f E H*

i=l

f . (ab) 8 1

n

( I @f ) (

C ( e f . a ) ( e ; . b) 8 e i e j )

i,j=l

( I 8 f )(p(a)p(b)), and so p(ab) = p(a)p(b). Finally,

We now have

Example 6.2.5 (Examples of coact.ions of Hopf algebras o n algebras)

245


246

1) Any Hopf algebra H is an H-comodule algebra (left and right) with ~ . we comodule structure given by A. Let us compute H ~ If h~ E HCoH, have A(h) = C hl 8 hp = h 8 1. Applying I 8 E to both sides, we obtain , HCoH2 k1. Since the reverse inclusion is clear, we have h = ~ ( h ) lhence HCoH = k1. 2) Let G be an arbitrary group, and A a graded k-algebra of type G (see Example 6.1.13, 2) ). Then A is a kG-comodule algebra with comodule structure given by

where a =

C a,,

a, E A, almost all of them zero. We also have AcOkG =

sEG

Al. 3) Let A#,H be a crossed product. This becomes an H-comodule algebra with P : A#uH A#uH 8 H , p(a#h) = x ( a # h l ) 8 h2.

-

We have (A#,H)COH = A#,l E A. Indeed, if a#h E ( A # , H ) " o ~ , then applying I @ I 8 E to the equality p(a#h) = (a#h) 8 1 we obtain a#h E A#,l, and the reverse inclusion is clear. Since the smash product is a particular case of a crossed product, the assertion also hold for a smash I product A# H. It is possible to associate different smash products to a right H-comodule algebra A. First, the smash product #(H, A) is the k-vector space Hom(H, A) with multiplication given by

-

Exercise 6.2.6 With the multiplication defined in (6.5), #(H, A) is an associative ring with multiplicative identity UHEH. Moreover, A is isomorphic ~(h)a. to a subalgebra of #(H, A) by identifying a E A with the map h Also H* = Hom(H, k) is a subalgebra of #(H, A).

Remark 6.2.7 If we take k with the H-comodule algebra structure given by UH, then the multiplication from (6.5) is just the convolution product. I We can also construct the (right) smash product of A with U, where U is any right H-module subring of H*, (i.e. possibly without a 1). This smash product, written A#U, is the tensor product A 8 U over k but with multiplication given by

6.2. COACTIONS O F HOPF ALGEBRAS ON ALGEBRAS

247

If H is co-Frobenius, H*Tatis a right H-module subring of H*,A#H*Tat makes sense and is an ideal (a proper ideal if H is infinite dimensional) of A#H*. In fact, A#H*Tat is the largest rational submodule of A#H* where A#H* has the usual left H*-action given by multiplication by l # H * . To see this, note that A#H* is isomorphic as a left H*-module to H* 8 A, where the left H*-action on H* @ A is given by multiplication by H* @ 1. The isomorphism is given by the H*-module map

with inverse q!J defined by q!J(a#h*) = C h* -- S-l(al) @ a0 . Since ( H *@A)Tat= H*Tat@A,(A#H*)Tat = 4(H*Tat @A) = A#H*Tat Thus we have

If H is finite-dimensional, then H is a ,left H*-module algebra, and these smash products are all equal (this is the usual smash product from the previous section). Note that the idea for the definition of (6.5) comes naturally by transporting the smash product structure from A#H* to Hom(H, A) via the isomorphism of vector spaces from Lemma 1.3.2. Exercise 6.2.8 I n general, A#H*Tat is properly contained i n #(H, A)Tat. Remark 6.2.9 Let us remark that for graded rings A over an infinite group G, A#(kG)*Tat is just Beattie's smash product (211. W e can adjoin a 1 to A#H*Tat in the standard way. Let (A#H*Tat)l = A#H*Tat x A with componentwise addition and multiplication given by

Then (A#H*Tat)l is an associative ring with multiplicative identity ( 0 , l ) and with A#H*Tat isomorphic to an ideal i n (A#H*Tat)l via i(x) = (x,O). Again, for graded rings A over an infinite group G , (A#(kG)*Tat)l is just Quinn's smash product (1841. I We define now the categories of relative Hopf modules (left and right). Definition 6.2.10 Let H be a Hopf algebra, A an H-comodule algebra. W e say that M is a left (A, H)-Hopf module if M is a left A-module and a right H-comodule (with m H C mo @I m l ) , such that the following relation holds.

248


We denote b y A ~ the H category whose objects are the left ( A ,H)-Hopf modules, and in which the morphisms are the maps which are A-linear and H - colinear. We say that M is a right ( A ,H)-Hopf module if M is a right A-module and a right H-comodule (with m H Em0 B m l ) , such that the following relation holds.

We denote by M z the category with objects the right ( A ,H)-Hopf modules, and morphisms linear maps which are A-linear and H-colinear. Similar definitions may be given for left H-comodule algebras. If A is such an algebra, the objects of the category ZM are left A-modules and left H comodules M satisfying the relation

for all a E A , m E M , and the objects of the category H M are ~ right A-modules and left H-comodules M satisfying the relation

for all a E A , m E M . I f M is a left H-module, we denote by

M~ = {

m M ~I h . m = ~ ( h ) m V , hE H).

I f A is a left H-module algebra, and M is also a left A#H-module, it may be easily checked that M~ is an AH-submodule o f M . If M is a right H-comodule with m H C mo @ ml, we denote by

I f A is a right H-comodule algebra, and M is also a right ( A ,H)-module, it may be checked that M~~~ is an ACoH-submoduleo f M . The following result characterizes the categories o f relative Hopf modules in case H is co-F'robenius. Proposition 6.2.11 Let H be a co-Frobenius Hopf algebra, and A a right H-comodule algebra. Then: i) The category A ~ isHisomorphic to the category of left unital A#H*Tatmodules (i.e. modules M such that M = (A#H*Tat). M ) , denoted by A#H*~atM U . iz) The category M : is isomorphic to the category of right unital A#H*Tatmodules, denoted b y M i # H . , , t .

6.2. COACTIONS OF HOPF ALGEBRAS ON ALGEBRAS

249

Proof: i) The reader is first invited to solve the following Exercise 6.2.12 Let H be co-Frobenius Hopf algebra and M a unital left A#H*T"t-module. Then for any m E M there exzsts an u* E H*Tatsuch u* . m = (l#u*) . m, so M is a unital left H*Tat-module, and that m therefore a rational left H*-module.

"

. Exercise shows that M is a rational left Let M E A # H * v a t M UThe H*-module, and therefore a right H-comodule. M also becomes a left A#H*-module via (a#h*) . m = (a#h*u*) . m

for a E A, h* E H*, m E M , u* E H*Tat, and m = u* . m . The definition is correct, because we can find a common left unit for finitely many elements in at . Now we turn M into a left A-module by putting a . m = (a#e).m. We have

so M E Conversely, if M E then M becomes a left H*-module with H*Tat. M = M , and a left A#H*-module via

Then

so M becomes a unital left A#H*Tat-module. It is clear that the above correspondences define functors (which are the identity on morphisms) establishing the desired category isomorphism. ii) The proof is along the same lines as the one above. It should be noted that H*Tatis stabilized by the antipode, which is an automorphism of H considered as a k-vector space, and so if h* E H* with Ker(h*) I, I a finite codimensional coideal, then Ker(h*S) 2 S - l ( I ) , which is also a

>


250

coideal o f finite codimension. W e also note that i f M E M?, then the right A#H*-module structure on M is given by

Exercise 6.2.13 Consider the right H-comodule algebra A with the left and right A#H*-module structures given by the fact that A E M ? and A E AMH:

Then A is a left A#H* and right AcoH-bimodule, and a left ACoHand right A# H* -bimodule. Consequently, the map

is a

ring

morphism.

Exercise 6.2.14 Let A be a right H-comodule algebra and consider A as a left or right A#H*Tat-module as in Exercise 6.2.13. Then: i ) ACoHE End(A#H*ratA) iz) AcoH End(AA#H*rat).

,

Example 6.2.15 1 ) If H is a Hopf algebra, H is a right comodule algebra as i n Example 6.2.5, I), then the categories H M H and M z are the usual categories of Hopf modules. 2) If G is a group, H = IcG, and A is a graded Ic-algebra of type G (see Example 6.2.5, 2) ), then the category (respectively M:) is the categorg I of left (resp. right) A-modules graded over G. Proposition 6.2.16 If H is a co-Frobenius Hopf algebra, and 0 # t E J, (resp. M E Then: let M E i ) t . M MCoH ii) If m E M~~~ and c E A , then t . ( c m ) = ( t . c ) m (resp. t . ( m c ) = m ( t . c)). In particular, the map M McoH, m ++ t . m is a morphism of AcoHbimodules.

c

MY).

-

Proof: W e prove only one o f the cases. i) I f h* E H * , then h* . (t . m) = ( h * t ). m = h * ( l ) t .m. ii) t . (cm)= C t ( c l m l ) c 0 m o = C t ( c 1 ) c o m = ( t . c)m.

6.3. THE MORlTA CONTEXT

251

Corollary 6.2.17 If H is a finite dimensional Hopf algebra, 0 # t E H is

a left integral, and A is a left H-module algebra, the map

I

is a morphism of AH-bimodules.

Definition 6.2.18 The map t r from Corollary 6.2.17 is called the trace

function. W e say that the the H-module algebra A has an element of trace I 1 if t r is surjective, i.e. there exists an a E A with t . a = 1. Example 6.2.19 1) Let G be a finite group acting on the k-algebra A as g is a left integral automorphisms (see Example 6.1.13, 1) ). Then t = sEG

i n H = k G , and the trace function is i n this case

I n case A is a field, a Galois extension with Galois group G , the trace function is then exactly the trace function defined e.g. in N. Jacobson [99, p.2841, which justifies the choice for the name. The connection with the trace of a matrix is the following: i n the Galois case, the trace of an element is the trace of the image of this element i n the matrix ring via the regular representation (cf. [99, p.4031). 2) If H is semisimple, then any H-module algebra has an element of trace 1. Indeed, if t is an integral with ~ ( t=) 1, then t . 1 = 1. I Exercise 6.2.20 (Maschke's Theorem for smash products) Let H be a

semisimple Hopf algebra, and A a left H-module algebra. Let V be a left A#H-module, and W an A#H-submodule of V . If W is a direct summand i n V as A-modules, then it is a direct summand i n V as A#H-modules.

6.3

The Morita context

Let H be a co-Frobenius Hopf algebra, t a nonzero left integral on H , and A a right H-comodule algebra. In this section, we construct a Morita context connecting A#H*Tat and ACoH. T h e n we will use the Morita context t o study the situation when A/ACoH is Galois. Recall from Exercise 6.2.13 that A is an A#H*rat - ACoH-bimoduleand an AcoH - A#H*Tat-bimodule with the usual modu module structure o n A, and for a , b E A, h* E H*Tat,the left and right A#H*Tat-module structures are given by: (a#h*) . b = C a b o h * ( b ~ ) ,


252

-

and

b . (a#h*) = (h*sW1) (ba) =

1boaoh*(S-'(blal)).

If g is the grouplike element of H from Proposition 5.5.4 (iii) (which was denoted there by a), we can also define a (unital) right A#H*'at-module structure on A by

b ., (a#h*) = b . (a#g

-

h*) =

boaoh'(~-'(bla~)~).

-

Since g defines an automorphism of A#H*rat, a#h* I-+ a#g h*, it follows that with this structure A is also an ACoH- A#H*Tat-bimodule. We define now the Morita context. Let P = A # H I T ~ ~A A c o ~with the standard bimodule structure given above. Let Q = ~ c o H AAWH*ratwhere now the right A#H*Tat-module structure on A is defined using the grouplike from Proposition 5.5.4 (iii), which we will now denote by g, as above. Define bimodule maps [-, -1 and (-, -) by

I-, -](a

abo#t

and

(-,-)

(-, -)(a

-

-

€3 b) = [a, b] =

bl,

: Q 8 P = A @ ~ ~ # ~A * r a tA ' o ~ , @Q

-

b) = (a, b) = t

-

(ab) =

aobOt(alb1).

Note that since t A C A " " ~ ,the image of (-, -) lies in AcoH. Then, with the notation above, we have Proposition 6.3.1 For H with nonzero left integral t, A, P, Q, [-, -1, (-, -) as above, the sextuple

is a Morita context.

-

Proof: We have to check that: 1. The bracket [-,-I: A B A c o ~ A A#H*rat satisfies [ab, c] = [a, bc] for b E A ' o ~ , which is clear, and that it is a bimodule map. Left A#H*Tat-linearity: [(a#l) . b, c] = C[abol(bl),c] = C abocol(bl)#t cl and

-

=

aboco#l(bl)t

-

cl since t is a left integral.

6.3. THE MORITA CONTEXT

253

Right A#H*rat-linearity: [a,b.,(c#l)] = C [ a ,b o ~ o l ( S - ~ ( b l c l ) g=) ]C aboco#(t and,

[a,b](c#l) = x ( a b o # t ='

=

-

- b1~1)1(S-~(b2~2)g),

- bl)(c#l)

C aboco#(t -- blcl)l C ab0co#(t -- l ( ~ - ' ( b z c l ) ) -) blcl

-

since X(l h ) = ( 1 h ) ( g )= l ( h g ) . 2. The bracket (-,-) : A @ A # H * T O ~A + t A C - ACuHis obviously r the definition is correct by Exercise 6.2.16. left and right ~ " " ~ - 1 i n e aand Moreover, ( a ,(b#l) . c) = C ( a ,bcol(c1)) = C aobocot(aibici)l(c2) = C aoboco((t -- a i b i ) l ) ( c i ) = C aoboco((t(1 S-l (a2b2))) a l b l ) ( c l ) = Caoboco(t a l b l ) ( c l ) ( l-- S-'(a2b2))(g) by X(m)= m ( g ) and (a.,(b#l), c ) = C(aobol(S-l (a1b l ) g ) ,c ) = C aobocot(alblcl)l(S-'(a2b2)g). 3. Associativity of the brackets. First note that we will use (g t)S-' = t from Proposition 5.5.4 (iii). Now,

-

-

-

-

-

u ,. [b,C ] =

C a ., (bco#t - c l )

=

C aoboco(t - ~ 2 ) ( S - l ( a i b l c l ) ~ ) C a o b o c t ( ~ - l( a l b l ) g ) C aobo((g t ) s - ' ) ( a l b i ) c C aobot(albl)cby the above

=

( a ,b)c.

= = =

Also [a,b]. c = C ( a b o # t bl)c = C a b o c o ( t b l ) ( c l ) = Cabocot(b1cl) = c). I If any of the maps of the above Morita context is surjective, then it is an isomorphism. While for the map (-, -) this is well known, there is a little problem with the other map, since A#H*rat has no unit. Although the proof is almost the same as the usual one, we propose the following

-

-

Exercise 6.3.2 If the map [-, -1 from the Morita context i n Proposition 6.3.1 is surjective, then it is bijective.


254

Now we discuss the surjectivity of the Morita map to AcoH,leaving the discussion on the other map for the section on Galois extensions.

Definition 6.3.3 A total integral for the H-comodule algebra A is an H comodule map from H to A taking 1 to 1. Since an integral for the Hopf algebra H is a colinear map from H to k, the H-comodule algebra k has a total integral if and only if H is cosemisimple.

Exercise 6.3.4 Let H be a finite dimensional Hopf algebra. Then a right H-comodule algebra A has a total integral if and only if the corresponding left H*-module algebra A has an element of trace 1. We give now the characterization of the surjectivity of one of the Morita context maps.

-

Proposition 6.3.5 The Morita context map to ACoH is onto if and only A. if there exists a total integral Q, : H Proof: (+) Let Q, be a total integral, i.e. Q, is a morphism of right Hcomodules, and cP(1) = 1. Then cP is also a morphism of left H*-modules, @ ( h )for any h E H . so Q,(t h ) = t Suppose h E H is such that t h = 1, then for (-, -) the map from Proposition 6.3.1,

-

-

-

-

which shows that (-, -) is onto. Since t H & HCoH= k l H , to find an . h E H with t h = 1, it is enough to prove that t H # 0. But if t H = 0 , then for any h,g E H we have that:

-

-

= C t ( g 2 h 3 ) g l h 2 ~ - ' ( h l= ) x(t

- -

-

-

( g h ~ ) ) S - l ( h i= ) 0

- -

and so ( H t) H = 0. But H t = H*Tat, so H*Tat H = 0. Finally, since H*Tatis dense in H*, this implies that H* H = 0 which is clearly a contradiction. (+) Choose a E A such that t a = 1, and define Q, : H ---+ A by

-

Then @ ( I ) = 1 and Q, is a morphism of left H*-modules since for h* E H * , ~H E,

6.4. HOPF-GALOIS EXTENSIONS

6.4

Hopf- Galois extensions

Let H be a Hopf algebra over the field k, and A a right H-comodule algebra. We denote by p:A-ABH, p(a)=~ao8al the morphism giving the H-comodule structure on A, and by ACoHthe subalgebra of coinvariants. We define the following canonical map

can : A g

A

A c o ~

-

A 8 H, can(a 63 b) = ( a 8 l ) p ( b )=

x

abo 8 bl

Definition 6.4.1 W e say that A is right H-Galois, or that the extension I A/AcoH is Galois, i f can is bijective.

-

We can also define the map

can' : A @

~

~

Ac

o

~

A 8 H , can(a 63 b) = p(a)(b 8 1) =

x

aob 8 a l .

Exercise 6.4.2 If S is bijective, then can is bijective if and only i f can' is

bijective. We give two examples showing that this notion covers, on one hand, the classical definition of a Galois extension, and, on the other hand, in the case of gkadings (Example 6.2.5, 2) ) it comes down to another well known notion. We will give some more examples after proving a theorem containing various characterizations of Galois extensions.

Example 6.4.3 (Examples of Hopf-Galois extensions) 1) Let G be a finite group acting as automorphisms on the field E

> k. We

know from Example 6.1.13, 2) that E is a left kG-module algebra, hence a right kG*-comodule algebra. Let F = E ~ It. is known that E / F is Galois with Galois group G if and only if [E : F] =I G I (see N. Jacobson


256

[99, Artin's Lemma, p. 2291). Suppose that E/F is Galois. Let n =I G 1, G = {TI,.. . , r],), ( ~ 1 ,... ,u,) a basis of E / F . Let {pl, . . . ,p,) c kG* be the dual basis for {qi) c kG. E is a right kG*-comodule algebra with p : E E €3 kG*, p(a) = C ( Q . a)@pi. c a n : E @ F E E€3kG* isgiven by can(a8b) = Ca(vi.b)€3pi. If w = Cxj@ u j E Ker(can), it follows that

-

.

-

(because pi are linearly independent). As in the proof of Artin's Lemma, it may be shown that if the system (6.9) has a non-zero solution, then all the elements x j are in F, which contradicts the fact that {ui) is a basis. Hence all x j are 0, so w = 0. It follows that can is injective. But can is F-linear, and both E @F E and E @ kG* are F-vector spaces of dimension n2, and therefore can is a bijection. Conversely, we use dimF@ € 3 E~ ) = [E : FI2and dimF(E €3 kG*) = [E : F] I G I. If can is an isomorphism, it follows that [E : F] =I G 1, so E / F is Galois. 2) Let A = @ A, be a graded k-algebra of type G. We know from Example sEG

"

6.2.5, 2) that A is a right kG-comodule algebra, and that A"" = Al . We recall that A is said to be strongly graded if A,Ah = Aghl Vg, h E G , or, equivalently, if AgAg-l = Al, Vg E G. We have that A/A1 is right kGGalois if and only if A is strongly graded. Assume first that A is strongly graded. Let

where ai E A,-I, bi E A,, C a i b i = 1. It may be seen immediately that (can o P)(a €3 g) = a €3 9. Moreover,

257

6.4. HOPE'-GA LOIS EXTENSIONS

Conversely, if can is bijective, it is in particular surjective. For each g E G, let ai, bi E A be such that

It follows that all b, may be assumed homogeneous of degree g, and C a,b, = 1. Since the sum of homogeneous components is direct, it follows that the I a, may be also assumed homogeneous of degree g-I. We remark that in the last example it was enough to assume that can is surjective to get Galois. As we will see below, in the main result of this section, this is due to the fact that kG is cosemisimple, in particular co-Frobenius. For any M E AM^, consider the left A#H*Tat-module map

where the A#H*Tat-module structure of A McoH is induced by the usual left A#H*Tat action on A. Thus 4~ is also a morphism in the category AM^. ~f 4 M is an isomorphism for all M E A M H , we say the Weak Structure Theorem holds for AM^. Similarly, if for any M E M z , the map 4h : McoHBAcon A M, $ h ( m 18a ) = m a @ A C ~ H

-

is an isomorphism, the Weak Structure Theorem holds for M z We prove now the main result of this section.

Theorem 6.4.4 Let H be a Hopf algebra with non-zero left integral t, A a right H-comodule algebra. Then the following are equivalent: i) A/ACoH is a right H-Galois extension. ii) The map can: A BAcon A A @ H is surjective. iii) The Morita map [-, -1 is surjective. iv) The Weak Structure Theorem holds for *M H . v) The map 4M is surjective for all M E AM^. vi) A is a generator for the category A ~ HA#H*TatMU.

-

Proof: Since H is co-Frobenius, the map

is bijective. Then, since [-, -1 = (IB r) o can, it follows that ii) + iii). This also shows that i ) ii),~ using Exercise 6.3.2. In order to show that v) =+ iii), we consider A#H*Tat E AM^, which is @ A as in (6.6). Therefore, (A#H*Tat)cOH= (H*Tut@ isomorphic to H*Tat ~ ) c o H= t @ A, and so 4 ~ # ~ - r a=t [-, -1.

258


iii) + iv). Let M E A ~ Hm, E M . We show first that 4~ is one to one. Suppose m = 4 ~ ( C a 8i mi) for ai E A, mi E MCoH.Let e* be an element of H*T"tthat agrees with E on the finite set of elements ai,, m l in H . Suppose E[ck,dk]= I#(?*. Then

=

C

Ot

-

(dr

aimi)) since mi E M~~~

So if m = 0 it follows that C ai @ A c o ~mi = 0, and so 4M is injective. To show that q5M is surjective, note that for m and e* as above

We have thus proved that iii), iv), and v) are equivalent. iv)+ vi). Let M E AM^. Since ACoHis a generator in Aco~M, for some set I, there is a surjection from ( A ~ O ~ ) (to' )McoH. Thus there is a A @ A c o ~( A ~ O ~ ) to ( ' )A M~~~II M . surjection from A(') vi) =+ v). Let M E AM^. Since A a generator, given x E M , there is an index set I, (fi)iEI, fi E Hom;(A, M), ai E A, with C fi(ai) = x. I Then fi(l) E M ' o ~ ,and z = 4 M ( Ca; @I f i ( l ) .

-

Remark 6.4.5 A similar statement holds with can' replacing can, and the category M z replacing

AM^.

Corollary 6.4.6 If H is co-Frobenius and the equivalent conditions of Theorem 6.4.4 hold, then the map n in Exercise 6.2.13 induces a ring isomorphism A#H*Tat 2 E n d ( A A c o ~ ) T a t ,

where the rational part is taken with respect with the right H*-module structure given by (f . h*)(b) = C h*(bl)f (bo). Proof:We prove first that the map

is injective. Let z E A#H*Tat be such that z . a = 0, Va E A. Let g* E H*T"t be such that z(l#g*) = z. Since [-, -1 is surjective, there exist ai, bi E A

6.4. HOPF-GALOIS EXTENSIONS

259

such that 1#g* = C [ a i , bi]. Then we have z = z(l#g*) = C [ z . ai, bi] = 0. Thus it remains to show that the corestriction of .rr to E n d ( A A c o ~ ) Tisa t a t , let h* E H*Tat such that f = f sh*. surjective. Let f E E n d ( A A c o ~ ) Tand Let t' be a right integral and C ai @ bi E A B A c o ~A such that 2

18(h* 0 S-I) =

aibio 8 S(bil)

-

t'.

Then, for any b E A we have

and the proof is complete. I If H is finite dimensional, then from the Morita theory it follows that A/ACoHis an H-Galois extension if and only if A is projective finitely generated as a right ACoH-moduleand the map .rr is an isomorphism. The behaviour of .rr in the general co-Frobenius case was exhibited in the previous proposition. The next result investigates the structure of A as a right ACoH-modulein the Galois case.

Corollary 6.4.7 If H is co-fiobenius, then any H-Galois H-comodule algebra A is a fiat right AcoH-module. Proof: A well known criterion for flatness ([3, 19.191) says that A is flat over AcoH if and only if for every relation

there exist elements cl, . . . , c, 1, . . . ,n) such that

E A and cij E AcoH

(2 =

1 , . . . ,m,j =


260 and

SO let a l , . . . , a n E A and bl,. . ., bn E AcoH such that sider the morphism in

AM^

n

C ajbj = 0.

j=1

Con-

Since A is a generator in AM^, there exist a set X and a surjective morphism 4 : A ( ~ -+ ) Ker( 0 and some A-module Q such that any direct summand of Q is not isomorphic

296

CHAPTER 7. FINITE DIMENSIONAL HOPF ALGEBRAS

to any Pi. Let r be the least common multiple of the numbers n l , . . . ,nt. We have that W(T)2

Let a = m i n ( y , . . . ,

p,('"l)

$ . ..$

pirrnt)Q(') @

y), say a = m.Then ni

,

This ends the proof if we denote F = A(*), which is free, and E = . . , pt(rmt -ant) @Q(r)lwhich is not faithful since it does not p:rrnl-anl) contain any direct summand isomorphic to Pi (note that rmi - a n i = 0 and that we have used again the Krull-Schmidt theorem). I

Proposition 7.2.6 Let W be a finitely generated left B-module such that w(') is a free B-module for some positive integer r . Then W is a free B-module.

Proof: Take B = B1@.. .@B,, a direct sum of indecomposable B-modules. If t is a nonzero left integral of B and t = tl . . . t, is the representation of t in the above direct sum, we see that

+ +

btl

+ . . . + bt,

= bt = &(b)tl

+ . . . + €(b)t,

=

~(b)t

for any b E B , showing that t l , . . . , t , are left integrals of B. Since the space of left integrals has dimension 1, we have that there exists a unique j such that t j # 0, and then t = tj. Hence B j is not isomorphic to any other Bi, i # j , since B j contains a nonzero integral and Bi doesn't, and if f : Bj --+ Biwere an isomorphism of B-modules we would clearly have that f (t) is a nonzero left integral. Let us take now PI = Bj, P2,. . . , P, the isomorphism types of principal @ . . . $P,("")the representation indecomposable B-modules, and B E of B as a sum of such modules. Note that n l = 1. We know that w(') is free as a B-module, say w(')1,B(P) for some positive integer p. Since

pin1)

the Krull-Schmidt theorem shows us that the decomposition of W as a sum of indecomposables is of the form W E @ . . . @ P , ( ~ for " ) some r n l , . . . ,m,. Moreover, since w ( ~E)P:'~') @ . . . @ P,("~"),we must have pni = rmi. In particular p = r m l . Then for any i we have that pni = r m l n i = rmi, so mi = mini. We obtain that W E B("'), a free Bmodule. I

pirn')

7.2. THE NICHOLS-ZOELLER THEOREM

297

Proposition 7.2.7 Let W be a finitely generated left B-module such that ( ~ ) ) there exists a faithful B-module L with L @ W = w ( ~ ~ as~ B-modules. Then W is a free B-module. Proof: We know from Exercise 5.3.5 that B is an injective left B-module. Then we can apply Proposition 7.2.5 and find that w ( ~=)F $ E for some positive integer r, some free B-module F and some B-module E which is not faithful. Proposition 7.2.6 shows that it is enough to prove that w(') is free. We have that

so we can replace W by ~ ( ~ and 1 , thus assume that W E F $ E . Similarly there exists a positive integer s such that L(") 2 F' $ E' with F' free and E' not faithful. Since L is faithful, L ( ~is) also faithful, so F' # 0. Since

. have reduced to the case L and we can replace L by ~ ( " 1 We Denote t = dim(L). Since L 8 W = w ( ~we ) , obtain' that

Since F is free, say F

LBF

2

F'

$ E'.

we see that

E~ ( 4 1 ,

L@B(~) = (L @ ~ ) ( 4 ) - ( B ( * ' ~ ( ~ ) ) ) (by ( ~ )Proposition 7.2.2) E

-

~ ( t 4 )

-

~ ( t )

-

Equation (7.9) and the Krull-Schmidt theorem imply now that E ( ~ ) L ~ E . If E # 0 we have that

Proposition 7.2.2 tells then that F' @ E is nonzero and free, hence it is , This shows that E = 0 and faithful, and then so is ~ ( ~ a1 contradiction. then W is free. a Lemma 7.2.8 If any finite dimensional M E gM is free as a B-module, then any object M E EM is free as a B-module.

298


Proof: We first note that any nonzero M E EM contains a nonzero finite dimensional subobject in the category E M . Indeed, let N be a finite dimensional nonzero H-subcomodule of M (for example a simple Hsubcomodule). Then B N is a finite dimensional subobject of M in the category E M . For a nonzero M E EM we define the set 3 consisting of all non-empty subsets X of M with the property that B X is a subobject of M in the category EM, and B X is a free B-module with basis X . By the first remark, 3 is non-empty. We order F by inclusion. If (Xi)iEI is a totally ordered subset of F, then clearly X = UicIXi is a basis of BXi is a subobject of M in E M , the B-module B X , and B X = CiEI so X E 3. Thus Zorn's Lemma applies and we find a maximal element Y E 3. If B Y # M , then M I B Y is a nonzero object of EM, so it contains a nonzero subobject S/BY, where B Y S 5 M and S E Z M . Let Z be a basis of S/BY and Y' C S such that .rr(Y') = Z, where n : S -,S/BY is the natural projection. Then S is a free B-module with basis Y U Y', so Y U Y' E F , a contradiction with the maximality of Y. Therefore we must have BY = M, so M is a free B-module with basis Y. I Theorem 7.2.9 (Nichols-Zoeller Theorem) Let H be a finite dimensional is free as Hopf algebra and B a Hopf subalgebra. Then any M E a B-module. I n particular H is a free left B-module and dim(B) divides dim(H).

EM

Proof: Lemma 7.2.8 shows that it is enough to prove the statement for finite dimensional M E g M . Let M be such an object. Since H is finitely generated and faithful as a left B-module, and from Proposition ~ B-modules, ( ~ ) ) we obtain from Proposition 7.2.7 7.2.1 H @ M E M ( ~ ~ as that M is a free B-module. The last part of the statement follows by taking

M = H.

I

Corollary 7.2.10 If H is afinite dimensional Hopf algebra, then the order of G ( H ) divides dim(H). I As an application we give the following result which will be a fundamental tool in classification results for Hopf algebras. If H is a Hopf algebra, , is an ideal of H . A Hopf subalgebra we will denote by HS = K e r ( ~ )which K of H is called normal if K + H = H K + .

Theorem 7.2.11 Let A be a finite dimensional Hopf algebra, B a normal Hopf subalgebra of A, and A/B+A the associated factor Hopf algebra. Then A is isomorphic as an algebra to a certain crossed product B#,A/B+A. Proof: We prove the assertion in a series of steps. Step I. B+A is a Hopf ideal of A.

7.2. T H E NICHOLS-ZOELLER THEOREM

299

It is clearly an ideal of A. Since c = (E@ & ) A ,if b E K e r ( ~ )then , A(b) E Ker(@ ~ c) = B + @ B B @ B + , so B+A is also a coideal. Finally, since S ( B + ) B + , it follows that the antipode stabilizes B+A. Step 11. A becomes a right H-comodule algebra via the canonical projection T : A -+ H = A/(B+A), and B = ACoH. The canonical inclusion i : B -t A is a morphism of left B-modules, and since B B is injective ( B is a finite dimensional Hopf algebra), i splits, i.e. there exists a left B-module map O : A -+ B with O i = IB. Let

+

We show that Q(B+A) = 0. Indeed, if b E B and a E A, then

g

It follows that there exists : H -t A a linear map such that Q n = Q. Thus from C a l Q ( a 2 ) = iO(a) we deduce that C a l Q ~ ( a 2 )=,iO(a). Let . a E AcoH, i.e. C a l @ 7r(a2) = a @ ~ ( 1 )Then

so a = iO(a) E B. Conversely, if b E B , then from C bl @ b2 = b @ 1 + C b i @ (b2 - c(b2)l) we obtain that C bl @ 4 b 2 ) = b @ n ( l ) , i.e. b E AcoH. Step 111. The extension A/B is H-Galois. Recall from Example 6.4.8 1) that the canonical Galois map

is bijective with inverse P-l(a @ b) = C a S ( b l ) @ b2. If M denotes the multiplication of A, then if we denote


300

we have P ( K ) = A 8 B+A. Indeed, if a,.al E A, b E B , and x = ab @I a' a 8 bat E K, then

which is in A @ B+A, since &(b)l- b E B + for all b E B . Conversely, for a , a' E A, b E B + , we have

-

Now p induces an isomorphism from (A@A)/K E A@BA to (AOA)/(A@ B+A) A 8 H , which is exactly the Galois map for the extension AIB. Step IV. We have that A 21 B @ H as left B-modules and right H-comodules. Recall first the multiplication in the smash product A#H*:

-

where (h* x)(y) = h*(xy). It follows that B 8H* is a subring of A#H*. We know from Nichols-Zoeller that BA is free of finite rank. Applying the same t o AOp,we get that B ~ ~ A OisPfree, then using the algebra isomorphism S : Bop 4 B it follows that A ~ free, S and thus AB is free (with the same rank as BA). Denote by 1 the rank of AB. Since the extension A / B is Galois, the map

is an isomorphism of A - B-bimodules, and hence we have that A$) r (B:) B B A)B E (A OB A)B ( A @ H ) B 2 ~ g ) By . Krull-Schmidt we get 1 = n, i.e. AB is free of rank n = dimk(H). By the above, we have that BA is also free and r a n k ( ~ A )= n. Now, A has an element of trace 1. Indeed, if t E H * is a left integral, and h E H is such that t(h)l = n(1) = hlt(h2), then an element a E A with n(a) = h

7.2. THE NICHOLS-ZOELLER THEOREM is an element of trace 1: t .a

Ct(li(u2))a1

= =

t(r(a1))f (.(a2))

=

C t ( T ( ~ ) l )(.(.)2) f C t ( h l ) f (h2) f (Ct(hlIh2)

=

f(n(l))=l.

= =

(f is a left inverse for T)

The extension A/B is also Galois, so the categories and AM^ are equivalent via the induced functor A @B -. If PI, P2,...,P, are the only projective indecomposables in E M it follows that A@BPI,A@BP2,...,A@B P, are the only projective indecomposables in AM^ N A # H * M .Write

Now A#H* is projective in

A

~ SOH

--

in this category, and in particular as left A-modules. But A#H* An as left A-modules, and again from this and from (7.11) we get as above that li = nlci and so A#H* = An (7.12)

as left A#H*-modules, therefore as left B @I H*-modules. We have now that A@H*=A#H*

(7.13)

as left B @ H*-modules, via

4 is bijective

-

Now it is clear that

with inverse

4-' : A#H* A @ H*, 4-'(a#h*) and 4 is left B €3 H*-linear because

=

C a.

@

h*

-

~ - l ( ~ ~ ) ,

302


Since BA is free of rank n, we get from (7.13) that

as left B 8 H*-modules. Combining this with (7.12) we obtain that

as left B 8 H*-modules. Since B 8 H* is finite dimensional, we can write

an indecomposable decomposition. Now A#H* A is projective, A#H* is free over B 8 H*, so A is also projective as a left B 8 H*-module. Thus

as left B 8 H*-modules. By (7.14) we have now

as left B 8 H*-modules, so by Krull-Schmidt we obtain that ki = ti, i = 1 , . . . ,s, i.e. A E B 8 H* as left B 8 H*-modules. But H* E H as left H*-modules by Theorem 5.2.3, so A B 8 H as left B-modules and right H-comodules. The result follows now from Theorem 6.4.12. I

--

7.3 Matrix subcoalgebras of Hopf algebras We say that a k-coalgebra C is a matrix coalgebra if C E Mc(n,k) for some positive integer n. This is equivalent to the fact that C has a basis (cij)lli,jln, with comultiplication A and counit E defined by

(dle2).

Proof: We use for the proof the well known commutation formula for traces Tr(AB) = Tr(BA) for any matrices A, B E Mn(k). (i) (cld) = Tr(t(c)t(d)) = Tr(t(d)t(c)) = (dlc). (ii) It is enough to check for basis elements. Take c = cij and d = c,,. Then c o d = &iiscTj and

The second formula can be proved similarly. (iii) Apply E to (ii) and use the counit property. (iv) We have that (XIYO z )

Tr(t(x)t(y O 2)) = ~r(t(x)t(t-l(t(~)~(z))) = Tr() = 1 in C , which means that the codimension of V in C is 1. Since t # 0 we have that 4 # 0 (otherwise the image o f t in C* through the isomorphism would be zero). Since both 4 and $J are zero on V, we obtain that there exists CY E k such that = cu4. The orthogonality relation shows that $(xc) = X(xcS(xc)) = 1. But

--

s(l)) =

X(l)€(A)

This ends the proof if we use the facts that H is semisimple if and only if € ( A ) # 0 (Theorem 5.2.10) and H is cosemisimple if and only if X(1) # 0 (Exercise 5.5.9). I We define for any h E H and p E H* the linear morphisms l ( h ) : H -+ H and l ( p ) :H 4 H by 1 ( h )( a ) = ha,

1 ( p )( a ) = p

-a

for any a E H . We have that

T r ( l ( h )o

soo 1 ) ) )

=

h ( h S 2 ( p-- A z ) S ( A l ) ) (by (7.22))

=

A(hS(AlS(p

= =

A2)))

C~ ( h s ( ~ l s ( ~ 2 ) ~ ( ~ . 3 ) ) ) C~ ( ~ s ( E ( A ~ ) P ( A ~ ) ) )

= X(h)p(A)

We have obtained

Exercise 7.4.2 Show that l(p)* = R ( p ) for any p E H*. In particular T r ( l ( p ) )= T r ( R ( p ) ) . Let us consider the element x E H such that

for any p E H*. Such an element x exists and is unique. Indeed, if i : H --+ H** is the natural isomorphism, and h** E H** is defined by h**(p) = T r ( l ( p ) )for any p E H * , we just take x = i-'(h*').

Exercise 7.4.3 Show that i f S 2 = Id and H is cosemisimple, then x is a nonzero right integral i n H .


314

Lemma 7.4.4 We have that Tr(l(x) o S 2 ) = A(l)&(A)= T r ( S 2 ) .

Proof: If we write (7.19) for f = I d we obtain

Then for any h E H (A

- h)(x)

=

- h)) (definition of x) V((X- hi) B h2)) (by (7.19)) C ( h - hlI(h2)

=

CNh2~(hl))

=

=

We obtain (A

Tr(R(X

- h)(z)

=

x

X(h2S(h1))

(7.24)

For h = 1, this shows that X(x) = X(1). We use now (7.23) for h = x and p = E and obtain Tr(1(z) o s 2 ) = X(x)&(A) = A(l)&(A) = T~(s~)

Lemma 7.4.5 The following formulas hold: 8x2 =EXZBXI. (i) (zi) x2 = dim(H)x = E(X)X. (iii) S2(x) = x. (iv) Tr(S2) = d i m ( ~ ) ~ r ( S i ~ ) . Proof: (i) Let ?1, : H*@I H* -+ ( H @H ) * be the linear isomorphism defined by $(pBq)(gB h) = p(g)q(h) for any p, q E H*, g, h E H. Then for proving that C x l B x2 = C 2 2 B x l it is enough to show that

for any p, q E H * . This can be seen as follows $(P @ q)(c x i @ 52)

(PB)(x) = Tr(l(pq)) (by the definition of x)

=

= T N P )0 l(9))

7.4. SEMISIMPLE AND COSEMISIMPLE HOPF ALGEBRAS

(ii) For any h E H we have that

(A

-

h)(x2) = (A

-

315

hS-l(x))(x)

=

A(~~S-'(X,)S(~IS-~(X~)))

=

A(~~s-~(xL)z-~S(~~))

= =

x

~(h2S-'(xz)xlS(hl)) (by

.(x)

(2))

C X(h2S(h1))

-

E(x)(A h)(x) (by (7.24)) = (A h) (E(x)x) =

for any h E H . Since H* = {A the other hand

-

-

hlh E H} we obtain that x2 = E(X)X.On

which completes the proof of (ii). (iii) Let p E H* and h E H. We have that

thus l(p o S 2 ) = S-2 o l(p) o S2. Then for any p E H* we have p(S2(x)) = ( P o S2)(x) = Tr(l(p o S2)) (definition of x) = TT(S-' o l ( p ) o s 2 )

316


which shows that S2(x) = x. (iv) Let T = 1(x) o S2. We know from Lemma 7.4.4 that Tr(T) = Tr(S2). We have that T(xh) = ( I (x) o s2)(xh) = x~'(xh) = xs2(x)s2(h) = x2s2(h) = dim(H)xs2(h) = dim(H)s2( x ) s 2(h) = dim(H)s2(xh) which shows that qxH= dim(H)SkH. Since obviously I m ( T ) & x H , we can regard TixH as a linear endomorphism of the space xH, and then

But since Im(T)

x H , we have that Tr(T) = T r ( q X H ) ,so we obtain

Theorem 7.4.6 (Larson-Radford) Let k be a field of characteristic zero and H a finite dimensional Hopf algebra over k, with antipode S . The following assertions are equivalent. (i) H is cosemisimple. (ii) H is semisimple. (zii) S2= Id. Proof: (iii)+(i) and (iii)+(ii) follow directly from Theorem 7.4.1 since T ~ ( S= ~ Tr(Id) ) = dim(H). (i)=+-(ii)We first use Exercises 4.2.17, 5.5.12 and 5.2.13 to reduce to the case where k is algebraically closed. Let C be a matrix subcoalgebra of H . We know that S2(C) = C (Theorem 7.3.7) and that there exists an invertible t in the algebra (C, o) such that S2(c) = t(-') o c o t for any c E C (Proposition 7.3.3). Let r be the order of S2 (which is finite by Theorem 7.1.7). Then c = S2r(c) = t ( - T ) o c o d T ) for any c E C, so dT) is in the

7.4. SEMISIMPLE AND COSEMISIMPLE HOPF ALGEBRAS

317

center of the algebra (C,0). Taking account of the algebra isomorphism [ : C + Mn(k) we have that the center of (C, o) is k x c , thus dT)= a X c for some a E k. Since t is invertible we have that a # 0, and then replacing t by &t (which still verifies S2(c) = t(-l) o c o t), we can assume that t(') = ~ c Then . [(t)' = I, the identity matrix, so the minimal polynomial of [(t) divides XT- 1, and hence it has only simple roots. This implies that the matrix [(t) is diagonalizable with eigenvalues r-th roots of unity. Since k has characteristic zero we may assume that the field of rational numbers Q k, and then, since k is algebraically closed, that the field Q (regarded as a subfield of the complex numbers) is contained in k. Since the inverse of a complex root of unity is the conjugate of that root, we obtain that [(t(-')) is diagonalizable with eigenvalues the (complex) conjugates of the eigenvalues of

Now we prove by induction on 1 5 a 5 pe - 1 that the set Sa = { hxi ( h E G, 0 5 i ( a ) is linearly independent. For a = 1, this follows from the Taft-Wilson Theorem. Assume that Sa-1 is linearly independent, and say that ah,jhxi = 0

+

for some scalars a h , i . Since A(x) = g 8 x x 8 1, and (x 8 l)(g 8 x) X(g 8 x)(x B l ) , we can use the quantum binomial formula, and get

=

Apply A to (7.31), then using (7.32) we see that

Fix some ho E H. Since Sa-1 is linearly independent, there exist $,$I E H* such that d(hoga-lx) = 1, 4(hxi) = 0 for any (h,i) # (hoga-', 1), $(hoxa-') = 1, and $(hxi) = 0 for any (h, i) # (ho, a - 1). Applying 4 B 11, to (7.33), we find that ( y ) A ~ h o , a= 0. AS a 5 pe - 1, we have # 0, thus a h o , a = 0. Now the induction hypothesis shows that all a h , i are zero, therefore Sa is linearly independent. But ISpe-l1 = pn-l+e shows that e must be 1, and Sp-l is a basis of H.

+

We prove now that H1 = kG+ ChEF P h , h g and P h , h g = k(hg - h) khx x 'I . Then as in (7.33) we have for any h E G. Let z = C ~ E Gc ~ ~ , ~Eh H Osisa

7.8. POINTED HOPF ALGEBRAS OF DIMENSION PN

341

Note that (hgZPS,s, h, i-s) = (h'gi'-S', s', h', it-s') if and only if s = s', i = i' and h = h'. Then for i 2 2, take s = 1, and hgi-sxs @ hxi-" has the coefficient ah,i in A(z). On the other hand, since A(z) E Ho@ H H@Ho, this coefficient must be zero, thus a h , i = 0. Therefore z E kG ChEC khx, which is what we want.

+

+

In particular, the only P,,, not contained in kG are P h , h g , h E G. On the other hand, xh E Ph,gh,thus P h , g h is not contained in kG. Thus we must have gh = hg for any h E G, i.e. g E Z(G). Also, xh E k(hg-h)+khx. Thus there exist a ( h ) E k*, P(h) E k such that xh = a(h)hx+P(h)(gh - h). We have that xgh

=

Xgxh

= Xg(a(h)hx =

Xa(h)ghx

+ P(h)(gh - h))

+ XP(h)g(gh - h)

and

showing that P(h) = 0. Now xh = cr(h)hx for any h E H, thus a is a linear character of G. Finally, taking the ptl' powers in A(x) = g@x+x@l, we obtain A(xP) = gP @ xp xp 8 1. Since gp # g, this implies that xp E PI,^^ = k(gp - 1). If XP = 0, then H E H1(G, g, a). If xp = P(gp - 1) for some nonzero scalar P, then by the change of variables y = ,B1'px (k is algebraically closed) we I see that H 2 H1(G, g, a ) .

+

Corollary 7.8.3 Let k be an algebraically closed field of characteristic zero and p a prime number. Then a pointed Hopf algebra of dimension p2 over I k is isomorphic either to a group algebra or to a Taft algebra.

Let p be a prime. Then a group of order p or p2 is abelian, therefore in order to find examples of non-cosemisimple pointed Hopf algebras of dimension pn with non-abelian coradical, we need n 2 4. We first investigate dimension p4, and the possibility of the coradical to be the group algebra of a non-abelian group of order p3. Proposition 7.8.4 Let H be a pointed Hopf algebra of dimension p4, p prime. Then either H is a group algebra or G ( H ) is abelian. Proof: It is enough to show that there do not exist Hopf algebras of dimension p4 with coradical the group algebra of a non-abelian group of

342


order p3. If we assume that such a Hopf algebra H exists, then the structure result given in Theorem 7.8.2 shows that there exist g E Z(G(H)) and a linear character a of G ( H ) such that a ( g ) is a primitive p t h root of unity. There exist two types of non-abelian groups with p3 elements. The first one is G1, with generators a and b, subject to

In this case Z(G1) =< ap >, and if cr E G;, then the relation bab-l = al+p shows that a(ap) = 1, thus a(g) = 1 for any g E Z(G1). The second type is G2, which is generated by a, b, c, subject to ap = bp = cp = 1, ac = ca, bc = cb, ab = bac Then Z(Gz) =< c >, and again the relation ab = bac shows that a(c) = 1 for any a E G;. Thus a(g) = 1 for any g E Z(Gz), which ends the proof. It is easy to see that there exist pointed Hopf algebras of dimension p5 with non-commutative coradical of dimension p4. We can take for example H = lcGl 8 Hp2, where Hp2 is a Taft Hopf algebra, and G1 is the first type of non-abelian group of order p3. Then H is pointed with G ( H ) = G I x Cp. We can also give examples of such Hopf algebras that are not obtained by tensor products as above. Example 7.8.5 i) Let M be the group of order p4 generated by a and b, subject to relations

Then Z ( M ) =< ap >. Let X be a primitive root of unity of order p2. Then a ( a ) = X and a(b) = 1 define a linear character of M , and a(ap) is a primitive root of unity of order p. We thus have a Hopf algebra H ( M , ap, a) with the required conditions. ii) Let E be the group of order p4 generated by a and b, subject to relations

Then Z ( E ) =< ap,bp >. Take a E E* such that a ( a ) = 1 and a(b) is a primitive root of unity of order p2. Then H ( E , bp, a ) is another example as we want. Now we show that if C = (Cp)n-l =< cl > x then a result similar to Corollary 7.8.3 holds.

-

< c2 > x . . . x < en-1 >

Proposition 7.8.6 If C = (Cp)"-l and H is a pointed Hopf algebra of dimension pn with G ( H ) = C , then H k(Cp)n-2 8 T for some Tajl Hopf algebra T. Moreover, there are exactly p - 1 isomorphism classes of such Hopf algebras.

7.9. POINTED HOPF ALGEBRAS OF DIMENSION P3

343

Proof: We know from Theorem 7.8.2 that H e H1 (C, g, a ) for some g E C and a E C * such that a(g) # 1. Regard C as a Zp-vector space. Then there exists a basis gl = g, g2, . . . ,gn-1 of C. Since a(g) # 1 we can find a ) 1. Then basis cl = g, c2, . . . , cn- 1 of C such that a(c2) = . . . = a ( ~ , - ~ = xc, = c,x for any 2 i < n - 1, the Hopf subalgebra T generated by g and x is a Taft algebra and we clearly have H 2 k(Cp)n-2 8 T. The second I part follows from Proposition 5.6.38.

2p.

Proof: Suppose dim(HI) 5 2p. Since Tl C H I , we have H1 = T I . In particular dim(P,,,) = 2 only for v = cu. Step 1. We prove by induction on n 5 p - 1 that Hn = Tn. Assume that Hn- 1 = Tn- 1 and Hn # Tn, and pick some h E Hn - T,, . Write h = h,.,, as in the Taft-Wilson Theorem and pick some h,,, E Hn - TT,. u,vEG(H)

Denoting g = u-lv we have that a = u-'h,,, E Hn - T,, and

with vi,j E Tn-1. Let b = a+vo,o E Hn -Tn. 7.9.2 shows that A(vo,,-1) = 9 €3 Vo,n-1 vo,n-I 8 cn-'. If g # cn we have V O , ~ -E~ Ho, and then A ( b ) E Ho €3 H H €3 Hn-2, which is a contradiction since b @ Hn-l. Hence g = cn and V C I , ~ = - ~a(cn - cn-l) + P c ~ - ~for x some a ,P E k , P # 0. We have that

+

+

-pCn-lX

€3 xn-l E H € 3 H n - 2 + H o @ H .

(7.37)

Since ( A ( x n - I ) - cn €3 xn-I - x 7 ~ - 1 €3 1) + (cn €3 xn-l - Cn-1 8 , p - 1 ) E H €3 Hn-2 Ho €3 H and ( A ( x n )- cn €3 xn - xn 8 1) - (;),cn-lx €3 X n - 1 E H 8 HnV2, relation 7.37 implies that b' = b oxn-' - n -1 Pxn satisfies A(bl)- cn 8 b' - b' €3 1 E H €3 Hn-2 HO€3 H . Therefore b' E HnP1 = Tn-l and b E T n Hn = Tn, providing a contradiction. Step 2. We have from Step 1 that Hp-1 = Tp-1 = T # H,. Using the Taft-Wilson Theorem and 7.9.2 as in Step 1, we find some b E H p - T with

+

+

+

P- 1

A(b) = 18 b

+ b €3 1 + X u , 8 xJ for some v, E T

(note that we need here ,=I CP = 1). We use induction to show that for any 1 5 m 5 p there exists b, E Hp - T such that

for some w, E T, oj E k. For m = 1, we see again as in Step 1 that some o,,B E k , ,B # 0. Observe that

up-1

= ~ ( 1 cp-I) -

+ pcp-lx

for

346


+

Applying (7.9.2) to a = b axp-' (in the case n = p - l),we obtain a bl as wanted. Assume that we have found b, for some 1 m 5 p - 1 satisfying (7.38). Applying relation (7.36) to bm and r = m we obtain

HOPF Algebras: An Introduction

Hopf algebras: an introduction

Hopf algebras: an introduction