Quantum Groups: A Path to Current Algebra (Australian Mathematical Society Lecture Series)

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Quantum Groups

Algebra has moved well beyond the topics discussed in standard undergraduate texts on \modern algebra." Those books typically dealt with algebraic structures such as groups, rings and fields: still very important concepts! However, Quantum Groups: A Path to Current Algebra is written for the reader at ease with at least one such structure and keen to learn the latest algebraic concepts and techniques. A key to understanding these new developments is categorical duality. A quantum group is a vector space with structure. Part of the structure is standard: a multiplication making it an \algebra." Another part is not in those standard books at all: a comultiplication, which is dual to multiplication in the precise sense of category theory, making it a \coalgebra." While coalgebras, bialgebras and Hopf algebras have been around for half a century, the term \quantum group," along with revolutionary new examples, was launched by Drinfel'd in 1986.

AUSTRALIAN MATHEMATICAL SOCIETY LECTURE SERIES Editor-in-chief: Professor Michael Murray, University of Adelaide, SA 5005, Australia Editors: Professor P. Broadbridge AMSI, The University of Melbourne, Victoria 3010, Australia Professor C. C. Heyde, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia Professor C. E. M. Pearce, Department of Applied Mathematics, University of Adelaide, SA 5005, Australia Professor C. Praeger, Department of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Introduction to Linear and Convex Programming, N . C A M E R O N Manifolds and Mechanics, A . J O N E S , A . G R A Y & R . H U T T O N Introduction to the Analysis of Metric Spaces, J . R . G I L E S An Introduction to Mathematical Physiology and Biology, J . M A Z U M D A R 2-Knots and their Groups, J . H I L L M A N The Mathematics of Projectiles in Sport, N . D E M E S T R E The Peterson Graph, D . A . H O L T O N & J . S H E E H A N Low Rank Representations and Graphs for Sporadic Groups, C. PRAEGER & L. SOICHER Algebraic Groups and Lie Groups, G . L E H R E R ( e d . ) Modelling with Differential and Difference Equations, G. FULFORD, P. FORRESTER & A. JONES Geometric Analysis and Lie Theory in Mathematics and Physics, A. L. CAREY & M. K. MURRAY (eds) Foundations of Convex Geometry, W . A . C O P P E L Introduction to the Analysis of Normed Linear Spaces, J . R . G I L E S The Integral: An Easy Approach after Kurzweil and Henstock, L. P. YEE & R. VYBORNY Geometric Approaches to Differential Equations, P. J. VASSILIOU & I. G. LISLE Industrial Mathematics, G . F U L F O R D & P . B R O A D B R I D G E A Course in Modern Analysis and its Applications, G . L . C O H E N Chaos: A Mathematical Introduction, J . B A N K S , V . D R A G A N & A . J O N E S

Quantum Groups A Path to Current Algebra ROSS STREET

Technical Editor: ROSS MOORE

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521695244 © R. Street 2007 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2007 ISBN-13 ISBN-10

978-0-511-26900-4 eBook (EBL) 0-511-26900-5 eBook (EBL)

ISBN-13 ISBN-10

978-0-521-69524-4 paperback 0-521-69524-4 paperback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To Oscar and Jack

Contents

Introduction 1 Revision of basic structures

page ix 1

2 3

Duality between geometry and algebra The quantum general linear group

5 9

4 5

Modules and tensor products Cauchy modules

13 21

6 7 8

Algebras Coalgebras and bialgebras Dual coalgebras of algebras

27 37 47

9 10

Hopf algebras Representations of quantum groups

51 59

11 12

Tensor categories Internal homs and duals

67 77

13 14

Tensor functors and Yang–Baxter operators A tortile Yang–Baxter operator for each

85

15 16

finite-dimensional vector space Monoids in tensor categories Tannaka duality

93 97 109

17 18

Adjoining an antipode to a bialgebra The quantum general linear group again

117 119

19 Solutions to Exercises References

121 133

Index

135

vii

viii

Bradshaw: “Ceremonial Figure”, Tassel Bradshaw Group, [Wal94, Plate 20].

Introduction Algebra has moved well beyond the topics discussed in standard undergraduate texts on “modern algebra”. Those books typically dealt with algebraic structures such as groups, rings and fields: still very important concepts! However, Quantum Groups: A Path to Current Algebra is written for the reader at ease with at least one such structure and keen to learn the latest algebraic concepts and techniques. A key to understanding these new developments is categorical duality. A quantum group is a vector space with structure. Part of the structure is standard: a multiplication making it an “algebra”. Another part is not in those standard books at all: a comultiplication, which is dual to multiplication in the precise sense of category theory, making it a “coalgebra”. While coalgebras, bialgebras and Hopf algebras have been around for half a century, the term “quantum group”, along with revolutionary new examples, was unleashed on the mathematical community by Drinfel
d [Dri87] at the International Congress in 1986. Before launching into an explanation of the duality required, I should mention here that an ordinary group gives rise to a quantum group by taking the vector space with the group as basis. When pushed to provide formal proofs of our claims, mathematicians generally resort to set theory. We build our structures on sets and feel satisfied when we can be explicit about the elements of our constructed objects. Up to the mid twentieth century, algebraic objects were sets with selected operations which assigned elements to lists of elements. Typically, we would have binary operations which might be called addition, multiplication or Lie bracket respectively assigning a sum, product or formal commutator to each pair of elements. In those days, the importance was recognized of dealing with the homomorphisms between algebraic structures: these were the functions which preserved the operations involved in the kind of structure at hand. The existence of a bijective homomorphism (isomorphism) between two algebraic objects meant that the two objects played the same role. So how could the literal elements be the defining ingredient? The important issue was the way the algebraic object related to others of its own kind by means of homomorphisms into it or out of it. Quite often the elements could be recaptured as homomorphisms from a particular object into the one of interest. For example, the elements of a vector space were in bijection with the linear functions from a selected one-dimensional space. ix

x

Introduction

Homomorphisms into an object might therefore be called “generalized elements” of the object. However, this notion of element of the object will depend on the kind of structure we are studying since that will determine what a homomorphism is (a group homomorphism, a linear function, a ring homomorphism, or whatever). We quite often wish to add more elements to our sets to improve the properties of the operations: as when we construct the integers from the natural numbers to obtain subtraction; or when we construct the rational numbers from the integers to obtain division; or when we construct the real polynomials from the real numbers to obtain an indeterminate. These constructions can be described explicitly as sets with operations that include the original ones. More importantly, each such construction is unique up to / C out isomorphism with a universal property: each homomorphism X of the original structure X, into a set C with the extra structure, extends / C out of the constructed object X ˆ ˆ. to a homomorphism X In this way it was realized that knowing the homomorphisms out of objects determined the objects just as surely as knowing the homomorphisms into them did. It is natural then to call homomorphisms out of an object “generalized co-elements”. Once this kind of duality principle is acknowledged, interesting facts appear. Let us take a simple example purely using sets. Consider two sets X and Y . Their cartesian product X × Y is the set whose elements are pairs (x, y) where x lies in X and y lies in Y . We are not studying any structure on these sets except for the property of being a set. So homomorphisms in / X ×Y this case are merely functions. It is clear that functions f : T into X × Y from a test object T are in bijection with pairs of functions / X and f2 : T / Y . In other words, T -elements (f1 , f2 ) where f1 : T of X × Y are in bijection with pairs consisting of a T -element of X and a T -element of Y . All that is fairly straightforward. Now suppose that our sets X and Y have no common elements; if they are not disjoint, replace them by isomorphic sets which are. Write X + Y for the union; we write X + Y rather than X ∪ Y to emphasize that it is the disjoint union (if X and Y were finite, the number of elements of X + Y would be the sum of the number of elements in X and the number / T is determined by its restriction to X in Y ). A function f : X + Y and its restriction to Y . In other words, the co-T -elements of X + Y are in bijection with pairs consisting of a co-T -element of X and a co-T -element of Y . We conclude that the constructions X × Y and X + Y are duals of one another. This is not something that was stressed when we were taught the more abstract multiplication and addition of numbers in infants’ school. If we now look at vector spaces or groups X and Y , the cartesian product X × Y as sets becomes a vector space or group by means of coordinatewise operations from X and Y ; again this has pairs as the generalized

Introduction

xi

/ X × Y . However, to obtain the dual constructions in these elements T cases is quite different from the disjoint union of sets: in the case of vector spaces, we have that X × Y is self-dual (called direct sum and denoted by X ⊕ Y ); in the case of groups, the dual notion is rather complicated (called the free product by group theorists). In order to formalize the way in which constructions such as these can be dual, we can use the notion of category. I intend to give a definition of this concept in this introduction. Before doing so, I would like to draw an analogy. It was noticed in projective plane geometry that theorems occurred in pairs: one such pair consists of Pascal’s Mystic Hexagram Theorem and Brianchion’s Theorem; both are about conics. Given one theorem in a pair, the other is obtained by interchanging the role of points and lines, reversing the incidence relation (“lies on” becomes “goes through”). To formally explain this duality, we abstract the notion of projective plane. Here is the essence of the definition. A projective plane P consists of two sorts of elements: one sort called points, the other called lines. It also consists of a relation between these elements, called incidence (this is a rule telling when a point is incident with a line). There are some axioms which include: 1.

for distinct points P and Q , there is a unique line L such that P and Q are both incident with L ; and,

2.

for distinct lines L and M , there is a unique point P such that P is incident with both L and M . Any system satisfying this is a projective plane! The points do not need to look like points and the lines do not need to look like lines in any sense. Of course, we still draw pictures to help our intuition. Now we are ready to formalize duality. Given a projective plane P , we obtain a projective plane P rev whose points are the lines of P , whose lines are the points of P , and whose incidence relation is the reverse of that of P . Notice that axioms (1) and (2) for P rev are respectively axioms (2) and (1) for P . This means that, if we prove a theorem about all projective planes, then the dual theorem is automatically true by applying the original theorem to P rev . It turns out that there are not too many interesting theorems assuming only axioms (1) and (2). A further axiom based on a theorem of Pappus can be added and the system remains self-dual. In fact, conics can be defined using an idea of Steiner, and Pascal’s Theorem can be proved. Let us now discontinue discussion of this analogy and return to the formalization of the duality at hand. A category A consists of two sorts of elements: one sort called objects, the other called morphisms (or arrows). It also consists of three functions. The first function assigns to each morphism f a pair (A, B) of objects in

xii

Introduction

which case A is called the domain (or source) of f while B is called the / B and A f / B are used. codomain (or target) of f ; the notations f : A /A The second function assigns to each object A a morphism 1A : A called the identity morphism of A . A pair (f, g) of morphisms is called composable when the codomain of f is equal to the domain of g . The third function assigns to each composable pair (f, g) of morphisms, a morphism g ◦ f , called the composite of f and g , whose domain is that of f and whose codomain is that of g . There are two axioms: 1.

if (f, g) and (g, h) are composable pairs of morphisms then (h ◦ g) ◦ f = h ◦ (g ◦ f ) ; and,

2.

if f : A

/ B is a morphism then f ◦ 1 = f = 1 ◦ f . A B

The standard argument shows that identity morphisms are unique. The notation A(A, B) (or HomA (A, B)) is used for the set of all morphisms in A from A to B . There is a category Set whose objects are sets, morphisms are functions, and composition is the usual composition of functions. There is a category Vectk whose objects are vector spaces over a fixed field k and morphisms are linear functions; composition is as usual. Similarly we have a category whose objects are groups and a category whose objects are rings. However, there are categories whose objects do not look like sets and whose morphisms do not look like functions. For example, there is a category whose objects are integers, whose morphisms are pairs (m, n) of integers such that the domain of (m, n) is m and the codomain is the product mn ; a pair ((m, n), (r, s)) of morphisms is composable when mn = r and the pair’s composite is (m, ns) . Now to duality. Given a category A , there is a category Aop whose objects are the objects of A , and morphisms are the morphisms of A ; however, the domain of a morphism is its codomain in A while its codomain in Aop is its domain in A . A pair (g, f ) of morphisms is composable in Aop if and only if (f, g) is composable in A ; its composite f ◦ g in Aop is the composite g ◦ f in A . We call Aop the dual or opposite of the category A . Perhaps it helps to say that Aop is the category obtained from A by / B in A is precisely a morphism reversing arrows: a morphism f : A op / A in A . Admittedly, if the objects of A look like sets (that is, f : B are sets with some structure), the same is true of Aop ; but the same cannot be said for morphisms that are functions, since formally reversed functions can scarcely be thought of as functions. The duality between cartesian product and disjoint union can now be made precise. In a category A , a product for two objects A and B / A , p : A× consists of an object A×B and two morphisms p1 : A×B 2 / B (called projections) with the following “universal” property: for all B

Introduction

xiii

/ A, b : T / B , there exists a unique objects T and morphisms a : T / A × B, denoted by (a, b) , such that p ◦ (a, b) = a and morphism T 1 p2 ◦ (a, b) = b . This means that T -elements of A × B are in bijection with pairs consisting of a T -element of A and a T -element of B . / D in a category A is called a right inverse for a A morphism h : C / morphism k : D C when k ◦ h = 1C ; we also say that k is a left inverse for h . A morphism h is invertible (or an isomorphism) when it has both a left and right inverse; in this case, a familiar argument shows that the left and right inverse agree and this common morphism is unique, being called the inverse of h and denoted by h−1 . If there exists an invertible morphism / D then we say C and D are isomorphic and write C ∼ C = D . In a category, we think of isomorphic objects as being essentially the same. Any two products of two objects A and B can be proved, by an easy argument, to be isomorphic. Now we have our duality between cartesian product and disjoint union of sets: cartesian product is the product in the category Set while disjoint union is the product in the category Setop. We can give an even simpler example. An object K of a category A is called terminal when, for all objects A of A , there is precisely one morphism / K . The singleton set 1 is terminal in the category Set while the A empty set ∅ is terminal in Setop. Any concept defined for all categories A has a dual concept which is the same concept translated to Aop : the prefix “co-” is used. So a product in Aop is called a coproduct in A . A terminal object in Aop , under this system, would be called a coterminal object in A ; but it is also called an initial object of A . In the spirit of category theory itself, we should consider appropriate /X morphisms of categories. These are called functors. A functor F : A between categories A and X consists of two functions. The first assigns to each object A of A an object F A of X . The second function assigns to each / F B of X . There / B of A a morphism F f : F A morphism f : A are two axioms: 1.

F 1A = 1F A for all objects A of A ; and,

2.

F (g ◦ f ) = F g ◦ F f for all composable pairs (f, g) in A . It is easy to see that functors preserve invertibility of morphisms: in fact they take inverses to inverses. Let us look at a couple of examples of functors. • Each object T of a category A determines a functor RT = A(T, ) : / Set called the functor represented by T ; the elements of A / A in A (that is, the RT A = A(T, A) are the morphisms a : T / R B takes T -elements of A), while the function RT f : RT A T such an a to f ◦ a .

xiv

Introduction • Suppose K is an object of A for which a product K ×A exists (and is /A chosen) for all objects A . There is a functor F = K × : A defined on objects by F A = K ×A and on morphisms by F f = K ×f / K ×B. where K × f = (p1 , f ◦ p2 ) : K × A

Categories were invented not only to formalize duality but to formalize the concept of “naturality” in mathematics. The idea was that a natural transformation should be one that involves no ad hoc choices. For example, if V is a vector space and V ∗ is the vector space of linear functions from V / V ∗∗ which into the base field k , there is a natural linear function V ∗ / k defined by evaluation takes v ∈ V to the linear function ev : V / V that depends on a choice of at v . However, any linear function V ∗∗ basis for V should not be natural. / X and G : A / X are functors between the same Suppose F : A / G is a function. The funccategories. A natural transformation θ : F / GA of X . tion assigns to each object A of A a morphism θA : F A / There is a single axiom: for each morphism f : A B, Gf ◦ θA = θB ◦ F f . There is an obvious componentwise composition of natural transformations.
 This defines a category A , X , called a functor category, where the objects / X and the morphisms are natural transformations. are functors F : A A natural isomorphism is an invertible morphism in the functor category. / Set is called representable when it is isomorphic to A functor F : A RT for some object T ; such a T is called a representing object for F . For / Set , which takes each vector space to example, the functor U : Vectk its underlying set and each linear function to that function, is representable: we have U ∼ = Rk since the linear functions from the field k to a vector space V are in natural bijection with elements of V . Many constructions in mathematics are designed to provide representing objects for interesting functors. Let us look at a couple of examples of natural transformations: / Set is a functor and T is an object of A . Each • Suppose F : A /F element x of F T determines a natural transformation x ˆ : RT defined by x ˆA (a) = (F a)(x) . The Yoneda Lemma states that this
 defines a bijection F T ∼ = A , Set (RT , F ) . The inverse bijection / F to is even easier: it takes the natural transformation θ : RT the element θT (1T ) of F T . / L is a morphism of a category A in which prod• Suppose h : K ucts of pairs of objects exist. Then we obtain a natural transforma/L× whose value at the object A is the tion f × : K × / L×A . morphism f × A = (f ◦ p1 , p2 ) : K × A

Introduction

xv

Modern algebra in the sense of the first half of the twentieth century dealt with sets equipped with operations. Soon after, the idea of co-operations crept into mathematics. The notion of coalgebra is dual to algebra. This is the main concept in this book. Now I turn to the book’s contents. Chapter 1 gives precise definitions of monoids and groups; the axioms are expressed in terms of diagrams ready to be imported to a general category. This importation is carried out in Chapter 2 where we provide the important example of 2 × 2 matrices in readiness for the quantum version. A duality between geometry and algebra is explained. In Chapter 3, we describe the quantum general linear group of 2 × 2 matrices as a coalgebra. This comes from lectures by Manin in Montr´eal. Chapter 4 is about modules over rings; we find it natural to take a 2-sided point of view so that our basic module M has a left action by a ring R and a right action by a ring S which compatibly interact. Chapter 5 concerns finitely generated, projective modules under the mysterious name of “Cauchy modules”. It turns out that F. W. Lawvere noticed a concept in enriched category theory which has Cauchy sequences as an example; when interpreted for additive categories, it leads to modules that are finitely generated and projective. Chapter 6 discusses algebras, Lie algebras and the Poincar´e–Birkhoff–Witt Theorem. Chapter 7 is about coalgebras and bialgebras. A coalgebra is a vector space with a comultiplication. A bialgebra is an algebra which is also a coalgebra subject to a compatibility condition. The dual vector space of a coalgebra is an algebra, however, the usual dual of an algebra need not naturally be a coalgebra. In Chapter 8 we describe Sweedler’s modification (see [Swe69] and [Abe80]) of the dual of an algebra which is a coalgebra. In Chapter 9, we look at Hopf algebras. These should be thought of as generalized groups. An important part of group theory is the theory of their representations: these are modules over the group ring. In Chapter 10 we look at modules over bialgebras. Then, in Chapter 11, we move to use categories more seriously. We discuss categories equipped with an abstract tensor product: monoidal or tensor categories. We discuss examples involving braids. A deep example, not treated here, is the subject of the paper [JS95]. An important property of the
tensor product U ⊗V of vector spaces is that it represents the vector space V , W of linear functions from V to W : 
 Vectk (U ⊗V , W ) ∼ = Vectk U, V , W . In Chapter 12, this idea is lifted to arbitrary tensor categories. Examples from knot theory are provided. The Yang–Baxter equation, from the branch of physics called statistical mechanics, had a major influence on the new examples of Hopf algebras called quantum groups. In Chapter 13, an algebraic concept of Yang– Baxter operator, which makes sense in any tensor category, is explained. A family of examples from linear algebra is provided in Chapter 14.

xvi

Introduction

In Chapter 15, the notion of monoid is lifted to the level of generality at which “algebras” and “coalgebras” become precise categorical duals. For the first time, I believe, in a text at this level, emphasis is placed on “2-cells” between monoid morphisms, providing the student with a gentle introduction to higher-dimensional category theory. Each bialgebra has a tensor category of representations. This correspondence is a modern formulation of Tannaka–Krein duality. The treatment of this topic in Chapter 16 makes use of the 2-dimensional structure of monoids from Chapter 15. There is by now a vast literature on Tannaka duality. We satisfy ourselves with a sketch in Chapter 17 of an application to construct universally a Hopf algebra from a bialgebra. Finally, in Chapter 18, the example of Chapter 3 is revisited in the light of what has been learned. There are exercises at the end of several chapters. Solutions to most of these are provided in Chapter 19. Acknowledgements ... Many ideas presented here are my version of joint research with Andr´e Joyal. I consider myself very fortunate to have experienced such exciting collaboration. I would like to thank the students and staff who attended the original course in the first half of 1990 at Macquarie University. I am very grateful to Paddy McCrudden (as a Vacation Scholar under my supervision in January–February 1995) for his careful reading of my handwritten notes, and particularly for his systematic checking of the exercises. It is a pleasure to acknowledge the support of grants from the Australian Research Council during the preparation of this work. ... the typing process Many of the chapters were carefully typed by Elaine Vaughan. Technology moved ahead incredibly from that word-processor available in our Department in 1990. Typesetting with TEX and LATEX was begun by Ross Moore and continued by several post-doc and graduate students; namely Sam Williams, Ross Talent (now deceased) and Simon Byrne. Each added further exercises and solutions, as these were encountered in lecture courses. Most important in this was the use of XY-pic1 to produce the commutative diagrams that appear throughout this book, and which play such an integral part in the visualization and understanding of much of the material. None of these diagrams has been imported as an image built using other software. All, including the braids, tangles, and 2-cell diagrams, are 1 Originally written by Kristoffer Rose; extended and enhanced by Rose and Moore for mathematical applications and higher quality output. The XY-pic package and documentation is now included with all TEX and LATEX distributions.

Introduction

xvii

specified within the LATEX source using the XY-pic package syntax. Indeed the syntax and coding to handle curves and 2-cells was written in 1993–94 by Ross Moore, specifically for use with this book. Since then the XY-pic package has become a useful tool for presenting diagrammatic material in Category Theory and other branches of mathematics, computer science and linguistics. As an application of Ross Moore’s work with the LATEX2HTML translation software, an earlier version of the present manuscript was made available via the “world-wide web”, now known as the internet. In that form it was used as a source of lecture notes for courses at Macquarie. A great deal of credit is also due to Simon Byrne (as a Vacation Scholar in January–February 2005) for finishing off the typing of exercises and for assembling the manuscript into a form ready to submit as a proposal for the Australian Mathematical Society Lecture Series. With this go-ahead, the final version of the manuscript, complete with up-to-date Bibliography, Index, front-matter and filler images was prepared by Ross Moore, who is acknowledged here as being the Technical Editor for this monograph. ... the illustrations The illustrations appearing at the end of some chapters are reproduced from Grahame Walsh, Bradshaws [Wal94]. I am very grateful to the Bradshaw Foundation and Edition Limit´ee for consenting to their inclusion. The original coloured rock paintings, which the silhouettes trace, are the work of Australian people living as much as 50 millennia before our time.

xviii

Introduction

These paintings have been mentioned already in the mathematico-scientific literature, in connection with knots and braids; viz. How old are knots? It has been suggested that the Stone Age should be called the Age of String. The extraordinary tasselled figures photographed and described by G. L. Walsh in Bradshaws: Ancient Rock Paintings of Western Australia (Edition Limit´ee, 1994) have been suggested to be 50,000 years old. Knots have been intimately linked with the development of humans, through weapons, fishing, hunting, clothing, housing, boating and a myriad of other ways. The metaphor of knots is found throughout literature, and knots and interlacing are featured in many forms of art. Ronald Brown, review of: “The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots” by Colin C. Adams (W. H. Freeman 1994), appeared in Nature, Vol. 371 (13 October 1994).

Suggested Further Reading [JS91b] Andr´e Joyal and Ross Street. An introduction to Tannaka duality and quantum groups. In Category Theory (Como, 1990), volume 1488 of Lecture Notes in Mathematics, pages 413–492. Springer, MR1173027 Berlin, 1991. [Kas95] Christian Kassel. Quantum Groups, volume 155 of Graduate Texts in Mathematics. Springer, New York, 1995. MR1321145 [Maj95] Shahn Majid. Foundations of Quantum Group Theory. Cambridge University Press, Cambridge, 1995, (paperback, 2000). MR1381692 [SS93]

Steven Shnider and Shlomo Sternberg. Quantum Groups: From CoAlgebas to Drinfel
d Algebras. Graduate Texts in Mathematical Physics, II. International Press, Cambridge, MA, 1993. MR1287162

[Yet01]

David N. Yetter. Functorial Knot Theory, volume 26 of Series on Knots and Everything. World Scientific Publishing Co. Inc., River MR1834675 Edge, NJ, 2001.

1 Revision of basic structures

The cartesian product of n sets X1 , . . . , Xn is the set X1 × · · · × Xn = {(x1 , . . . , xn ) | xi ∈ Xi } . There is a canonical bijection (X1 × · · · × Xm ) × (Xm+1 × · · · × Xn ) ∼ = X1 × · · · × Xn given by deleting the inside brackets. The diagonal function δ:X

/ X × ···× X

is given by δ(x) = (x, . . . , x) . The cartesian product of no sets is the special set 1, with precisely one element, which should technically be denoted by empty parentheses ( ) . Particular cases of the canonical bijections are X ×1 ∼ = X ∼ = 1×X . / 1 will be denoted by ε rather than δ ; it is the only The diagonal X / Y1 , . . . , f : Xn / Yn induce / 1 . Functions f1 : X1 function X n a function / Y1 × · · · × Yn f1 × · · · × fn : X1 × · · · × Xn     given by f1 × · · · × fn (x1 , . . . , xn ) = f1 (x1 ), . . . , fn (xn ) . / X on a set X is given by 1 (x) = x . The identity function 1X : X X / 1 is uniquely determined. Similarly the diagonal We noted that ε : X / X × X is unique, determined by commutativity of the diagram δ: X X ∼ =

(Identity)

∼ = δ

1×X

ε×1X

X ×X 1

1X ×ε

X ×1.

2

Quantum Groups: A Path to Current Algebra

Furthermore, the following diagram commutes (Associativity)

δ

X

X ×X

δ×1X

X ×X ×X .

1X ×δ

/ X × X × X so determined is none other than the

The function X ternary diagonal.

A monoid is a set M together with special purpose functions η : 1 / M such that the following diagrams commute. µ : M ×M

/M,

M ∼ =

(Id)

∼ = µ

1×M (Assoc)

µ

M

M ×M

η×1M

M ×M

1M ×η

µ×1M 1M ×µ

M ×1

M ×M ×M

If we write 1 for the value of η at the only element of 1 and we write x y for µ(x , y) then the above diagrams translate to the equations 1x = x = x1 (x y) z = x (y z)

for all x , y , z ∈ M .

This time functions η and µ are not uniquely determined by the set M . However given µ , condition (Id) uniquely determines η while the condition /M (Assoc) gives an unambiguous ternary operation µ : M × M × M which we write as µ(x , y , z) = x y z . Generally there is an unambigu/ M determined by the ous multiple product function µ : M × · · · × M binary µ . An element x ∈ M is called invertible when there exist y , z ∈ M such that y x = 1 and x z = 1 . Notice that y = y 1 = y(x z) = (y x)z = 1 z = z so each invertible element x determines uniquely an element, denoted x−1 , satisfying x−1 x = 1 = x x−1 . A group is a monoid in which each element is invertible. Then we have a / M such that this next diagram commutes. function ι : M M

δ

M ×M

(Invertibility)

ι×1M 1M ×ι

ε

M ×M η

I

µ

M

Revision of basic structures

3

Note carefully the dependence of this axiom on the diagonal structure of cartesian product. For a set X, the n-fold cartesian product X × · · · × X is denoted by X n . Each permutation ξ on {1, . . . , n} induces a bijection / Xn

σξ : X n

given by σξ (x1 , . . . , xn ) = (xξ(1) , . . . , xξ(n) ) . In particular, we have the switch coming from the non-identity permutation of {1 , 2} : / X ×X

σ : X ×X

,

σ(x , y) = (y , x) .

Each σξ is a composite of bijections of the form 1X × · · · × σ × · · · × 1X . Notice that the following diagram commutes. X (Commutativity)

δ

δ σ

X ×X

X ×X

  A monoid M , η , µ is called commutative when the following diagram commutes. M µ

µ σ

M ×M It follows that the composite M n permutation ξ .

σξ

M ×M Mn

µ

M is independent of the

Suppose M and N are monoids. A monoid morphism (or homomorphism / N such that the following diagrams of monoids) is a function f : M commute. 1 η

M ×M

η

f ×f

µ

M

f

N

M

N ×N µ

f

N

Expressed in terms of elements, these diagrams merely say that f (1) = 1 and f (x y) = f (x)f (y) . If N has left-cancellation (i.e., ab = ac implies b = c ; e.g., if N is a group) then f (1) = 1 is redundant.

4

Quantum Groups: A Path to Current Algebra

Monoid morphisms preserve invertibility : if x ∈ M is invertible, f (x−1 ) = f (x)−1 . So for groups M and N we have commutativity of the square M

f

ι

M

N ι

f

N.

A rig is a set R enriched with two monoid structures, a commutative one written additively and the other written multiplicatively, such that the following equations hold:

(Distributive)

a0 = 0 = 0a a(b + c) = a b + a c , (a + b)c = a c + b c .

The natural numbers N = {0 , 1 , 2 , . . . } provide an example of a rig. A ring is a rig for which the additive monoid is a group. The integers Z provide an example. A rig is commutative when the multiplicative monoid is commutative. A field is a commutative ring for which each element is either 0 or has a multiplicative inverse. / S is a function which is a For rigs R and S a rig morphism f : R monoid morphism for both the additive and multiplicative structures. Let k denote a field. A k-algebra is a ring A together with a ring morphism / A . Notice that either A is trivial (i.e., 1 = 0) , or η is injective η: k
[ κ = κ ⇒ κ − κ
= 0 ⇒ κ − κ
is invertible ⇒ η(κ − κ
) is invertible 1=0

==⇒ η(κ − κ
) = 0 ⇒ η(κ) = η(κ
) ] . We can define scalar multiplication / A by κ a = η(κ) a . k×A / B is a ring For k-algebras A and B, a k-algebra morphism f : A morphism such that the next diagram commutes; k η

η

A

f

B   that is, f (κ a) = κ f (a) . We write Algk A , B for the set of k-algebra / B. morphisms f : A An isomorphism is a bijective morphism; automatically its inverse function is also a morphism.

2 Duality between geometry and algebra

The purpose of this section is to convince you that commutative algebras are really spaces seen from the other side of your brain. For a compact hausdorff space X, we have the algebra C(X) of continuous, / C . The addition and multiplication complex-valued functions a : X are obtained pointwise from C . / Y gives rise to an algebra morphism A continuous function f : X / C(f ) : C(Y ) C(X) (note the reversal of direction!) via C(f )(b) = a , / 1 gives the algebra where a(x) = b(f (x)) . In particular, the unique X / X of the / C(X) , while each point x : 1 morphism η : C = C(1) / space gives an algebra morphism C(X) C. Actually C(X) is more than just a C-algebra; it is what is called a commutative C ∗ -algebra (there is a norm and an involution ( )∗ coming from conjugation). With this extra structure the duality becomes precise: Each commutative C ∗ -algebra A is isomorphic to C(X) for some compact hausdorff space X; each C ∗ -algebra morphism / C(X) has the form C(f ) for a unique continuous C(Y ) /Y. function f : X This result is commonly referred to as Gelfand duality. Algebraic geometry is the study of spaces called varieties : the solutions to polynomial equations in several variables. In studying the variety given by x2 + 2y 3 = z 4 over the field k, we pass to the k-algebra A = k[ x , y , z ] / (x2 + 2y 3 = z 4 ) . By k[ x , y , z ] we mean the k-algebra of polynomials in three commuting indeterminates x , y , z ; the elements are expressions  αijk xi y j z k i,j,k

5

6

Quantum Groups: A Path to Current Algebra

where αijk ∈ k and (i , j , k) runs over a finite subset of N3 . The quotient algebra A is obtained from k[ x , y , z ] by identifying elements when they may be transformed one into another by means of the equation x2 +2y 3 = z 4 and the algebra axioms. / B is Given a k-algebra B , a k-algebra morphism f : k[ x , y , z ] uniquely determined by its values on x , y , z . In fact we have a bijection   Algk k[ x , y , z ] , B ∼ = B3 . Similarly, we have a bijection     Algk A , B ∼ = (u , v , w) ∈ B 3 | u2 + 2v 3 = w4 /B where A is as above. Again we see that a k-algebra morphism A corresponds to a map of varieties in the reverse direction. For general k-algebras A and B, it is suggestive to call a morphism f : / B a B-point of A . A point of (the space corresponding to) A is a A k-point, not to be confused with an element of the algebra A itself. 

commutative k-algebras

op

spectrum



spaces



coordinate algebra

Let X denote a category. I am thinking of the objects of X as spaces X / Y as the maps appropriate to that kind and Y say, and the arrows  X of space. Write X X , Y for the set of arrows from X to Y in X . Let X1 , . . . , Xn be arbitrary objects of X . A product for this list of objects consists of an object X1 × · · · × Xn together with arrows pi : X1 × · · · × Xn

/ Xi

for i = 1, . . . , n

such that, given any other object K and arrows / Xi

f :K

there exists a unique arrow f : K

for i = 1, . . . , n / X1 × · · · × Xn with p ◦ f = f . i i pi

X1 × · · · × Xn f

Xi

fi

K This means we have a bijection       X K , X1 × · · · × Xn ∼ = X K, X1 × · · · × X K, Xn .

Duality between geometry and algebra

7

In particular, the empty product is called a terminal object, denoted by 1 . We have   X K, 1 ∼ = 1. Products are unique up to isomorphism (if they exist). / X × · · · × X is defined by p ◦ δ = 1 for all i . The The diagonal δ : X i X canonical isomorphisms f1 × · · · × fn and isomorphisms σξ can be defined as for sets. The diagrammatic definition of monoid and group can be carried into the category X (provided the products exist;  1 and  M × M are enough). If M is a monoid (group) in X then each X K, M becomes a monoid (group) using the multiplication ∗ given by f ∗ g = µ ◦ (f × g) ◦ δ K

δ

K ×K

f ×g

M ×M

µ

M.

A group in the category of topological spaces and continuous maps is called a topological group. A group in the category of smooth manifolds and smooth maps is called a Lie group. We are more interested here in groups in the category (Comm Algk )op of commutative k-algebras and reversed morphisms; these are called affine groups over k . This is the variety point of view. On the algebraic side they are called commutative Hopf algebras over k . Product of varieties becomes tensor product A⊗k B of k-algebras (more on this later). A commutative Hopf algebra H thus has structure given by the k-algebra morphisms ε:H

/k ,

δ:H

/H⊗ H , k

ν:H

/H

called counit , comultiplication , antipode (corresponding respectively to the unit, multiplication, inversion for the group).  Now for each commutative k-algebra A , we obtain a group Algk H, A of A-points in H. It will also be necessary to consider the algebraic version of affine monoids over k . These are called commutative bialgebras over k . They have a counit and comultiplication, but generally no antipode. Example 2.1 Let M(2) denote k[ a , b , c , d ] as a commutative k-algebra. / k is defined by ε(a) = ε(d) = 1 , ε(b) = ε(c) = 0 . A counit ε : M(2) Clearly k[ a , b , c , d ] ⊗k k[ a , b , c , d ] ∼ = k[ a , b , c , d , a
, b
, c
, d
]

8

Quantum Groups: A Path to Current Algebra

with the coprojections k[ a , b , c , d , a
, b
, c
, d
] k[ a , b , c , d ]



a , b , c , d and a , b , c , d a,b,c,d . / The comultiplication δ : M(2) M(2)⊗k M(2) is given by k[ a , b , c , d ] a,b,c,d

a,b,c,d

a a
+ b c
, a b
+ b d
, c a
+ d c
, c b
+ d d
.

This makes M(2) into a commutative k-bialgebra. Notice that we have a monoid isomorphism     Algk M(2) , A ∼ = Mat 2 , A where on the right we have the multiplicative monoid of 2 × 2 matrices with entries in A . Thus M(2) is the coordinate k-algebra of the variety of 2×2 matrices. To obtain the coordinate k-algebra of the general linear group, we take GL(2) = k[ a , b , c , d , x ] / (x a d − x b c = 1) . / GL(2) which induces There is an epimorphic k-algebra morphism M(2) a bialgebra structure on GL(2) from that on M(2). The antipode ν : GL(2) a,b,c,d,x

GL(2) x d , −x b , −x c , x a , a d − b c

makes GL(2) into a commutative Hopf algebra.

Bradshaw: Tassel Bradshaw Group, [Wal94, Plate 22].

3 The quantum general linear group

The passage from quantum to classical mechanics is quite well defined by taking the limit as Planck’s constant  tends to 0. The passage in the other direction is not so clear cut, and may not be uniquely determined. On the algebraic side, “quantization” involves deforming commutative algebras to non-commutative ones: e.g.,

xy = yx

becomes

x y = e
y x .

Usually we deal with q = e
rather than  , so classical results correspond to the case q = 1 . Quantum spaces correspond to more general k-algebras, not necessarily commutative. Let k be a fixed field and fix q ∈ k with q = 0 . Write k x1 , . . . , xn  for the k-algebra of polynomials in non-commuting indeterminates x1 , . . . , xn . As a vector space over k , a basis is given by those elements m

m

m

1 2 r xξ(2) · · · xξ(r) xξ(1)

for which r ∈ N and m1 , . . . , mr ∈ Z+ and ξ : {1, . . . , r} any function. Notice that

/ {1, . . . , n} is

k[ x , y ] = k x , y /( x y = y x ) . The coordinate algebra of the space of quantum 2 × 2 matrices is defined by Mq (2) = k a , b , c , d /R where R is the system of equations ab = q −1 ba , ac = q −1 ca , cd = q −1 dc , bd = q −1 db bc = cb , ad − da = (q −1 − q) bc . a

b

c

d

(mnemonic)

9

10

Quantum Groups: A Path to Current Algebra

The monomials am1 bm2 cm3 dm4 form a basis for the algebra, as a vector space over k .

    a b Algk Mq (2) , A ∼ ∈ Mat (2 , A) | R holds . = c d  
a b a b
be two A-points of Mq (2) such Theorem 3.1 Let and

c d c d that each entry of the first commutes with each entry of the second.  
a b a b
(as matrices) is an A-point of Mq (2) . (i) The product c
d
c d  a b = (ad − q −1 bc) commutes with (ii) The “q-determinant” detq c d each of a , b , c , d and  
 
a b a b
a b a b
× det q
. det q = detq


c d c d c d c d  a b is invertible in A then (iii) If det q c d   −1  a b −1 a b d −qb = det q −1 c d a c d −q c is an A-point of Mq−1 (2) . The above result can be proved by direct calculation, but this gives little insight into the special nature of the relations R . Examples such as this arose in work of L. D. Fadde
ev [FRT88] and his school on the quantum inverse scattering transform (QIST) method. The version I present here comes from some lectures of Yu. Manin [Man88] given at Universit´e de Montr´eal in June 1988. The following “explanation” of Theorem 3.1 is due to Yu. Kobyzev (Moscow, winter 1986–87). Introduce the quantum plane, as defined by the k-algebra = k x , y /(xy = q −1 yx) . A2|0 q The monomials xm y n with m , n ∈ N form a basis for this as a vector space. We also need to consider a quantized version of a Grassmannian algebra in two variables: = k ξ , η /(ξ 2 = η 2 = 0 = ξη + q ηξ) . A0|2 q The monomials ξ m η n with m , n ∈ {0, 1} form a basis for this algebra. The reason for the funny superscripts 2|0 and 0|2 comes from “supergeometry” where dimensions are represented by pairs d | d
of numbers. This A0|2 q is a quantum superplane. /A An A-point of B is called generic when the algebra morphism B is injective.

The quantum general linear group

11

Theorem 3.2 Suppose (x, y) is a generic A-point of A2|0 q and (ξ , η) is a generic A-point of A0|2 q . Suppose a , b , c , d ∈ A all commute with x , y , ξ , η . Put 
  

  
  x a b x x a c x ξ a b ξ = , = , = .



y y c d y b d η y c d η If q 2 = −1 , the following conditions are equivalent : (i) (x
, y
) and (x

, y

) are points of A2|0 q ; 2|0



(ii) (x , y ) is a point of Aq and (ξ , η ) is a point of A0|2 q ;  a c (iii) is a point of Mq (2) . b d [For q 2 = −1 we only have (iii) ⇒ (i) & (ii).] −1



Proof. (i) ⇔ (iii). (x
, y
) is a point of A2|0 q iff x y = q y x ; that is, iff −1 (a x + b y)(c x + d y) = q (c x + d y)(a x + b y) . Multiply out the products using the fact that a , b , c , d each commute with x and y ; since (x , y) is generic, we can equate coefficients of x2 , y 2 , x y . So the single equation is in fact equivalent to the following set of three equations:

(∗)

ac = q −1 ca

,

bd = q −1 db

,

ad − da = q −1 cb − q bc .

Interchanging b and c we see that (x

, y

) is a point of A2|0 q iff (∗∗)

ab = q −1 ba

,

cd = q −1 dc

,

ad − da = q −1 bc − q cb .

Taking the last equations in (∗) & (∗∗) we get q −1 cb − q bc = q −1 bc − q cb ; that is, (q + q −1 )(bc − cb) = 0 hence bc = cb, provided q 2 = −1 . So (iii) ⇔ (∗) & (∗∗), which together are equivalent to (i). 2 2 (ii) ⇔ (iii). (ξ
, η
) is a point of A0|2 q iff 0 = (a ξ + b η) = (c ξ + d η) = (a ξ +b η)(c ξ +d η)+q (c ξ +d η)(a ξ +b η) . Using ξ 2 = η 2 = 0 these become ab ξη+ba ηξ = 0 and cd ξη+dc ηξ = 0 and ab ξη+bc ηξ+q (cb ξη+da ηξ) = 0 . Using ξη = −q ηξ and the linear independence of η and ξ in A, we get that −q ab+ba = 0 and that −q cd+dc = 0 and also −q (ad+q cb)+bc+q da = 0 . These are equivalent to (∗∗). So (ii) ⇔ (∗) and (∗∗) ⇔ (i).


a b c d and its transpose to both transform the quantum plane into itself; or for  a b to transform both the plane and superplane into themselves. c d

In other words, the relations R are precisely what is needed for

Proof of Theorem 3.1. (i) Let B be the free k-algebra containing the indeterminates a , b , c , d , a
, b
, c
, d
, x , y subject to the relations on these variables in the hypotheses of Theorems 3.1 and 3.2. Then (x , y) is generic;

12  a c

Quantum Groups: A Path to Current Algebra b a
b
and are B-points of Mq (2) . By Theorem 3.2, we have

d c d    
x a c x a b
are B-points of A2|0 that and
q . Each coordinate
y c d b d y



in the first of these commutes with all of a , b , c , d while coordinates in the   a c x second commute with a , b , c , d . Also is generic since when b d y 2|0 / Aq for which (a , b , c , d , x , y)  / (1 , 0 , 0 , 1 , x , y) composed with B 
 a b
x we get (x, y), which is generic. Similarly is generic. So by

y c d 
  

 a c x a c x a b a b
Theorem 3.2 we have and



b d c d y b d y c d  
a b a b
2|0 both being B-points of Aq . Again by Theorem 3.2, is a c
d
c d B-point of Mq (2). / A for which To obtain the result for the given A apply the morphism B 

/ (a , b, . . . , d , 0 , 0) . (a , b , . . . , d , x , y)



(ii) We now get a natural definition of the quantum determinant which immediately yields its multiplicativity: in the notation of Theorem 3.2, ξ
η
= (a ξ + b η)(c ξ + d η) = det q



a c

(iii) This is left as an exercise for the reader.

b d

ξη .


The quantum general linear group is defined from 2×2 matrices by inverting the determinant: GLq (2) = Mq (2)[t]/(t a = a t , t b = b t , t c = c t , d t = t d , t det q = 1) . Similarly, the quantum special linear group is defined by requiring that the determinant be equal to 1: SLq (2) = Mq (2)/(det q = 1) . Theorem 3.2 describes the representations of these “groups” on quantum 0|2 spaces A2|0 q and Aq . Exercise 3.1 Give a direct proof of Theorem 3.1 applied to quantum 2×2 matrices.

4 Modules and tensor products

Let R be a ring (not necessarily commutative). We write Rop for the ring with opposite multiplication σ

R×R

R×R

µ

R.

(To say R is commutative is to say Rop = R .) A left R-module is an abelian group M (written additively) together with a function R×M

/M

whereby (r , m) 

/ rm

called scalar multiplication, such that 1m = m

,

(r s) m = r (s m)

(r + r
) m = r m + r
m

,

r (m + m
) = r m + r m
.

/M. A right R-module is defined similarly, with multiplication M × R op A left R -module structure on an abelian group M “is the same” as / M is a scalar a right R-module structure. More precisely, µ : R × M σ

µ

multiplication for a left Rop -module iff M × R R×M M is one for a right R-module. In this way, we can deal only with left R-modules and omit “left”, unless we explicitly stipulate otherwise. If R is commutative, R = Rop and there is no need to distinguish left and right modules. If R is a field, an R-module is precisely a vector space over R . Furthermore, Z-modules are precisely abelian groups since each abelian group A admits a unique Z-scalar multiplication given by  n a = a + · · · + a (n terms) for n ≥ 0 and n a = − (−n)a for n < 0 . A subset X of an R-module M is said to generate M (or span M ) when, for each m ∈ M , there exist r1 , . . . , rn ∈ R and x1 , . . . , xn ∈ X such that (∗)

m = r1 x1 + · · · + rn xn .

Call M finitely generated when it is generated by some finite subset. 13

14

Quantum Groups: A Path to Current Algebra

A (not necessarily finite) subset X of M is linearly independent when for x1 , . . . , xn ∈ X distinct elements, having a relation of the form r1 x1 + · · · + rn xn = 0 with r1 , . . . , rn ∈ R implies that r1 = · · · = rn = 0 . Then each expression (∗) is unique up to order of factors (with x1 , . . . , xn distinct). An R-module F is said to be free when it is generated by some linearly independent subset. Every vector space is free, but this is peculiar to R being a field. It is easy to see that Z/(2) is not a free abelian group. Each set X determines an R-module FR (X) = {r1 x1 + · · · + rn xn | ri ∈ R , xi ∈ X , n ∈ N} with addition and scalar multiplication defined in the obvious way. We can identify x ∈ X with 1 x ∈ FR (X) and see easily that X is linearly independent and generates FR (X) . So FR (X) is free. / N is (left)R-linear (or an For R-modules M and N , a function f : M
R-module morphism) when f (m+m ) = f (m)+f (m
) and f (r m) = r f (m) for all m , m
∈ M and r ∈ R . Write HomR (M, N ) for the abelian group / N ; the addition is given by (f + g)(m) = of R-linear functions f : M f (m) + g(m) . Warning: You may think HomR (M, N ) becomes an R-module by defining (rf )(m) = r f (m) . But this rf does not preserve scalar multiplication when R is non-commutative. / Y . An RFor sets X and Y , write Y X for the set of all functions f : X / M is uniquely determined by its restriction linear function f : FR (X) to X . Indeed, this gives an isomorphism of abelian groups HomR (FR (X) , M ) ∼ = MX where the addition on M X is pointwise. A submodule H of an R-module M is a subset which is closed under addition and scalar multiplication. This gives an equivalence relation ≡H on M whereby m ≡H m
if and only if m − m
∈ H . The equivalence class containing m ∈ M is m + H = {m + h | h ∈ H} , called the H-coset containing m . The set M/H of H-cosets becomes an R-module via (m + H) + (n + H) = (m + n) + H

,

r(m + H) = r m + H .

/ M/H for which ρ(m) = We have a surjective R-linear function ρ : M / N with g(m) = 0 for all m ∈ H, m + H . For each R-linear g : M

Modules and tensor products

15

/ N with gˆ ◦ ρ = g . The kernel there exists a unique R-linear gˆ : M/H / N is a submodule ker f = {m ∈ M | f (m) = 0} of any R-linear f : M of M ; we have a commutative diagram f

M

N

ρ

M/ ker f

∼ =

im f

of R-modules, where im f = {f (m) | m ∈ M } is the image of f , the bottom arrow is an R-linear isomorphism, and the right arrow is an inclusion of a submodule. The submodule (X) generated by a subset X of an R-module M is the smallest submodule of M which contains X . As such it is the image of / M whose restriction to X is the inclusion the R-linear function FR (X)  / X M . Of course (X) is generated by X, but in general not freely. Suppose that M is a right R-module and N is a left R-module. A function / A into an abelian group A is R-bilinear when it satisfies f : M ×N f (m , n + n
)

= f (m , n) + f (m , n
)

f (m + m
, n) = f (m , n) + f (m
, n) f (m r , n) = f (m , r n) . Write BilR (M, N ; A) for the abelian group, which is a subgroup of AM×N , / A . Our main goal is to construct a of R-bilinear functions f : M × N / M⊗ N . “universal” bilinear function λ : M × N R Let B denote the subset of the abelian group FZ (M × N ) consisting of all elements of the form (m + m
, n) − (m , n) − (m
, n) , (m, n + n
) − (m , n) − (m , n
) , (m r , n) − (m , r n) for m , m
∈ M with n , n
∈ N and r ∈ R . Put M ⊗R N = FZ (M × N )/(B) . Then we have abelian group isomorphisms   HomZ (M ⊗R N , A) = HomZ FZ (M × N )/(B) , A   ∼ {g ∈ Hom F (M × N ) , A | g is zero on B} = ∼ = =

Z

Z

{f ∈ AM×N | f is R-bilinear} BilR (M, N ; A) .

16

Quantum Groups: A Path to Current Algebra

/A In particular by taking A = M ⊗R N we get the identity morphism A corresponding, under the composite of the above string of isomorphisms, / M ⊗ N . Then we easily see that to a bilinear morphism λ : M × N R / A uniquely determines an abelian group each R-bilinear f : M × N / A with g ◦ λ = f . morphism g : M ⊗R N For (m , n) ∈ M × N , we put m ⊗ n = λ(m , n) . A typical element of M ⊗R N then has the form k  mi ⊗ n i i=1

where m1 , . . ., mk ∈ M and n1 , . . ., nk ∈ N . These elements satisfy (m + m
) ⊗ n

= m ⊗ n + m
⊗ n

m ⊗ (n + n
) m r ⊗n

= m ⊗ n + m ⊗ n
= m ⊗r n .

 / With R and S rings, a module M from R to S, written M : R S , is an abelian group M enriched with a left R-module structure and a right S-module structure related by (r m)s = r(m s) for all r ∈ R , m ∈ M and s ∈ S . (In the literature this structure is also known as a left R-/right S-bimodule.) In this notation, tensor product can be viewed as a kind of “composition of modules”. S M

R

N

M⊗S N

T

For M and N as above, M ⊗S N becomes a module from R to T by defining r(m ⊗ n)t = (r m) ⊗ (n t) . This composition of modules is not strictly associative, but is associative up to canonical isomorphisms much like cartesian product of sets. This can be seen by defining a multiple tensor product as we now proceed to do. For rings R and S and any set X , there is a free module from R to S generated by X . It is denoted by FRS(X) and its elements have the form r1 x1 s1 + · · · + rn xn sn

for ri ∈ R , si ∈ S , xi ∈ X , n ∈ N .

Modules and tensor products

17

 / S we have   HomRS FRS(X) , M ∼ = MX

For each module M : R

where HomRS (N, M ) is the abelian group which has as elements the left R/M. /right S-module morphisms N Given rings and modules as in the diagram M2 M1

R2

M3

...

R1

..

. Mn

L

R0 a function f : M1 × · · · × Mn the equations

Rn

/ L is called multilinear when it satisfies

f (m1 , . . . , mi + m
i , . . . , mn ) = f (m1 , . . . , mi , . . . , mn ) + f (m1 , . . . , m
i , . . . , mn ) r0 f (m1 , . . . , mn ) = f (r0 m1 , m2 , . . . , mn ) f (m1 , . . . , mi ri , mi+1 , . . . , mn ) = f (m1 , . . . , mi , ri mi+1 , . . . , mn ) f (m1 , . . . , mn ) rn

= f (m1 , . . . , mn−1 , mn rn )

for ri ∈ Ri and mi , m
i ∈ Mi . Write Mult (M1 , . . . , Mn ; L) for the abelian group of such functions f . It should now be clear how to construct a module M1 ⊗R1 M2 ⊗R2 · · · ⊗Rn−1 Mn : R0

 / Rn

and multilinear function λ : M1 × · · · × Mn

/ M ⊗ ··· ⊗ 1 R1 Rn−1 Mn

having the universal property that, for each multilinear function f : M1 × / L , there exists a unique left R -/right R -module morphism · · · × Mn 0 n / L for which g ◦ λ = f . This describes an g : M1 ⊗R1 · · · ⊗Rn−1 Mn abelian group isomorphism n Mult (M1 , . . . , Mn ; L) ∼ = HomR R0 (M1 ⊗R1 · · · ⊗Rn−1 Mn , L)

(where HomRS (M, N ) = Mult (M, N ) is the abelian group of left R-/right / N ). When there is no ambiguity about the S-module morphisms M

18

Quantum Groups: A Path to Current Algebra

rings, we write M1 ⊗ · · · ⊗Mn instead of M1 ⊗R1 · · · ⊗Rn−1 Mn . As with cartesian product we have canonical isomorphisms (M1 ⊗ · · · ⊗Mk )⊗(Mk+1 ⊗ · · · ⊗Mn ) ∼ = M1 ⊗ · · · ⊗Mn . / M ⊗M in which m  / m ⊗ m , does not preHowever, the diagonal M serve addition. The empty tensor product M1 ⊗ · · · ⊗Mn for n = 0 is just  / R0 as a module R0 R0 , using multiplication in R as scalar multiplication on both sides. We have canonical isomorphisms R⊗R M ∼ = M ∼ = M ⊗S S . Given M, M
: R

 / S , we write M

 / S

f : M ⇒ M
: R

or

R

f

S

M


/ M
is a left R- and right S-module morphism. Given

to mean f : M the data M1

R0

f1 M1


M2

R1

f2

Mn

...

R2

Rn−1

M2


fn

Rn


Mn

we obtain f1 ⊗ · · · ⊗ fn : M1 ⊗R1 · · · ⊗Rn−1 Mn ⇒ M1
⊗R1 · · · ⊗Rn−1 Mn
:  / R0 Rn given by (f1 ⊗ · · · ⊗ fn ) ◦ λ = λ ◦ (f1 × · · · × fn ) . We have seen that tensor products allow us to represent bilinear functions as module morphisms. Another way of doing this uses Hom instead of tensor. Given a triangle of modules M

R

S N

L

T we can enrich the abelian group HomR (M, L) (resp. HomT(N , L)) of left R(resp. right T -) module morphisms with a module structure  / HomR (M, L) : S T  / T S) (resp. Hom (N , L) : R

Modules and tensor products

19

using the scalar multiplications (s f t)(m) = f (ms) t

(resp. (r g s)(n) = r g(sn) ) .

We then have abelian group isomorphisms HomTS (N , HomR (M, L)) ∼ = Mult (M, N ; L) ∼ = HomSR (M , HomT(N , L)) induced by the canonical bijections  M N  M ∼ L = LM×N ∼ = LN . Combining these with the earlier results, we have HomTS (N , HomR (M, L))

∼ = ∼ =

HomTR (M ⊗S N , L) HomSR (M, HomT(N , L)) .

These isomorphisms are determined by the evaluation morphisms evM : M ⊗S HomR (M, L) T

evN : Hom (N , L)⊗S N

/L /L

,

m⊗f 

,

g ⊗n 

Explicitly, the first isomorphism takes any u : N composite M ⊗S N

1M ⊗u

M ⊗S HomR (M, L)

/ f (m) / g(n) .

/ Hom (M, L) to the R evM

L.

Exercise 4.1 For rings R , S , T and any sets X , Y prove that T = FRS(X)⊗S FST(Y ) FR (X × Y ) ∼ (x , y ) x⊗y .

Hint:

Look at left R-/right T -module morphisms into M : R

 / T .

Exercise 4.2 Describe Z/(2)⊗Z Z/(5) . Exercise 4.3 (a) If R, S are rings, describe a canonical ring structure on R⊗Z S . (b) Is the function from R to R⊗Z S taking R to r ⊗ 1 a ring morphism? Why? (c) Show that R⊗Z S is the coproduct of R, S in the category of commutative rings.

20

Quantum Groups: A Path to Current Algebra

Exercise 4.4 Show that a module M from R to S amounts to the same thing as a left R⊗Z S op -module. Exercise 4.5 Describe explicitly the construction of M ⊗S N ⊗T L .

Bradshaw: Tassel Bradshaw Group, [Wal94, Plate 23].

5 Cauchy modules

 /  / A module M : R S gives rise to a module M ∗ = HomR (M, R) : S R called the left dual of M . There is a canonical module morphism / Hom (M, L) R

∗ ρM L : M ⊗R L

given by ρM L (u ⊗ l)(m) = u(m)l , for each left R-module L .  / Call an M : R S Cauchy when ρM L is an isomorphism for all left R-modules L . Our goal in this section is to characterize Cauchy modules more intrinsically. A module P is called projective when, for all surjective module morphisms / L
and all module morphisms f : P / L
, there exists some e: L / module morphism g : P L with f = e ◦ g . P f

g

L

e

L


/ N is said to be a retraction when there exists a A morphism r : M / M with r ◦ i = 1 . When a retraction exists from M morphism i : N N to N , we call N a retract of M . Proposition 5.1 A module P is projective iff P is a retract of some free module F . Proof. (1) A retract Q of a projective P is projective. To see this take / P and r : P / Q with r ◦ i = 1 . Suppose e : L / / L
is a i:Q Q

/ / L . Then f ◦ r : P L , and since surjective morphism and f : Q / L with e ◦ h = f ◦ r. But P is projective, there is a morphism h : P then e ◦ (h ◦ i) = (e ◦ h) ◦ i = f ◦ r ◦ i = f ◦ 1Q = f , so g = h ◦ i has e ◦ g = f . / L
surjective and (2) Free modules F (X) are projective. Take e : L
/ L . Then we can choose (using the axiom of choice) any f : F (X) 21

22

Quantum Groups: A Path to Current Algebra

an element g(x) ∈ L for each x ∈ X such that e(g(x)) = f (x) . Since / L; F (X) is free, we can extend g uniquely to a morphism g : F (X) and furthermore e ◦ g = f since they agree on X . (3) For each module M there is a free module F and a surjective morphism / M . Just take F to be the free module F (M ) on the underlying e: F / M we only have to give it on M , set of M . To give a morphism e : F so we take e(m) = m . Clearly this e is surjective. / P is surjective and P projective then e is a retraction. (4) If e : F For we have i as in: P 1P

i e

F

P .


This brings us to the fundamental theorem of “Morita theory”.  / Theorem 5.2 The following conditions on a module M : R S are equivalent. (i) M is Cauchy.  / S such that both the (ii) There exists a morphism d : S ⇒ M ∗ ⊗R M : S following two composites are identity morphisms M ∼ = M ⊗S S

1M ⊗d

M∗ ∼ = S⊗S M ∗

d⊗1M ∗

M ⊗S M ∗ ⊗R M M ∗ ⊗R M ⊗S M ∗

(iii) There exists a module N : S e : M ⊗S N

/R

evM ⊗1M

R⊗R M ∼ = M

1M ∗ ⊗evM

M ∗ ⊗R R ∼ = M∗ .

 / R and morphisms ,

d:S

/ N⊗ M R

such that the following composite is the identity morphism M ∼ = M ⊗S S

1M ⊗d

M ⊗S N ⊗R M

e⊗1M

R⊗R M ∼ = M.

(iv) M is a finitely generated projective left R-module. Proof. (i) ⇒ (ii). Since ρM M is an isomorphism, there is an element of ∗ M / M . This element of M ∗ ⊗ M now M ⊗R M taken by ρM to 1M : M R / M ∗ ⊗ M whose value at 1 ∈ S determines a unique morphism d : S R  is the element. Write d(1) = i ui ⊗ mi . The condition ρM M (d(1))(m) = m  becomes i ui (m) mi = m for all m ∈ M . This immediately gives that the

Cauchy modules

23

∗ first composite of (ii) takesm to m . To see  that the second takes u ∈ M to itself we use u(m) = u( ui (m) mi ) = ui (m) u(mi ) .

(ii) ⇒ (iii). Just take N = M ∗ , e = evM and d as in (ii). k (iii) ⇒ (iv). Just put d(1) = i=1 ni ⊗ mi ∈ N ⊗ R M . From the fact that the composite in (iii) is the identity, we have i e(m ⊗ ni ) mi = m for all m ∈ M . So M is generated by m1 , . . . , mk . It remains to see that / L
surjective and f : M / L
. Then M is projective. Take s : L /L g: M we can choose l1 , . . . , lk ∈ L with s(li ) = f (mi ) . Define  by g(m) = i ) li and we get s(g(m)) = i e(m ⊗ n i e(m ⊗ ni ) s(li ) =  ⊗ ⊗ e(m n ) f (m ) = f ( e(m n )m ) = f (m) , as required. i i i i i i (iv) ⇒ (i). It is easy to see that a retract of a Cauchy module is Cauchy (Exercise 5.3). So it suffices to show that M = FR (X) is Cauchy for X a finite set {x1 , . . . , xk }. But then M ∗ = HomR (F (X) , R) ∼ = Rk and k ∼ HomR (M , L) = HomR (F (X) , L) = L . Under these isomorphisms ρM L / Lk with (r , . . . , r ) ⊗ l  / carries across to the morphism Rk ⊗R L k k 1  / (r1 l , . . . , rk l) which has inverse (l1 , . . . , lk ) i=1 ui ⊗ li , in which ui ∈ Rk projects to 0 in all components except the i-th where it projects to 1 . So ρM
L is an isomorphism. / R we obtain two Given rings R and S , from any ring morphism f : S  /  / modules f R : S R and Rf : R S , which have R as underlying abelian group. They have scalar multiplicatons /

S × fR Rf × S

fR

, ,

/ Rf

×R R × Rf

/ fR /R f

(r , r
)  (r
, r) 

/ r r
/ r
r .

fR

given by, respectively (s , r)  (r , s)  For any module L : R

/ f (s) r / r f (s)

, ,

 / T we have canonical isomorphisms

f R ⊗R L

∼ =

r ⊗l

L

∼ =

HomR (Rf , L)

l u(1)

u.

It follows easily from this that (Rf )∗ ∼ = and that Rf is Cauchy.

fR

24

Quantum Groups: A Path to Current Algebra

 / A module M : R S is called convergent when there exists a ring / R and a module isomorphism M ∼ morphism f : S = Rf .   /  / The product i∈I Mi : R S of a family of modules Mi : R S with i ∈ I , has as elements the families m = (mi )i∈I with mi ∈ Mi ; addition and scalar multiplication are given by m + m
= (mi + m
i )i∈I

,

r m s = (r mi s)i∈I .

There are projections 

prj :

/M j

Mi

for each j ∈ I

i∈I

given by prj (m) = mj . There are also injective module morphisms /

inj : Mj



for each j ∈ I

Mi

i∈I

given by inj (m) = m where mj = m and mi = 0 for  all i = j ; we can use these to identify each Mj with the submodule of i∈I Mi consisting of those m with mi = 0 for all i = j .    / The direct sum S is the submodule of i∈I Mi which i∈I Mi : R consists of those m for which mi = 0 for all but finitely many i ∈ I . This is the submodule generated by the union ∪i∈I Mi , hence we can write   i∈I miinstead of m ∈ i∈I Mi . Of course the injections inj actually land in i∈I Mi . Proposition 5.3 There are module isomorphisms: (a)

HomR





i∈I

f (b)

 i∈I





/

(f ◦ ini )i∈I , 

∼ = o

HomR (Mi , L)

i∈I

 o

Mi ⊗S N

i mi ⊗ n



∼ =

Mi , L



Mi ⊗S N

/

i∈I



i (mi ⊗ n)

.

Proof. (a) Injectivity. If f ◦ ini = 0 for all i ∈ I then f is zero on each Mi  and hence on Mi .

Cauchy modules

25

/ L for all i ∈ I , define f :  M / L by Surjectivity. Given fi : Mi i   fi (mi ) . f ( mi ) =    ∼ (b) HomR Mi ⊗S N , L Mi , HomR (N, L) = HomS i

∼ =



i

  Hom Mi , HomR (N, L) S

i

∼ =



HomR (Mi ⊗S N , L)

i

∼ =

HomR



(Mi ⊗S N ) , L



i

and the composite isomorphism is induced by the given map in (b). This proves it. (Why?)
  When I is finite, notice that i∈I Mi = i∈I Mi . This is also frequently written ⊕i∈I Mi . So M ⊕ N = M × N = M + N . Exercise 5.1 Show that a module P is finitely generated and projective if and only if P is a retract of a free module on a finite set. Hint: In part (3) of the proof of Proposition 5.1 we did not need F (M ); only F (X) for any X generating M . Exercise 5.2 Suppose M is a finitely generated projective module over a commutative ring R. Show that M ∗ is a finitely generated projective module and that the canonical morphism M → M ∗∗ is bijective. Exercise 5.3 Prove directly from the definition of “Cauchy module” that a retract of a Cauchy module is Cauchy. Exercise 5.4 Re-examine the proof of Theorem 5.2 to show that a module  / M :R S is Cauchy if and only if ρM M is surjective.

26

Bradshaw: Tassel Bradshaw Group, [Wal94, Plate 30].

6 Algebras

Let R be any ring. An algebra over R (or R-algebra) is a module A : R together with module morphisms µ : A⊗R A

/A

,

η:R

 / R

/A

such that A⊗R A⊗R A

(Associativity)

(Identity)

A

η ⊗1A 1A ⊗η

µ⊗1A

µ

A⊗R A

1A ⊗µ

A

µ

A⊗R A

A.

1A

Notice that A becomes a ring with multiplication a b = µ(a ⊗ b) and identity 1 = η(1) .  / For R-algebras A , B : R R an algebra morphism f : A module morphism satisfying A⊗R A

f ⊗f

R

B⊗R B

η

η µ

A

/ B is a

µ f

A

B

f

B.

We write AlgR (A , B) for the set of algebra morphisms from A to B .

27

28

Quantum Groups: A Path to Current Algebra

 / Example 6.1 For any module M : R S , the endomorphism algebras, over S and R respectively, are given by  / S EndR (M ) = HomR (M, M ) : S  / S S R . End (M ) = Hom (M, M ) : R In each case the multiplication is given by composition. A module morphism µ ˆ : A ⇒ EndS(M ) : R

 / R

corresponds to a module morphism  / S.

µ : A⊗R M ⇒ M : R

To say that µ ˆ is an algebra morphism is precisely to say that µ is a scalar multiplication enriching M with the structure of left A-module.

Example 6.2 For any module M : R

 / R , write

M ⊗n = M ⊗R · · · ⊗R M

(n terms) .

The tensor algebra on M is defined by the “geometric series” T (M ) =

∞ 

M ⊗n

n=0

/ T (M ) induced by the canonical

with multiplication µ : T (M )⊗R T (M ) isomorphisms M ⊗p ⊗R M ⊗q and unit η : R

∼ =

M ⊗(p+q)

/ T (M ) equal to the injection in0 : R = M ⊗0

/

∞ 

M ⊗n .

n=0

Composition with the injection in1 : M

/ T (M ) gives a bijection

AlgR (T (M ) , A) ∼ = HomR R (M, A) / A to g : T (M ) /A for any algebra A . The inverse takes f : M given by g(m1 ⊗ · · · ⊗ mr ) = f (m1 ) · · · f (mr ) . In particular, if we take M = A and f = 1A , we obtain an algebra morphism µ : T (A)

/A

with

µ(a1 ⊗ · · ·

⊗ ar )

= a1 · · · ar .

Algebras

29

Example 6.3 Let G be any monoid. There is an R-algebra R(G) which is just the free module FRR(G) on the underlying set of G together with the multiplication µ which extends that of G in the sense that R R R(G × G) ∼ (G)⊗R FR (G) = FR FR

G×G µ

µ

R (G) . FR

G

This R(G) is called the monoid R-algebra of G ; or when G is a group, / A into the the group R-algebra of G . Each monoid morphism G multiplicative monoid of A extends uniquely to an R-algebra morphism / A. R(G) A representation of G on M is an R(G)-module. Scalar multiplication / M can be viewed as a monoid morphism R(G)⊗R M G

/ End (M ) . R

The subset of M given by {gm−m | g ∈ G , m ∈ M } generates a submodule (gm − m | g ∈ G , m ∈ M ) and we write M/G for the quotient module M/(gm − m | g ∈ G , m ∈ M ) . An ideal in an algebra A is a submodule I such that a x b ∈ I for all x ∈ I and a , b ∈ A . There is a unique structure of algebra on the quotient module / A/I is an algebra morphism. The A/I for which the canonical ρ : A / B is an ideal in A . kernel of any algebra morphism f : A If X is a subset of an algebra A , we write (X) for the smallest ideal of A containing X . This should not cause confusion with the module notation; the ideal (X) is precisely the submodule (A X A) generated by the subset A X A = {a x b | a , b ∈ A , x ∈ X} of A . Now given any algebra morphism / B satisfying g(x) = 0 for all x ∈ X , then an algebra morphism g: A / B is uniquely determined via the equation f ◦ ρ = g . f : A/(X) Now suppose that R is a commutative ring. Then left R-modules are “the same thing” as right R-modules. Moreover, each left R-module M can  / R by defining be naturally regarded as a module M : R r m s = (r s) m

for all r , s ∈ R and m ∈ M .

In dealing with modules over a commutative ring, we happily regard left modules as two-sided via this process. Thus for R-modules M1 , . . . , Mn we have a tensor product R-module M1 ⊗R · · · ⊗R Mn .

30

Quantum Groups: A Path to Current Algebra

Furthermore every permutation ξ on the set {1 , . . . , n} induces a canonical module isomorphism ∼ =

σξ : M1 ⊗R · · · ⊗R Mn m1 ⊗ · · ·

Mξ(1) ⊗R · · · ⊗R Mξ(n) mξ(1) ⊗ · · ·

⊗ mn

⊗m

ξ(n)

.

Given an algebra A over R with multiplication µ and unit η , we obtain an opposite algebra Aop on the same module A , with multiplication σ

µop : A⊗R A

A⊗R A

µ

A

/ A . Call A commutative when Aop = A and with the same unit η : R as algebras. It follows that the composite A⊗R · · · ⊗R A

σξ

A⊗R · · · ⊗R A

µ

A

is independent of the permutation ξ . Example 6.4 For any set X, the set RX of all functions from X into the commutative ring R becomes a commutative R-algebra after defining addition, scalar multiplication and multiplication as acting pointwise. The / RX is given by η(r)(x) = r for all r ∈ R and x ∈ X . unit η : R

Example 6.5 Let M be any module over the commutative ring R . There is a natural representation of the symmetric group Sn on M ⊗n given by / End (M ⊗n ) ; that is, ξ · (m ⊗ · · · ⊗ m ) = m σ : Sn 1 n R ξ(1) ⊗ · · · ⊗ mξ(n) . The symmetric R-algebra on M is given by the “exponential series” S(M ) =

∞ 

M ⊗n /Sn .

n=0

Another way of constructing this is as follows. For any R-algebra A we can form a commutative R-algebra by taking the quotient of A by the ideal (a b − b a | a , b ∈ A ) . Applying this construction to the tensor algebra T (M ) gives S(M ) . For every commutative R-algebra A , we have that AlgR (S(M ) , A) ∼ = HomR (M , A) . In particular, corresponding to the identity map 1A : A / A. R-algebra morphism µ : S(A)

/ A there is an

Algebras

31

The following diagram of “forgetful” and “free” constructions summarizes some of the above. Comm AlgR

S

AlgR R

T forget multiplication

forget module

ModulesR

Monoids word monoid

forget multiplication

Sets

FR forget module structure

Skew commutativity a b + b a = 0 for an R-algebra is too strong as a requirement for all a , b ∈ A . For example taking b = 1, it would give (1 + 1)a = 0 . Hence if R is a field of characteristic other than 2 (meaning 1 + 1 = 0 in R), we would get a = 0 , and so A = {0} . An R-algebra A is said to be skew commutative when for all a ∈ A either a2 = 0 or a ∈ η(R) . Then, provided none of a , b and a + b is in the / A , we have image of η : R a b + b a = (a + b)2 − a2 − b2 = 0 . Example 6.6 For any R-algebra A we can form the quotient by the ideal (a2 | a ∈ / η(R)) to obtain a skew commutative algebra. If we do this to the tensor algebra T (M ) we obtain the exterior algebra Λ(M ) . Alternatively, let Λn (M ) be the quotient module of M ⊗n by the submodule generated by the elements m1 ⊗ · · · ⊗ mn with mi = mj for some i = j (this submodule is {0} when n = 0 or 1); then Λ(M ) =

∞ 

Λn (M ) .

n=0

We write m1 ∧ · · · ∧ mn for the image of m1 ⊗ · · · ⊗ mn in Λ(M ) . For all x , y , z ∈ M we have x∧x (rx + sy) ∧ z

= 0 therefore x ∧ y = − y ∧ x = r(x ∧ z) + s(y ∧ z) .

32

Quantum Groups: A Path to Current Algebra

If M = FR {x1 , . . . , xk } is a free module on a k-element set then Λn (M ) is   a free module on a nk -element set; so Λ(M ) is a free module on a set with 2k -elements. In particular Λk (M ) is free on the singleton set {x1 ∧· · ·∧xk }, so if k  rij xj yi = j=1

then y1 ∧ · · · ∧ yk must be a unique scalar multiple y1 ∧ · · · ∧ yk = det(rij ) x1 ∧ · · · ∧ xk of x1 ∧ · · · ∧ xk . This can be taken as a definition of the determinant of (rij ) ∈ Mat (k , R) . If A is a skew commutative algebra then we have a bijection AlgR (Λ(M ) , A) ∼ = HomR (M, A) . An R-Lie algebra is an R-module L together with a module morphism / L satisfying the conditions β : L⊗R L

(Jacobi identity)

β(x , x) = 0       β β(x , y) , z + β β(z , x) , y + β β(y , z) , x = 0 .

Call such a β a Lie bracket on the module L . Example 6.7 For any R-algebra A the commutator [ a , b ] = a b − b a defines a Lie bracket on the underlying R-module of A : [[a,b],c]+ [[c,a],b]+ [[b,c],a] = [a,b]c − c[a,b] + [c,a]b− b[c,a] + [b,c]a− a[b,c] = (a b c − b a c) − (c a b − c b a) + (c a b − a c b) − (b c a − b a c) + (b c a − c b a) − (a b c − a c b) = 0 . So A becomes a Lie algebra, denoted by AL . It turns out (at least when R is a field) that every Lie algebra is a submodule, closed under commutator, of such an example.

 / Example 6.8 Let A be an R-algebra and M : A A a module. Then / M is an R-module morphism satisfying a derivation D : A (Leibniz rule)

D(a b) = D(a) b + a D(b) .

Algebras

33

Notice that a = b = 1 gives D(1) = 2 D(1) , so D(1) = 0 . Let DerR (A , M ) denote the submodule of HomR (A, M ) consisting of the derivations. We write DerR (A) for DerR (A , A) . It is easy to check that DerR (A) is closed /A under commutator in the algebra EndR (A) ; that is, if D1 , D2 : A are derivations then so is [ D1 , D2 ] = D1 ◦ D2 − D2 ◦ D1 .

Example 6.9 The tangent space at the identity of each Lie group is a Lie algebra. The pioneering work of Sophus Lie and Eli Cartan showed how much information about the Lie group is obtainable from the Lie algebra (especially in the compact case). The Lie groups GL (n , R) and SL (n , R) and O(n , R) consist of those matrices x ∈ Mat(n , R) for which respectively x is invertible , det x = 1 and x xt = 1 . They have associated Lie algebras gl(n , R) = sl(n , R) = o(n , R)

=

Mat(n , R)   x ∈ gl(n , R) | trace(x) = 0   x ∈ gl(n , R) | xt = −x .

(We shall not stop to prove this here.) The Lie bracket is [ x , y ] = x y−y x in each case. As an exercise the reader should check that sl(n , R) and o(n , R) are closed under commutator.

/ L
is an R-module Suppose L and L
are R-Lie algebras and f : L morphism. Then f is a Lie algebra morphism when it satisfies     f β(x , y) = β f (x) , f (y) . Write LieR (L , L
) for the set of Lie algebra morphisms f : L

/ L
.

We saw in Example 6.7 above that each R-algebra A gives rise to an R-Lie algebra AL using the commutator. We shall describe an “adjoint” for this process: for each R-Lie algebra L we obtain an R-algebra U(L) , called the universal enveloping algebra of L , such that there is a natural bijection   AlgR U(L) , A ∼ (∗) = Lie(L , AL ) . For this we use the tensor algebra T (L) on the underlying R-module of L , and take the quotient by the appropriate ideal:   U(L) = T (L)/ x ⊗ y − y ⊗ x − β(x , y) | x , y ∈ L .

34

Quantum Groups: A Path to Current Algebra

We have a Lie algebra morphism i as in: i

L in1

U(L)L ρ

T (L) and it is composition with i that induces the bijection (∗) . The direct sum L1 ⊕L2 of Lie algebras L1 , L2 is their direct sum as modules together with the Lie bracket     β (x1 , x2 ) , (y1 , y2 ) = β(x1 , y1 ) , β(x2 , y2 ) .

Proposition 6.10 There is an algebra isomorphism U(L1 ⊕ L2 ) ∼ = U(L1 ) ⊗R U(L2 ) whose composite with i : L1 ⊕ L2 to x1 ⊗ 1 + 1 ⊗ x2 .

/ U(L ⊕ L ) takes the pair (x , x ) 1 2 1 2

Proof. It is left to the reader to check that / U(L ) ⊗ U(L ) / x1 ⊗ 1 + 1 ⊗ x2 L1 ⊕ L2 (x1 , x2 )  1 2 L R  / / (x1 , 0) L1 U(L1 ⊕ L2 )L x1  / / (0 , x2 ) L2 U(L1 ⊕ L2 )L x2 are Lie algebra morphisms. These three must therefore be composites with i of algebra morphisms φ : U(L1 ⊕ L2 ) ψ1 : U(L1 ) ψ2 : U(L2 )

/ U(L ) ⊗ U(L ) 1 2 R / U(L ⊕ L ) 1

2

/ U(L ⊕ L ) . 1 2

/ U(L ⊕L ) by ψ(a ⊗ b) = ψ (a) ψ (b) . Then Define ψ : U(L1 ) ⊗R U(L2 ) 1 2 1 2 we have that   ψ φ(x1 , x2 ) = ψ(x1 ⊗ 1 + 1 ⊗ x2 ) = (x1 , 0) + (0 , x2 ) = (x1 , x2 )   = φ(x1 , 0) = x1 ⊗ 1 φ ψ1 (x1 )   = φ(0 , x2 ) = 1 ⊗x2 . φ ψ2 (x2 ) Hence φ and ψ are mutually inverse.




Algebras

35

A deeper result which we shall not prove here is: Proposition 6.11 (Poincar´e–Birkhoff–Witt) If the R-Lie algebra L is / U(L)L is injective. free as an R-module then i : L A Lie algebra L is called commutative when β(x , y) = 0 for all x , y ∈ L . So an algebra A is commutative iff AL is commutative. Notice that, for any module M , we can make M into a commutative Lie algebra. Then the universal enveloping algebra of M is precisely the same as the symmetric algebra of M ; that is, U(M ) = S(M ) . In particular, we have (Proposition 6.10): S(M ⊕ M
) ∼ = S(M ) ⊗R S(M
) . Exercise 6.1 Let R be a commutative ring and G be a group. Consider left modules M, N, L over the group algebra R(G). (a) Show that M ⊗R N becomes an R(G)-module on defining: g(m ⊗ n) = (gm) ⊗ (gn) for g ∈ G, m ∈ M, n ∈ N . (b) Show that HomR (M, L) becomes an R(G)-module on defining: (gu)(n) = gu(g −1 m) for g ∈ G, u ∈ HomR (M, L), m ∈ M . (c) Show that evaluation evM : M ⊗R HomR (M, L) R(G)-module morphism.

/ L is an

(d) Prove that evaluation induces an isomorphism of R-modules     HomR(G) N, HomR (M, L) ∼ = HomR(G) M ⊗R N, L .

36

Bradshaw: Stylized Bradshaw Group, [Wal94, Plate 40].

7 Coalgebras and bialgebras

Let R be any ring. By a coalgebra over R (or R-coalgebra) we mean a  / module C : R R together with module morphisms δ:C

/ C⊗ C R

and

ε:C

/R

such that C

C

δ

δ

δ ⊗1X

C⊗R C

C⊗R C⊗R C

1X ⊗δ

C⊗R C

ε⊗1X 1X ⊗ε

C .

1C

We call δ the comultiplication and ε the counit. This structure provides a module with “formal diagonals”. There is a uniquely determined / C⊗ C⊗ · · · ⊗ C R R R  where for each c ∈ C we have δ(c) = c1i ⊗ · · · ⊗ cni . The notation δ:C

i

δ(c) =



c(1) ⊗ · · ·

⊗ c(n)

(c)

is sometimes used even though the representation of δ(c) in the tensor product is not uniquely determined — we act as though a choice of this representation has been made for each c ∈ C . Given a multilinear function / A we also write f : C × ···× C    f (c(1) , . . . , c(n) ) . f δ(c) = (c)

37

38

Quantum Groups: A Path to Current Algebra

In terms of this notation the axioms can be rewritten as    δ(c(1) ) ⊗ c(2) = c(1) ⊗ c(2) ⊗ c(3) = c(1) ⊗ δ(c(2) ) (c)

c =



(c)

ε(c(1) ) ⊗ c(2) =

(c)



(c)

c(1) ⊗ ε(c(2) ) .

(c)

Suppose C and D are coalgebras. A coalgebra morphism f : C module morphism such that f

C

D

δ

f

C

/ D is a

D;

δ ε

C⊗R C

f ⊗f

ε

D⊗R D

R

that is, 

f (c(1) ) ⊗ f (c(2) )



=

(c)

f (c)(1) ⊗ f (c)(2)

(f (c))

ε(f (c))

=

ε(c) .

We write CogR (C , D ) for the set of coalgebra morphisms from C to D . Suppose R is commutative. A coalgebra C over R is cocommutative when C δ

δ

C⊗R C

σ

C⊗R C .

Return now to a general ring R . Suppose that A is an R-algebra and C is an R-coalgebra. Then HomR (C , A ) becomes an R-algebra under the following convolution structure: HomR (C , A )⊗R HomR (C , A ) ⊗

HomR (C⊗R C , A⊗R A ) µ◦ ◦δ

HomR (C , A )

R ∼ =

HomR (R , R ) η◦ ◦ε

HomR (C , A ) .

Coalgebras and bialgebras

39

In terms of elements, for left R-module morphisms f , g : C convolution product is given by f ∗ g = µ ◦ (f ⊗ g) ◦ δ

and

/ A their

1 = η◦ε.

Using the notation for comultiplication this becomes the formula  f (c(1) ) g(c(2) ) . (f ∗ g)(c) = (c)

In particular, with A = R each R-coalgebra C gives rise to a convolution R-algebra structure on the dual C ∗ = HomR (C , R ) . However, we prefer / C∗ to regard C ∗ as an R-algebra via the multiplication µ : C ∗ ⊗R C ∗ defined by the following diagram. µ⊗1

C ∗ ⊗R C ∗ ⊗R C

C ∗ ⊗R C

1⊗1⊗δ

e

C ∗ ⊗R C ∗ ⊗R C⊗R C

R e

1⊗e⊗1

C ∗ ⊗R C (This works even for non-commutative R.) Example 7.1 Suppose X is any category which admits finite products. / ModR is a functor into the category of modules from Suppose F : X R R to R . Suppose there are natural module morphisms φX

1 , ... , Xn

: F (X1 × · · · × Xn )

/ FX ⊗ FX ⊗ ··· ⊗ FX 1 R 2 R n R

compatible with the canonical associativity isomorphisms for product and tensor product. Then for each object X of X we obtain a coalgebra F X, with comultiplication and counit FX FX

Fδ Fε

F (X × X) F1

φX,X φ0

F X⊗R F X R.

If furthermore R is commutative and F is compatible with the switches, then this coalgebra is cocommutative.

40

Quantum Groups: A Path to Current Algebra

Sub-example (a). (R commutative) The free R-module construction gives a functor / Mod FR : Set R from the category of sets to ModR . We have isomorphisms φ : FR (X1 × · · · × Xn )

∼ =

FR X1 ⊗R · · · ⊗R FR Xn x1 ⊗ · · · ⊗ xn .

(x1 , . . . , xn )

Proof. (n = 2)   ∼ HomR FR (X × Y ) , M = M X×Y ∼ = (M X )Y ∼ = HomR (FR Y , M X )   ∼ = HomR FR Y , Hom(FR X , M ) ∼ = Hom (F X⊗F Y , M ) . R

R

R




So each FR X becomes an R-coalgebra.

Sub-example (b). The universal enveloping algebra provides a functor U : LieR

/ Mod R

and we have already observed U (L ⊕ L
) ∼ =
(x, x )

U (L)⊗R U (L
) x ⊗ 1 + 1 ⊗ x
.

and

U ({0}) ∼ = R

Since direct sum is product in LieR we have another standard example. Thus each universal enveloping algebra U (L) becomes an R-coalgebra. The comultiplication here is determined by L x

i

U (L)

δ

U (L)⊗ U (L) x ⊗ 1+1 ⊗x .

Sub-examples (a), (b) suggest two definitions that we can make for any coalgebra C . • Say that c ∈ C is set-like when δ(c) = c ⊗ c and ε(c) = 1 . (In the case of C = FR (X) the set-like elements are precisely the elements of X .) Write D(C) for the set of set-like elements of C .

Coalgebras and bialgebras

41

• Say that c ∈ C is primitive when δ(c) = c ⊗ 1 + 1 ⊗ c . (In the case of C = U (L) each element of L is primitive.) Write P(C) for the submodule of primitive elements of C . Proposition 7.2 (with R a field.) The set-like elements of any coalgebra C form a linearly independent subset D(C) . Proof. Suppose D(C) is linearly dependent. Let n + 1 be the first natural number for which there is a linearly dependent subset of D(C) with that many elements. Then any set of n elements of D(C) must necessarily be linearly independent, while there exist distinct g, g1 , . . . , gn ∈ D(C) which are linearly dependent. Then we can write g = λ1 g1 + · · · + λn gn with the λi ∈ R all non-zero. Then n 

λi gi ⊗ gi

=

i=1

=

n 

λi δ(gi ) = δ(g) = g ⊗ g

i=1 n 

λi λj gi ⊗ gj .

i,j=1

Since {g1 , . . . , gn } is linearly independent in C then { gi ⊗ gj } is linearly independent in C⊗C , so we can equate coefficients: λi λj = 0 for i = j and λi = λ2i . Since λi = 0 this means n = 1 and λi = 1 . But g = g1 was not allowed.
We shall come back to set-like and primitive elements in the context of bialgebras. Example 7.3 (with R commutative.) Let C = FR N be the free R-module on the countable set N . Define

 1 for n = 0 δ(n) = p ⊗q and ε(n) = 0 for n > 0 . p+q=n This defines a cocommutative coalgebra structure on C . Take an R-algebra A and look at an example of convolution with this coalgebra C . The convolution structure transports across the R-module isomorphism ( sequences in A ) HomR (C , A ) ∼ = AN

42

Quantum Groups: A Path to Current Algebra

to give the multiplication ab =

 

ap b q

p+q=n

for sequences a = (an ) = (a0 , a1 , . . . ) and b = (bn ) = (b0 , b1 , . . . ) in A . The unit sequence is (1 , 0 , 0 , 0 , . . . ) . A precise definition of indeterminate can be taken to mean the sequence x = (0 , 1 , 0 , 0 , 0 , 0 , . . . ) ∈ AN in A . Each u ∈ A is identified with u 1 = (u , 0 , 0 , 0 , . . . ) ∈ AN . Then each a ∈ AN can be written as a formal (no convergence requirements!) power series ∞  an xn . a = n=0 N

Write A[[x]] for A with this algebra structure. It is the R-algebra of formal power series in A . If A is commutative so is A[[x]] . In particular when A = R we obtain the commutative R-algebra C ∗ = R[[x]] . Example 7.4 Let n = {1, 2, . . . , n} and put C = FR (n × n) . Then C becomes an R-coalgebra on defining δ(i , j) =

n 

(i , k) ⊗ (k , j)

and

ε(i , j) =

k=1

1 0

for i = j otherwise .

Given any R-algebra A , the convolution structure simply transports across the R-module isomorphism HomR (C , A ) ∼ = An×n

( n × n matrices in A )

to give the usual matrix multiplication (aij ) (bij ) =

n 

aik bkj .

k=1

In this way we obtain the R-algebra Mat(n , A ) ∼ = EndR (An ) of n × n matrices with entries in A .

Suppose that R is a commutative ring. An R-bialgebra is an R-module B together with algebra and coalgebra structures /B µ : B⊗R B / B⊗ B δ: B R

and

η: R

/B

and

ε: B

/R

Coalgebras and bialgebras

43

satisfying the conditions µ

B⊗R B

B

δ ⊗δ

δ ∼ = 1⊗σ⊗1

B⊗R B⊗R B⊗R B

B⊗B

ε⊗ε

η ⊗η

R⊗R

ε

B⊗B

B⊗B

∼ =

µ

B

µ⊗µ

B⊗R B⊗R B⊗R B

δ η

R

B

B η

ε 1R

R

R.

Notice the complete duality between (µ , η ) and (δ , ε ) . When expressed in terms of elements the duality is not so apparent:  δ(x , y) = x(1) y(1) ⊗ x(2) y(2) (x) (y)

ε(x y) = ε(x) ε(y) and δ(1) = 1 ⊗ 1

and ε(1) = 1 .

For R commutative, the tensor product A⊗R A
of R-algebras A and A
becomes an R-algebra via the multiplication (A⊗R A
)⊗R (A⊗R A
)

∼ = 1⊗σ⊗1 σ1324

(A⊗R A)⊗R (A
⊗R A
)

µ⊗µ

A⊗R A


and unit R ∼ = R⊗R R

η ⊗η

A⊗R A
.

Also the tensor product C⊗R C
, of R-coalgebras C and C
, becomes an R-coalgebra via the comultiplication C⊗R C


δ ⊗δ

(C⊗R C)⊗R (C
⊗R C
)

and counit C⊗R C


ε⊗ε

∼ = 1⊗σ⊗1

(C⊗R C
)⊗R (C⊗R C
)

R⊗R R ∼ = R.

44

Quantum Groups: A Path to Current Algebra

With this, we can make the observation: Proposition 7.5 Suppose (µ , η ) and (δ , ε ) are respectively, algebra and coalgebra structures on the R-module B . Then the following conditions are equivalent: (i) B is a bialgebra; (ii) µ : B⊗R B (iii) δ : B

/ B and η : R

/ B⊗ B and ε : B R

/ B are coalgebra morphisms; / R are algebra morphisms.

/ B
is a function For bialgebras B and B
a bialgebra morphism f : B which is both an algebra and coalgebra morphism. Write BigR (B , B
) for the set of such functions f . Before giving examples of bialgebras we prove some extra results on the set-like and primitive elements for the bialgebra case. Proposition 7.6 If B is a bialgebra then the set-like elements are closed under multiplication: so D(B) becomes a monoid. Proof. δ(bb
) = δ(b) δ(b
) = (b ⊗ b)(b
⊗ b
) = bb
⊗ bb
for b , b
∈ D(B) ; also
ε(bb
) = ε(b) ε(b
) = 1 · 1 = 1 .

Proposition 7.7 If B is a bialgebra then the set of primitive elements is closed under commutator, so P(B) becomes a Lie algebra. Also ε(x) = 0 for all x ∈ P(B) . Proof. For x , y ∈ P(B) we have δ([ x , y ])

= δ(x) δ(y) − δ(y) δ(x) = (x ⊗ 1 + 1 ⊗ x)(y ⊗ 1 + 1 ⊗ y) − (y ⊗ 1 + 1 ⊗ y)(x ⊗ 1 + 1 ⊗ x) = xy ⊗ 1 + x ⊗ y + y ⊗ x + 1 ⊗ xy − (yx ⊗ 1 + y ⊗ x + x ⊗ y + 1 ⊗ yx) = [ x , y ]⊗1 + 1 ⊗[ x , y ]

so that [ x , y ] ∈ P(B) . Also x = (1 ⊗ ε) δ(x) = (1 ⊗ ε) (x ⊗ 1 + 1 ⊗ x) = x + ε(x) ; hence ε(x) = 0 .
Example 7.8 Return to the situation of coalgebras in Example 7.1. There / Mod which give rise to are two conditions on the functor F : X R bialgebras F X .

Coalgebras and bialgebras

45

(a) When the morphisms φ are all invertible. Then F takes each monoid G in X to a bialgebra F G . The multiplication and unit for G give an algebra structure F G ⊗R F G ∼ = F (G × G)

Fµ

FG

and

R ∼ = F1

Fη

FG

on F G . These are coalgebra morphisms since all arrows in X “commute with diagonals”. By Proposition 7.5, each F G becomes a bialgebra. This is / Mod , so for each monoid G the situation for the functor FR : Set R the monoid algebra R(G) is a cocommutative bialgebra. Notice here that   G∼ = D R(G) as monoids (see Proposition 7.6). / Alg . (b) When F lifts to F : X R In this case each F X is clearly a bialgebra since the comultiplication and counit are algebra morphisms (Proposition 7.5). In particular, for the func/ Alg this is indeed the situation. Thus we have that: tor U : LieR R each universal enveloping algebra U (L) is a cocommutative bialgebra. Example 7.9 Return to Example 7.3 of a coalgebra. This time, to use the symbol N to denote our countable set would be confusing. Instead we denote it by E = {e0 , e1 , e2 , e3 , . . . } . Then the coalgebra structure on FR (E) is

 0 for n > 0 , δ(en ) = ep ⊗ eq and ε(en ) = 1 for n = 0 . p+q=n We now make FR (E) into an algebra via ep eq =

(p + q)! ep+q p! q!

with

e0 = 1 .

(The binomial coefficient is an integer and so “lives” in any ring R .) Then FR (E) is a bialgebra. If R is a field of characteristic 0 (i.e., 1 + · · ·+ 1 = 0 in R for any non-zero number of terms) put x = e1 so one easily sees that 1 n x . Hence, as an algebra, FR (E) is isomorphic to the polynomial en = n! algebra R[x] in one variable. For general R we can think of FR (E) as the algebra of Hurwitz polynomials in one indeterminate: k  an xn n! n=0

with each an ∈ R .

Example 7.10 Return to Example 7.4 and form the symmetric algebra S(C) of the coalgebra C = FR (n × n) . Since using n × n can be confusing we replace it by any set X = {xij | i, j ∈ n} of cardinality n2 . Then we

46

Quantum Groups: A Path to Current Algebra

identify S(FR (X)) with the polynomial R-algebra R[(xij )] in n2 commuting indeterminates xij for i , j ∈ n . In Example 7.1 we saw that this becomes a bialgebra by virtue of the fact that it is the universal enveloping algebra of a commutative Lie algebra FR (X), but this is not the structure of interest here. The coalgebra C induces the bialgebra structure   1 for i = j δ(xij ) = xik ⊗ xkj and ε(xij ) = 0 for i = j k

which we call the matrix bialgebra M(n) over R . This must not be confused with the matrix algebra Mat(n , R) ∼ = AlgR (M(n) , R ) (which is the algebra of “points” of M(n) ). Exercise 7.1 For any R-coalgebra C, prove the following identities:  (a) δ(c) = ε(c(2) ) ⊗ δ(c(1) ) =

(c) 

δ(c(2) ) ⊗ ε(c(1) )

(c)

(b)

 (c)

(c)



ε(c(1) ) ⊗ c(3) ⊗ c(2) =



c(1) ⊗ ε(c(3) ) ⊗ c(2)

(c)



c(2) ⊗ c(1)

(c)

ε(c(1) ) ⊗ ε(c(3) ) ⊗ c(2) = c .

(c)

Bradshaw: “Transitionary Figures”, Clothes Peg Figure Period, [Wal94, Plate 66].

8 Dual coalgebras of algebras

We have seen that the dual C ∗ of a coalgebra has a natural structure of an algebra. One might expect the dual A∗ of an algebra to be a coalgebra in an obvious way, but this is not true because of the failure of the canonical morphism / (M ⊗ N )∗ M ∗ ⊗R N ∗ R to be always invertible. If M is Cauchy the morphism is invertible since (M ⊗R N )∗

= HomR (M ⊗R N, R ) ∼ = HomR (M, N ∗ ) ∼ = M ∗ ⊗R N ∗ .

So for an algebra A which is Cauchy (as a module) we obtain a coalgebra, denoted by A∗ , via δ : A∗ ε : A∗

µ η

∗

/ (A⊗ A )∗ ∼ = A∗ ⊗R A∗ R / R∗ ∼ =R.

However, instead of restricting A, which is unsatisfactory since many of the examples are not Cauchy, we modify the definition of the dual A∗ . Let’s call an ideal I of an algebra A coCauchy when the quotient algebra A/I is Cauchy (as a module). Define A0 = { u ∈ A∗ | u is zero on some coCauchy ideal of A } . Proposition 8.1 (with R a field.) (a) A0 is a submodule of A∗ . (b) If f ∈ AlgR (A , B ) then f ∗ : B ∗ with f , takes B 0 into A0 .

/ A∗ , given by composition

(c) For any R-algebra B the canonical morphism A∗ ⊗B ∗ ∼ induces an isomorphism A0 ⊗B 0 = / (A⊗B)0 . 47

/ (A⊗B)∗

48

Quantum Groups: A Path to Current Algebra / A0 ⊗A0 satisfying

(d) There exists a unique δ : A0 δ

A0

A∗

∼ =

A0 ⊗A0

µ∗

(A⊗A)0

(A⊗A)∗ .

Proof. (a) If u ∈ A0 and r ∈ R then ker(r u) ⊇ ker u , so that r u ∈ A0 . Take u , v ∈ A0 zero on coCauchy ideals I and J respectively. We can find subspaces U , V and W of A with A = (I ∩ J) ⊕ U ⊕ V ⊕ W and I = (I ∩ J) ⊕ U and J = (I ∩ J) ⊕ V . So A/I ∼ = V ⊕ W and A/J ∼ = U ⊕W ∼ are finite dimensional. Thus A/I ∩ J = U ⊕ V ⊕ W is finite dimensional. Hence I ∩ J is a coCauchy ideal on which u + v is zero. (b) Take v ∈ B 0 zero on coCauchy J in B . Then f −1 (J) ⊆ ker(vf ) = f / B/J so ker f ∗ (v) is an ideal of A ; but f −1 (J) is the kernel of A B −1 that A/f (J) is isomorphic to a subspace of B/J . So f −1 (J) is coCauchy. (c) We shall use Exercise 8.1. Before beginning the proof of Proposition 8.1(c) notice that for any coCauchy ideal K in A⊗B we have coCauchy ideals I = {a ∈ A | a ⊗ 1 ∈ K} = ( A J = {b ∈ B | 1⊗b ∈ K} = ( B  A⊗R J + I⊗R B = ker A⊗R B

A⊗B )∗ (K)

⊗1

A⊗B )∗ (K) / (A/I)⊗ (B/J)  R

1⊗

of A and B and A⊗B respectively. Now take w ∈ (A⊗B)0 which is zero on some coCauchy K as above. Then w is zero on A⊗J + I⊗B ⊆ K . However, A⊗B/(A⊗J + I⊗B) ∼ = A/I ⊗ B/J so there exists a unique w ¯: w

A⊗B

R w ¯

ρ⊗ρ

(A/I)⊗(B/J) Furthermore since A/I and B/J are finite dimensional, so that we have ∼ = ∗ ∗ (A/I⊗B/J)∗ is invertible, there is some element (A/I)  ¯ ¯⊗(B/J) ∗ ¯ . In particular, for x ∈ A/I hi ⊗ki ∈ (A/I) ⊗(B/J)∗ corresponding to w and y ∈ B/J we have that w(x⊗y) ¯ =

¯ hi (x) k¯i (y) . i

(∗)

Dual coalgebras of algebras

49

/ A/I / R and similarly define ¯i ◦ ρ : A Define the composite hi = h / B/J / R . These are in A0 and B 0 since they are ki = k¯i ◦ ρ : B zero on I and J respectively. Hence we have  hi ⊗ ki ∈ A0 ⊗B 0 i

which is the image of w ∈ (A⊗B)0 because of (∗) . Conversely, if h ∈ A0 and k ∈ B 0 vanish on coCauchy I and J (ideals of A and B) then h ⊗ k vanishes on A⊗J + I⊗B which is a coCauchy ideal of A⊗B . (d) Suppose u ∈ A0 vanishes on a coCauchy ideal I . Then µ∗ (u)(a ⊗ b) = (uµ)(a ⊗ b) = u(ab) , so µ∗ (u) vanishes on A⊗I + I⊗A which is a coCauchy ideal of A⊗A . Hence µ∗ takes A0 into (A⊗A)0 and δ exists as desired.
Corollary 8.2 (with R a field.) For each algebra A a coalgebra structure ∗  / A∗ η on A0 is given by the δ in 8.1(d) and ε = (A0  R∗ ∼ = R). / B induces a coalgebra morphism Also each algebra morphism f : A / A0 given by restriction of f ∗ (see Proposition 8.1(b) ). f 0 : B0 Proof. Draw the diagrams expressing the axioms on µ , η and f . Simply apply ( )∗ then restrict to ( )0 .
For each algebra A we obtain a left and right A-module structure on A∗ given as follows, for a ∈ A and u ∈ A∗ : (au)(x) = u(xa) In fact A∗ : A

,

(ua)(x) = u(ax) .

 / A . For any f ∈ A∗ write Af

and f A and Af A

for the R-submodules of A∗ consisting of those elements of the form af and f a and af b respectively with a , b ∈ A . Proposition 8.3 (with R a field.) For f ∈ A∗ these are equivalent: (1) (2) (3) (4)

f ∈ A0 ; / (A⊗A )∗ ; µ∗ (f ) is in the image of A∗ ⊗A∗  / (A⊗A )∗ ; µ∗ (f ) is in the image of (A⊗A )0  Af is Cauchy; (5) f A is Cauchy; (6) Af A is Cauchy.

Proof. (1) ⇒ (3) by Proposition 8.1(d). Also (3) ⇒ (2) is trivial.  (2) ⇒ (4). Letµ∗ (f ) be the image of i ui ⊗ vi ∈ A∗ ⊗A∗ . Then we have that f (ab) = i ui (a) vi (b) so bf = i vi (b) ui ∈ A∗ . Thus bf is in the subspace of A∗ spanned by the ui . Hence Af is finite dimensional.

50

Quantum Groups: A Path to Current Algebra

(4) ⇒ (1). Suppose Af is finite dimensional. Then also EndR (Af ) is finite/ End (Af ) given by dimensional, so the kernel I of the morphism A R   / / (bf bf a) , is a coCauchy ideal of A . But a ∈ I implies 1f a = 0 , a so f (a) = 0 . Hence f is zero on I so that f ∈ A0 . (5) ⇒ (1) is similar to (4) ⇒ (1) and (6) ⇒ (5) is trivial. (1) ⇒ (6). Take f ∈ A0 zero on the cofinite ideal I . Then for c ∈ I we have (a f b)(c) = f (b c a) = 0 . Thus A f A ⊆ I ⊥ = {u ∈ A∗ | u(I) = 0} ∼ = (A/I)∗ which is finite-dimensional since A/I is so.
Corollary 8.4 For any coalgebra C the canonical injection d : C given by d(c)(u) = u(c) , has image in (C ∗ )0 .

/ C ∗∗

Proof. Take c ∈ C . Then C ∗ d(c) = {u d(c) | u ∈ C ∗ }⊆ C ∗∗ . Now (u d(c))(v) = d(c)(v ∗ u) = (v ∗ u)(c) = (v ⊗ u)δ(c) = (v ⊗ u) (c) c(1) ⊗ c(2) =  ) u(c(2) ) using the definition of multiplication v ∗ u in C . Thus (c) v(c(1)  u d(c) = (c) u(c(2) ) d(c(1) ) , which is in the subspace of C ∗∗ spanned by the d(c(1) ) . Hence C ∗ d(c) has finite dimension and by using (4)⇔(1) of
Proposition 8.3 it follows that d(c) ∈ (C ∗ )0 . Theorem 8.5 (with R a field.) For all algebras A and coalgebras C there is a bijection AlgR (A , C ∗ )   f A C∗

∼ =  /

CogR (C , A0 )  d C (C ∗ )0

f0

A0



.

/ A∗ induces i∗ : A∗∗ / A0 ∗ , while the Proof. The inclusion i : A0  / f 0 ◦ d takes g ∈ Cog (C, A0 ) to the composite inverse to f R A

d

A∗∗

i∗

A0

∗

g∗

C∗ .

The remaining details are left to the reader.




/ N is an injective module Exercise 8.1 (with R a field.) When f : M / morphism then f ⊗R 1 : M ⊗R L N ⊗R L is injective. Furthermore we / (M ⊗ N )∗ is injective. Show this. have that M ∗ ⊗R N ∗ R Exercise 8.2 Show that an algebra morphism f : A element of A0 .

/ R is a set-like

/ R which factor (in Exercise 8.3 Show that A0 consists of those u : A the category of R-modules) as u = w ◦ f , where f is surjective with Cauchy codomain. Use this to simplify some proofs in this chapter.

9 Hopf algebras

Our base ring R will always be assumed commutative, and whenever ( )0 appears we happily suppose it to be a field. An R-Hopf algebra is an R-bialgebra H together with R-module morphism ν:H

/H

called the antipode, which satisfies the following diagram. H

δ

ν ⊗1

H⊗H

1⊗ν

H⊗H

ε

µ

H

η

R For any Hopf algebra H let H op denote the Hopf algebra obtained by µ / σ / replacing µ with µ ◦ σ : H⊗H H⊗H H and replacing δ with δ / σ / σ◦δ : H H⊗H H⊗H while keeping the same η , ε and ν. There is also a bialgebra H
obtained more simply by just replacing δ δ / σ / with σ ◦ δ : H H⊗H H⊗H while keeping the same µ , η , ε and ν. In general however, this H
is not a Hopf algebra. Proposition 9.1 Let H be a Hopf algebra. Then (a) the antipode ν is uniquely determined; (b) ν : H op

/ H is a bialgebra morphism;

(c) H
is a Hopf algebra if and only if ν is bijective (moreover the antipode for H
is the inverse for ν ); (d) if H is commutative or cocommutative then ν ◦ ν = 1H (that is, ν is an involution). Proof. (a) Since H is a coalgebra and an algebra, we have the convolution algebra structure on HomR (H, H) . An antipode is precisely an inverse for 1H ∈ HomR (H, H) under convolution. For any monoid, inverses are unique. 51

52

Quantum Groups: A Path to Current Algebra

(b) To show ν : H op 

µ

H⊗H

ν

H

/ H preserves multiplication we must show that H



=



σ

H⊗H

/ H⊗H

ν ⊗ν

H⊗H

/ H .

µ

We do this by showing that, under convolution, the left-hand side is a left inverse for µ ∈ HomR (H⊗H, H) while the right-hand side is a right inverse. H⊗H δ ⊗δ

(ν◦µ)∗µ

H µ

1⊗σ⊗1

⊗4

H

⊗4

bialgebra axiom

ε⊗ε bialgebra axiom

δ

H ε

definition of convolution

µ⊗µ

H ⊗2

ν ⊗1

H

antipode axiom η

R

δ ⊗δ

H ⊗2

µ

H ⊗2

H ⊗4

1⊗σ⊗1

H ⊗4

µ⊗1⊗1

H ⊗3

δ ⊗1

1⊗σ

H ⊗3

1⊗ε

1⊗σ

H ⊗3

1⊗1⊗ε

H

δ

1⊗δ ⊗1

H ⊗4

1⊗ε⊗1

H ⊗4 1⊗η ⊗1

1⊗1

H ⊗3

1⊗1⊗ν ⊗1

H ⊗2

ε⊗ε

µ⊗1⊗1

1⊗ν ⊗1

µ⊗1⊗1

H ⊗3

1⊗µ⊗1

H ⊗3

1⊗1⊗ν

µ⊗1

H ⊗3

µ⊗1 µ⊗1

ε

H ⊗2

1⊗ν

1⊗µ

H ⊗2 µ

R

η

H

While the second commutativity is perhaps more easily seen by looking at elements, the bonus we get on using diagrams is that, formally reversing all the arrows and replacing µ and η by δ and ε , we have the proof that

Hopf algebras

53

/ H preserves comultiplication. The following diagram proves ν : H op ν preserves unit, while the dual diagram proves ν preserves counit. η

R

ν

H

H 1⊗η

η ⊗η η

H

1

δ

ν ⊗1

H ⊗2

H ⊗2

1

ε

µ η

R

H.

(c) ν
is a (composition) right inverse for ν ⇔ ν ◦ ν
= 1H ⇔ ν ◦ ν
is a convolution left inverse for ν (since 1H is the convolution inverse for ν )

⇔

H

δ

H⊗H

1⊗ν


H⊗H

ν ⊗ν

ε

µ

H⊗H

H

η

(using (b))

R

⇔

H

δ

H⊗H

1⊗ν


H⊗H

σ

µ

H⊗H

H

ν

H

η

ε

R

⇔

H

δ

H⊗H

σ

H⊗H

ε

ν
⊗1

H⊗H

µ

H

ν

H .

η

R The last condition is the condition that ν
should be a left convolution inverse for 1H
in Hom(H
, H
) , except that ν is applied to the condition. Similarly, we get that ν
is a left (composition) inverse for ν if and only if ν
satisfies the condition to be a right convolution inverse for 1H
in Hom(H
, H
) with ν applied to the condition. It follows then that ν and ν
are mutually (composition) inverse precisely when ν
and 1H
are mutually convolution inverse; that is, if and only if ν
is an antipode for H
. (d) If H is cocommutative then H
= H so that H
is a Hopf algebra with antipode ν
= ν . So ν is its own (composition) inverse; that is, ν ◦ ν = 1H .
For the commutative case replace H by H op . Remark. Proposition 9.1(d) can also be seen from the observation that commutative Hopf algebras are groups in the opposite of the category of commutative algebras, while cocommutative Hopf algebras are groups in the category of cocommutative coalgebras; the antipode is inversion so is clearly involutory.

54

Quantum Groups: A Path to Current Algebra

Proposition 9.2 Let H and K be any Hopf algebras. Then each bialgebra / K preserves antipode. morphism f : H f

H

K

ν

ν f

H

K / B are coalgebra and algebra

/ C and g : A Proof. Clearly if f : D morphisms respectively, then Hom( C , A )

/ Hom(D , B )

whereby u 

/ g◦u◦f

is a monoid morphism for the convolution structures. In particular, here we have two monoid morphisms ◦f

and f ◦

: Hom(H, H)

/ Hom(H, K)

that both take 1H to f . Monoid morphisms take inverses to inverses. So   ν◦f = convolution inverse of f in Hom(H, K) = f ◦ν .
Using other fancier words, the category HopfR of Hopf algebras is a full subcategory of the category BigR of bialgebras. For any algebra H we have seen that H 0 becomes a coalgebra. If H is a bialgebra then H 0 becomes a bialgebra using the multiplication H 0 ⊗H 0 ∼ = (H⊗H)0 and unit R ∼ = R0

ε0

δ0

H0

H0

(recall Proposition 8.1). Furthermore, if H is a Hopf algebra then so is H 0 with antipode / H0 . ν0 : H 0 What we have here is a contravariant “self-adjoint” functor ( )0 : HopfRop

/ Hopf . R

What “self-adjoint” means in this context is that BigR (H, K 0 ) ∼ = BigR (K , H 0 ) .

Hopf algebras

55

Proposition 9.3 If H is any Hopf algebra then the monoid D(H) of setlike elements is a group. Proof. For g ∈ D(H) we have   ν(g) g = µ ◦ (ν ⊗ 1) (g ⊗ g)   = µ ◦ (ν ⊗ 1) δ(g)     = µ ◦ (ν ⊗ 1) ◦ δ (g) = η ε(g) = η(1) = 1 .
An A-point of a Hopf algebra H is an algebra morphism f : H

/ A.

/ A are commuting A-points of H Proposition 9.4 (a) If f , g : H / A is (meaning that [ f (h) , g(k) ] = 0 for all h , k ∈ H ) then f ∗ g : H an A-point of H. / A is an A-point of H then f has a convolution inverse (b) If f : H / A which is an A-point of H op . f ◦ ν : H op Proof. (a) The commuting property yields that H⊗H

f ⊗g

A⊗A

µ

A

/ H⊗H is an algebra morphism is an algebra morphism. But δ : H since H is a bialgebra. So f ∗ g ∈ Alg(H, A ) . (b) Clear from Proposition 9.1(b).
Example 9.5 For a monoid G , we have seen that the monoid algebra R(G) is a bialgebra. If G is a group then the group algebra R(G) becomes / R(G) given by ν(g) = g −1 . a Hopf algebra with antipode ν : R(G) −1 / G expressed diagrammatically in Set are (The axioms for ( ) : G / Mod into the axioms which define taken by the functor FR : Set R the antipode.) Example 9.6 For a Lie algebra L , write Lop for the Lie algebra with the same module L but with Lie bracket β op given by β op (x , y ) = β( y , x ) . For any algebra A we have (Aop )L = (AL )op . It follows (why?) that we have a canonical algebra isomorphism U(Lop ) ∼ = U(L)op . / Lop taking x to −x (note that We have a Lie algebra isomorphism L / U(L)op by [ −x , −y ] = [ x , y ] = −[ y, x ] ). So we define ν : U(L)

56

Quantum Groups: A Path to Current Algebra

L

−

Lop

i

i

U(L)

U(L)op ∼ = U(Lop ) .

ν

One easily checks that for x1 , . . . , xn ∈ L ν(i(x1 ) · · · i(xn )) = (−1)n i(x1 ) · · · i(xn ) . With this antipode U(L) becomes a Hopf algebra. Example 9.7 The matrix bialgebra M(n) (Example 7.10 of a bialgebra) is not a Hopf algebra. We need to  “adjoin  an inverse for the determinant”. Recall that M(n) = R[ X ] = S FR (X) where X = {xij | i , j = 1, . . . , n} has cardinality n2 . Define  (−1)|ξ| x1 ξ(1) x2 ξ(2) . . . xn ξ(n) det(X) = ξ∈Sn

where |ξ| is the least number of simple transpositions required to obtain the permutation ξ . Form the following  commutative  polynomial R-algebra: R[ X ∪ {t} ] = R[ (xij ) , t ] = S FR (X ∪ {t}) , in n2 + 1 (commuting) indeterminates t and xij with (1 ≤ i , j ≤ n) . Put GL(n) = R[ X ∪ {t} ]/( t det(X) − 1 ) as a commutative R-algebra. We make GL(n) into a bialgebra by defining δ(xij ) =

n 

xik ⊗ xkj

δ(t) = t ⊗ t

k=1

ε(xij ) = δij

ε(t) = 1

modulo ( t det(X) − 1 ) . Put Xij = {xrs | r = i , s = j } . Now define the / GL(n) by morphism ν : GL(n) ν(xij ) = ν(t)

=

t det(Xji ) det(X)

modulo ( t det(X) − 1) . Then GL(n) becomes a Hopf algebra. For any commutative R-algebra A we have a canonical isomorphism of groups AlgR (GL(n) , A) ∼ = GL(n , A) . Examples 9.5 and 9.6 above exhibit cocommutative Hopf algebras R(G) and U(L) , while Example 9.7 is a commutative Hopf algebra GL(n) . It is only recently that the importance of Hopf algebras which are neither commutative nor cocommutative has been properly understood.

Hopf algebras

57

Example 9.8 We now describe a “quantum deformation” of Example 9.7. This is a generalization to n×n, from the 2×2 case discussed in Chapter 3. Take X = {xij | i , j = 1 , . . . , n} as in Example 9.7. First we form the   free algebra R X = T FR (X) on the (non-commuting) indeterminates xij . Let Mq (n) denote the quotient of R X by the ideal generated by the following elements: xir xjk − xjk xir xir xjk − xjk xir − (q − q xik xjk − q xjk xik

for i < j and k < r −1

) xik xjr

for i < j and r < k for i < j

xik xir − q xir xik

for k < r .

This becomes a coalgebra with comultiplication δ(xij ) =

n 

xir ⊗ xrj

( modulo the ideal )

r=1

and counit ε(xij ) = δij

( Kronecker delta ) .

Define the “quantum determinant” by  det q (X) = (−q)|ξ| x1 ξ(1) x2 ξ(2) · · · xn ξ(n) ξ∈Sn

which is a central element of Mq (n) (that is, it commutes with all other elements). The quantum general linear group is defined by GL q (n) = Mq (n)[ t ]/( t det q (X) − 1 ) . We adjust the comultiplication and counit of Mq (n) by defining δ(t) = t ⊗ t and ε(t) = 1 . Then we have a bialgebra epimorphism ρ : Mq (n) Define ν : GL q (n)

/ GL (n) . q

/ GL (n) by q

ν(xij ) = t det q (Xji )

and

ν(t) = det q (X) .

Then GL q (n) becomes a Hopf algebra. Notice that GL q (n)op = GL q-1(n) . Many claims have been made in this section. For n = 2 the calculations in Theorem 3.1 prove them all. (This should be compared with Proposition 9.4 in the present section.) The general case can be verified similarly, but will follow from later work.

58

Quantum Groups: A Path to Current Algebra

Exercise 9.1 [Swe69] Assume our base ring R is a field and write ⊗ for ⊗R . An ideal in an algebra A is a submodule I such that µ(I⊗A + A⊗I) ⊆ I . We know that A/I becomes an algebra. A coideal of a coalgebra C is a submodule I such that δ(I) ⊆ I⊗C + C⊗I and ε(I) ⊆ 0 . (a) If I is a coideal of a coalgebra C, describe a coalgebra structure on / C/I becomes a coalgebra morphism. If C is C/I for which ρ : C a bialgebra and I is also an ideal, show that C/I is a bialgebra. What condition on I ensures C/I has an antipode if C has? I is called a Hopf ideal when this holds. (b) Verify that the polynomial R-algebra B = R x, y, z on three noncommuting indeterminates becomes a bialgebra with δ(x) = x⊗x ,

δ(y) = y⊗y ,

ε(x) = ε(y) = 1 ,

δ(z) = 1⊗z + z⊗x ε(z) = 0 .

(c) Verify that the ideal (xy − 1 , yx − 1) is a coideal in B. Let H denote the quotient bialgebra. (d) Show that H is a Hopf algebra with antipode ν given by ν(x) = y , ν(y) = x , ν(z) = −zy (modulo the ideal of (c) ). Show further that ν 2n (z) = xn zy n , ν 2n+1 (z) = −xn zy n+1 . Hence, this antipode has infinite order. (e)

i. Show that the ideals In = (xn zy n − z), Jn = (xn − 1) are Hopf ideals in H. ii. Show that the antipodes of both H/In and H/Jn have order 2n .

Bradshaw: Elegant Action Figure Group, [Wal94, Plate 84].

10 Representations of quantum groups

We mentioned in Example 6.3 that a representation of a group G was an R(G)-module. One kind of representation for a Hopf algebra H therefore suggests itself: a module over H . We begin by discussing modules over bialgebras. / A is a ring morphism then each (left) First note that if f : E A-module M becomes a (left) E-module via the action for e ∈ E , m ∈ M .

e m = f (e)m

This is called restriction of scalars along f . Let A be an R-algebra. Then each module is automatically an R/ A . Alternamodule via restriction of scalars along the unit η : R tively, we can view an A-module as an R-module M with a ring morphism / End (M ) . Later, we want to look at “comodules”, and so we µ ˆ: A R want a definition of A-module which dualizes. The good version is: an R/ M , called the action module M with a module morphism µ : A⊗R M of A on M , satisfying A⊗R A⊗R M

µ⊗1

A⊗R M

1⊗µ

µ

M

A⊗R M η ⊗1

M

µ 1

M.

We write ModR (A) for ModA just to emphasize that we build it up from ModR . Suppose M and N are (left) modules over the R-algebra A. Re / op  / op  / R and N : R A . We see that M ⊗R N : A A , gard M : A which means M ⊗R N becomes an A⊗A-module. If A is a bialgebra then we 59

60

Quantum Groups: A Path to Current Algebra

/ A⊗ A to obtain an A-module structure can restrict scalars along δ : A R on M ⊗R N . Explicitly, the action is the composite A⊗R M ⊗R N

δ ⊗1⊗1

A⊗R A⊗R M ⊗R N

1⊗σ⊗1

A⊗R M ⊗R A⊗R N

µ ⊗µ

M ⊗R N .

This generalizes to multiple tensor products (over R) of A-modules. In particular, the empty tensor product R becomes an A-module by restricting / R. scalars along the counit ε : A  / op A With M and N left A-modules as before, we can regard M : R  / op op  / op and N : R A , so that HomR (M, N ) : A A ; or in other words HomR (M, N ) becomes an Aop ⊗A-module. Thus if A = H is a Hopf algebra, we can restrict scalars along the R-algebra morphism δ

H

H⊗R H

ν ⊗1

H op ⊗R H

to make HomR (M, N ) into an H-module. Explicitly, the action of H on HomR (M, N ) is the composite H⊗R HomR (M,N )

δ ⊗1

H⊗R HomR (M,N )⊗R H

1⊗σ

H⊗R H⊗R HomR (M,N )

µ1 ⊗ν

HomR (M,N )⊗R H

µ2

HomR (M,N )

where µ1 and µ2 are the left and right actions: H⊗R HomR (M,N )

µ ˆ ⊗1

HomR (N,N )⊗R HomR (M,N )

◦

µ1 : h ⊗ f HomR (M,N )⊗R H

1⊗µ ˆ

HomR (M,N )⊗R HomR (M,M )

µ2 : f ⊗ h

◦

HomR (M,N ) / h(f m)) (m  HomR (M,N ) / f (h m)) . (m 

Proposition 10.1 For left modules M and N over the Hopf algebra H, the canonical R-module morphisms /N e : HomR (M, N )⊗R M / Hom (N, M ⊗ N ) d:M R R

where where

/ f (m) f ⊗m  / (n  / m ⊗ n) m 

are left H-module morphisms. Proof. Omitting ⊗R and HomR from the notation, we obtain the first of

Representations of quantum groups

61

these from the following diagram. The second we leave to the reader. δ11

H(M N )M

HH(M N )M 1δ11

1σ1

δ111

δ111

1ε11

11σ1

HHH(M N )M

1

H(M N )HM

HH(M N )HM

1ν111

H(M N )M

HHH(M N )M 1η11

11η1

1σ11 11σ1

HH(M N )HM

H(M N )HHM 1σ11

1µ11

HH(M N )M

11ν11

H(M N )HHM 11µ1

1σ1

1µ2 11

1

H(M N )HM

H(M N )HM 1µ2 1

1µ2 1

H(M N )M

µ1 1

1e

HN

(M N )M

µ1 11 1µ

(M N )HM

e µ

N


Corollary 10.2 For modules M , N and L over a Hopf algebra H, the canonical isomorphism HomR (M ⊗R N , L) ∼ = HomR (M , HomR (N, L)) restricts to an isomorphism HomH (M ⊗R N , L) ∼ = HomH (M , HomR (N, L)) . Proof. The canonical isomorphism is obtained from the evaluation e and the canonical d of Proposition 10.1.
In other words, we have a nice tensor–hom situation for the category ModR (H) of (left) H-modules. Both the tensor and the hom are preserved by the functor / Mod ModR (H) R given by ignoring the H-action.

62

Quantum Groups: A Path to Current Algebra

Although modules over the group algebra are representations of the group, so that the study of modules over a Hopf algebra does suggest itself, the point of view of Chapter 2 (i.e. space–algebra duality) leads more naturally / H⊗H of the to “comodules”. For here, it is the comultiplication δ : H Hopf algebra which corresponds to the spatial multiplication. Suppose C is an R-coalgebra. A (left) C-comodule is an R-module / C⊗ M , called the coaction of C M with a module morphism δ : M R on M , satisfying M

δ

δ ⊗1

C⊗R M

1⊗δ

C⊗R C⊗R M

C⊗R M ε⊗1

δ 1

M

M.

We write ComR (C) for the category whose objects are C-comodules and whose arrows are C-comodule morphisms; that is, R-module morphisms / N such that f : M M

δ

C⊗R M 1⊗f

f

N

δ

C⊗R N .

Each C-comodule M becomes a C ∗ -module with the action C ∗ ⊗R M

1⊗δ

C ∗ ⊗R C⊗R M

e⊗1

M.

See Chapter 7 for the algebra structure on C ∗ . By the fundamental theorem of Morita theory (Theorem 5.2), if C is Cauchy (as an R-module) then this gives a bijection between C-coactions δ and C ∗ -actions µ on each R-module M : recover δ as the composite M

d ⊗1

C⊗R C ∗ ⊗R M

1⊗µ

C⊗R M .

So for C Cauchy, we have an isomorphism of categories ComR (C) ∼ = ModR (C ∗ ) . If C is an R-bialgebra not necessarily Cauchy we obtain, in a manner dual to that for modules, a coaction on the tensor product (over R) of

Representations of quantum groups

63

C-comodules. Explicitly for C-modules M and N , the coaction for M ⊗R N is given by the composite M ⊗R N

δ ⊗δ

C⊗R M ⊗R C⊗R N

σ1324

C⊗R C⊗R M ⊗R N

The empty tensor product R has the coaction η : R

µ⊗1⊗1

C⊗R M ⊗R N .

/ C⊗ R . R

When it comes to Hom our formal duality fails: in reversing arrows we have maintained ⊗, yet Hom does not maintain its universal property. However, if M is Cauchy, HomR (M, N ) does have the reverse-arrow universal property: there is a bijection between R-module morphisms HomR (M, N )

L and R-module morphisms

M ⊗L

N

since HomR (M, N ) ∼ = M ∗ ⊗R N and M ⊗R L ∼ = HomR (M ∗ , L). Proposition 10.3 Each Cauchy R-module M gives rise to an R-coalgebra / R and comultiplication M ⊗R M ∗ with counit e : M ⊗R M ∗ / M ⊗ M ∗⊗ M ⊗ M ∗ R R R

1 ⊗ d ⊗ 1 : M ⊗R M ∗

(see Theorem 5.2). For any R-coalgebra C, the assignment δˆ = (M ⊗R M ∗

δ ⊗1

C⊗R M ⊗R M ∗

1⊗e

C)

determines a bijection between coactions δ:M

/ C⊗ M R

of C on M and coalgebra morphisms δˆ : M ⊗R M ∗

/C .

M ∗ ⊗R M  HomR (M, M ) has the universal Proof. M ⊗R M ∗ σ ρ property of Hom under reversal of arrows; so the diagrammatic proof that EndR (M ) is an algebra and that an action is an algebra morphism of the / End (M ) , dualizes. kind A
R Take M = Rn in the above proposition and let e1 , . . . , en be the standard basis. Now let e∗1 , . . . , e∗n be the dual basis for Rn ∗ so e∗i (ej ) = δij

64

Quantum Groups: A Path to Current Algebra

(Kronecker-δ). A coaction of C on Rn thus amounts to a coalgebra morph/ C , and this is determined by its values on the basis ism δˆ : Rn ⊗R Rn ∗ ∗ n elements ei ⊗ ej of R ⊗R Rn ∗ : ˆ ⊗ e∗ ) = x ∈ C . δ(e j i ij So C-comodule structures on Rn are in bijection with multiplicative matrices in C; that is, matrices x = (xij ) in C satisfying  δ(xij ) = xik ⊗ xkj , ε(xij ) = δij . k

Following Manin [Man88], we write the last two equations as ε(x) = i  where i is the identity matrix and x ⊗y = ( k xik ⊗ ykj ) is not the usual tensor product of matrices.

x 0 Example 10.4 In the situation of Example 9.8, x = (xij ) and 0 t are multiplicative matrices for Mq (n) and GL q (n), respectively. δ(x) = x ⊗x ,

Now suppose C = H is a Hopf algebra and M is a Cauchy R-module. By applying Proposition 10.1 to M ∗ (and using the canonical M ∗∗ ∼ = M ), we see that M ∗ ⊗R M becomes a coalgebra with counit σ

M ∗ ⊗R M

e

M ⊗R M ∗

R

and comultiplication M ∗ ⊗R M

1⊗d⊗1

M ∗ ⊗R M ∗ ⊗R M ⊗R M

σ

M ∗ ⊗R M ⊗R M ∗ ⊗R M .

/ H is a coalgebra morphism if and

Proposition 10.5 (a) δˆ : M ⊗R M ∗ only if the composite M ∗ ⊗R M

1⊗σ⊗1

δˆ

M ⊗R M ∗

H op

is a coalgebra morphism. (b) Suppose M is an H-comodule and put  δˆk = M ⊗R M ∗

δˆ

νk

H

 H .

For k even, δˆk is a coalgebra morphism and has convolution inverse δˆk+1 in HomR (M ⊗R M ∗ , H) .

Representations of quantum groups

65

Proof. (a) M ∗M

σ

MM∗

1⊗d⊗1

M ∗M ∗M M

H

1⊗d⊗1 σ4231

M M ∗M M ∗

δ δˆ⊗δˆ

σ3412

1⊗σ⊗1

M ∗M M ∗M

δˆ

σ ⊗σ

M M ∗M M ∗

H⊗H σ

δˆ⊗δˆ

H⊗H

M ∗M

σ

MM∗

δˆ

H

σ

MM∗

e

ε

e

R

/ H op is a coalgebra morphism, by Proposition 9.1(b); (b) Since ν : H / H is a coalgebra morphism for k even; then it is also true that ν k : H k ˆ ˆ so δk = ν ◦ δ is a coalgebra morphism, as required. This also means that right composition with δˆk preserves convolution. Since 1H and ν are convolution inverses, so are 1H ◦ δˆk and ν ◦ δˆk ; that is, so are δˆk and δˆk+1 .
Proposition 10.6 Suppose that M is a comodule over the Hopf algebra H and that M is Cauchy as an R-module. Then M ∗ becomes an H-comodule via   δˆ σ ν H . δˆ = M ∗ ⊗ M M⊗ M∗ H R

R

Moreover, the R-module morphisms /R e : M ⊗R M ∗ / M ∗⊗ M d:R R become H-comodule morphisms. One might therefore say that M becomes a Cauchy H-comodule. Proof. By Propositions 10.5(a) and 9.1(b), the stated δˆ is a coalgebra morphism. So by Proposition 10.5, it determines a coaction of H on M ∗ . Tracing through, one sees that this is dual to the situation for HomR (M, R) as in Proposition 10.1; so the proof dualizes, but it can also be shown directly that e and d are comodule morphisms.
Remark. To obtain results as in Proposition 10.5(b) for k odd, apply Proposition 10.5(b) to the M ∗ of Proposition 10.6; compare with [Man88, p.14]. Exercise 10.1 Give a direct proof of Proposition 10.3, concerning the coalgebra structure on M ⊗R M ∗ where M is a Cauchy R-module.

66

Bradshaw: Sash Bradshaw Group, [Wal94, Plate 52].

11 Tensor categories

It is clear that specific categories have entered explicitly into the above discussion, but we have made little use of them as categories apart from diagrams and duality. For what follows it is hard to imagine how to express the results without categories. A tensor category (also called “monoidal category” [EK66]) is a category V / V called tensor product, an object I together with functor ⊗ : V × V of V called the unit object , and natural families of isomorphisms / A⊗(B⊗C) aA,B,C : (A⊗B)⊗C / A , l : I⊗A /A rA : A⊗I A called respectively the associativity constraint, the right unit constraint and the left unit constraint, subject to the two conditions: (A⊗B)⊗(C⊗D) aA⊗B,C,D

aA,B,C⊗D

((A⊗B)⊗C)⊗D

A⊗(B⊗(C⊗D)) 1⊗aB,C,D

aA,B,C ⊗1

(A⊗(B⊗C))⊗D

A⊗((B⊗C)⊗D) aA,B⊗C,D

(A⊗I)⊗C

aA,I,C

A⊗(I⊗C) 1⊗lC

rA ⊗1

A⊗C . Define A1 ⊗ · · · ⊗An to be the object obtained by inserting brackets in some chosen preassigned way, such as from the left ((· · · (A1 ⊗A2 )⊗ · · · )⊗An ) . 67

68

Quantum Groups: A Path to Current Algebra

It is an important fact (Mac Lane’s coherence theorem) that, in general, the only automorphism which is a composite of isomorphisms of the form 1 ⊗ (x ⊗ 1) or (1 ⊗ x) ⊗ 1 , where x is a component of a, r, l or their inverses, is the identity arrow of A1 ⊗ · · · ⊗An . This essentially allows one to work as if the a, r, l are all identities. If all the a, r, l are indeed identities, then the tensor category is called strict . The opposite V op of a tensor category V consists of the opposite category of V (obtained by reversing the direction of arrows of V) and the reverse tensor product, so that A⊗B in V op is just B⊗A in V. A braiding for a tensor category V is a natural family of isomorphisms / B ⊗A : A⊗B c A,B

subject to the conditions c

A⊗(B⊗C)

A,B ⊗ C

(B⊗C)⊗A aB,C,A

aA,B,C

(A⊗B)⊗C

B⊗(C⊗A)

cA,B ⊗1

1⊗cA,C

(B⊗A)⊗C (A⊗B)⊗C

aB,A,C

c ⊗ A B,C

B⊗(A⊗C) C⊗(A⊗B)

−1 aA,B,C

−1 aC,A,B

A⊗(B⊗C)

(C⊗A)⊗B cA,C ⊗1

1⊗cB,C

A⊗(C⊗B)

−1 aA,C,B

(A⊗C)⊗B .

A braided tensor category is a tensor category with a chosen braiding (see [JS93]). A symmetry for a tensor category is a braiding which satisfies the following extra condition: 1 A⊗B A⊗B cA,B

cB,A

B ⊗A .

Tensor categories

69

A symmetric tensor category is a tensor category with a chosen symmetry. Example 11.1 The braid category B has as objects the natural numbers / n the braids on n strings; there are no 0, 1, 2, . . . and as arrows α : n / n for m = n . A braid α arrows m 5

•

•

•

•

•

5

•

•

•

•

•

α

on n strings can be regarded as an element of the Artin braid group Bn with generators s1 , . . . , sn−1 subject to the relations si sj

for j < i − 1

= sj si

si+1 si si+1

= si si+1 si

where si is the braid depicted as: n

1 •

i−1 •

i •

n

•

•

•

n •

i+1 i+2 • •

si •

•

•

Composition of braids is just multiplication in this group, represented diagrammatically by vertical stacking of braids with the same number of strings. • • • 3 β 3

•

•

•

3

•

•

•

γ

Tensor product of braids adds the number of strings by placing one braid next to the other longitudinally. •

•

•

•

•

•

⊗ •

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

= •

•

70

Quantum Groups: A Path to Current Algebra

This makes B a strict tensor category. A braiding c m,n : m + n is given by crossing the first m strings over the remaining n . m 4+2

/ n+m

n

•

•

•

•

•

•

•

•

•

•

•

•

c4,2 2+4

The axioms that show B is braided are easily checked diagrammatically. Example 11.2 The category ModR of modules over a commutative ring R is a symmetric tensor category with tensor product ⊗R , with the canon/ B⊗ A . ical constraints, and with symmetry σ : A⊗R B R Example 11.3 Let A be an R-bialgebra. If M and N are A-modules, we have an A-module structure on M ⊗R N given by  a · (m ⊗ n) = a(1) m ⊗ a(2) n (a)

as seen in the last chapter. So ModR (A) becomes a tensor category with tensor product ⊗R . / N⊗ M If A is cocommutative, the switch morphism σ : M ⊗R N R is a symmetry for ModR (A) . However, as in the rest of this book, we are more interested in non-cocommutative A . We ask: what are the possible braidings on the tensor category ModR (A)? / N ⊗ M gives, for each A , a morphism A braiding cM,N : M ⊗R N R / A⊗ A which gives an element γ = c (1 ⊗ 1) ∈ A⊗A . cA,A : A⊗R A R A,A  Conversely, each element γ = i ui ⊗ vi ∈ A⊗A determines a natural / N ⊗ M via the formula morphism cM,N : M ⊗R N R  cM,N (m ⊗ n) = (ui n) ⊗ (vi m) . i

This is a bijection, as can be seen from the following diagram in which m ˆ : / M is the unique module morphism with m(1) A ˆ = m. R⊗R R

η ⊗η

A⊗R A

cA,A

m ˆ ⊗n ˆ

M ⊗R N

A⊗R A n ˆ ⊗m ˆ

cM,N

N ⊗R M .

Tensor categories

71

In order for each cM,N to be an isomorphism it is necessary for γ ∈ A⊗R A to be invertible. In order for each cM,N to be a module morphism we need    c a · (m ⊗ n) = c (a(1) m) ⊗ (a(2) n) (a)

=

 (ui a(2) n) ⊗ (vi a(1) m) i

(a)

to be equal to a · c(m ⊗ n) = a ·



(ui n) ⊗ (vi m)

i

=

 (a)

(a(1) ui n) ⊗ (a(2) vi m) .

i

This is equivalent to the requirement   (ui a(2) ) ⊗ (vi a(1) ) = (a(1) ui ) ⊗ (a(2) vi ) . i,(a)

i,(a)

/ A⊗ A whose value at Regarding γ ∈ A⊗R A as a morphism γ : R R 1 ∈ R is the given γ , we can express this condition diagrammatically as (B0)

A

γ ⊗δ

A⊗4

σ1423 σ3142

A⊗4

µ⊗µ

A⊗2 .

For a braiding, we require two more conditions: cM,N ⊗L (m ⊗ n ⊗ l) = (1N ⊗ cM,L )(cM,N ⊗ 1L )(m ⊗ n ⊗ l) , cM ⊗N,L (m ⊗ n ⊗ l) = (cM,L ⊗ 1N )(1M ⊗ cN,L )(m ⊗ n ⊗ l) ; that is, 

ui(1) n ⊗ ui(2) l ⊗ vi m =

ui n ⊗ uj l ⊗ vj vi m ,

i,j

i,(ui )





ui l ⊗ vi(1) m ⊗ vi(2) n =



uj ui l ⊗ vj m ⊗ vi n .

i,j

i,(vi )

These are equivalent to the two conditions   ui(1) ⊗ ui(2) ⊗ vi = ui ⊗ uj ⊗ vj vi , i,j

i,(ui )



i,(vi )

ui ⊗ vi(1) ⊗ vi(2) =

 i,j

uj ui ⊗ vj ⊗ vi .

72

Quantum Groups: A Path to Current Algebra

Diagrammatically, these conditions become: (B1)

γ ⊗γ

R

A⊗4

σ1342

γ

1⊗1⊗µ δ ⊗1

A⊗2

(B2)

A⊗4

γ ⊗γ

R

A⊗4

A⊗3 σ3142

γ

A⊗4 µ⊗1⊗1

1⊗δ

A⊗2

A⊗3 .

Hence, we define a braiding element for a bialgebra A to be an invertible element γ ∈ A⊗R A which satisfies (B0), (B1), (B2). We have proved above that braiding elements for A are in bijection with braidings on the tensor category ModR (A) . A braided bialgebra (also called “quasitriangular bialgebra”) is a bialgebra equipped with a braiding element γ ∈ A⊗R A. A braiding element γ is called a symmetry element when γ 2 = 1 ∈ A⊗R A; these are in bijection with symmetries on ModR (A). A symmetric bialgebra (also known as a “triangular algebra”) is a bialgebra equipped with a symmetry element. Before leaving this example, we point out that conditions (B1), (B2) can be put in a more familiar form in thecase where A is Cauchy as an Rmodule. For in this case, elements γ = i ui ⊗ vi ∈ A⊗R A are in bijection / A via the formula with R-module morphisms g : A∗ γ=



R

d

A∗ ⊗R A

g⊗1A

A⊗R A



.

Condition (B1) precisely says that g preserves comultiplication, while condition (B2) says that g reverses multiplication. In fact, if γ is a braiding / A
op is a bialgebra morphism; preservation of unit and element, g : A∗ counit follows from cM,I = cI,M = 1M . We shall just look at the translation of (B2) to g. Begin with the defining diagram R

d

A∗ ⊗R A

A∗ ⊗R A∗ ⊗R A⊗R A δ ∗ ⊗1⊗1

d

A∗ ⊗R A

1⊗d⊗1

1⊗δ

A∗ ⊗R A⊗R A

Tensor categories

73

for δ ∗ , which is the multiplication for A∗ . To prove g reverses multiplication is to prove A∗ ⊗R A∗

g ⊗g

A⊗R A

σ

A⊗R A

d∗

µ g

A∗

A.

This is equivalent to proving the legs are equal after applying and composing with R

d

A∗ ⊗R A∗

1⊗d⊗1

⊗R A⊗R A

A∗ ⊗R A∗ ⊗R A⊗R A .

From the defining diagram for δ ∗ , this amounts to R

d

A∗ ⊗R A∗

1⊗d⊗1

A∗ ⊗R A∗ ⊗R A⊗R A

g⊗g⊗1⊗1

A⊗4

σ

A⊗4

µ⊗1⊗1

1⊗δ g⊗1⊗1

A∗ ⊗R A⊗R A

A⊗3 .

Using γ = (g ⊗ 1A ) ◦ d , we easily see that this is equivalent to (B2). Although a braiding is as useful as a symmetry for most purposes, there is sometimes further structure on a braiding which makes it even more like a symmetry without actually forcing it to be one. Suppose V is a braided tensor category. A twist for V is a natural family of isomorphisms /A θA : A such that θI = 1I and A⊗B

cA,B

B⊗A θB ⊗θA

θA⊗B

A⊗B

cB,A

B⊗A .

A balanced tensor category is a braided tensor category with a chosen twist. (A braiding is a symmetry if and only if the identity arrows provide a twist.)

74

Quantum Groups: A Path to Current Algebra

Example 11.4 The braid category B is canonically balanced. The twist / n is obtained by taking n vertical parallel strings with ends tied θn : n to two horizontal parallel rods, and rotating the bottom rod through a full 2π twist in the right-hand screw direction with thumb vertical. Then θ0 , θ1 are identities, while θ2 (which can be written as (s1 )2 using the notation from Example 11.1) is: 2

•

•

2

•

•

θ2

Example 11.5 There is a tensor category B˜ which is defined similarly to B, except that the arrows are braids on ribbons (instead of on strings) and it is permissible to twist the ribbons through full 2π turns (as in the following diagram). 3

α

3 ˜ n) = B˜n are groups under composition. A presentaThe homsets B(n, tion of this group B˜n is given by generators s1 , . . . , sn where s1 , . . . , sn−1 satisfy the relations as for Bn . These are depicted by thickened versions of the diagrams in Example 11.1, along with the extra relation sn−1 sn sn−1 sn = sn sn−1 sn sn−1 where sn is depicted as follows 1

2

···

n

n−1

n

···

sn n

Composition in B˜ is vertical stacking of diagrams, and tensor product for B˜ is horizontal placement of diagrams, much as for B. The braiding

Tensor categories

75

/ n+m for B˜ is obtained by placing the first m ribbons over c m,n : m+n /n the remaining n without introducing any twists. Now the twist θn : n for B˜ is obtained by regarding the two boundary edges of the ribbons as extra / 2n in B. Then in B˜ we have strings and taking θ2n : 2n 1

2

θ1

θ2

1

2

/ B be a Example 11.6 Let A and B be abelian groups and f : A × A bilinear function. There is a balanced strict tensor category Cf constructed as follows. The objects are the elements of A. The homset Cf (x, y) is empty unless x = y , in which case Cf (x, x) = B . The tensor product is given by β / α+β α / (x x) ⊗ (y y) = (x + y x + y) . / The braiding is c x,y = f (x, y) : x + y y + x and the twist is given by / θ = f (x, x) : x x. X

Example 11.7 Let A be a braided R-bialgebra with braiding element γ =  i ui ⊗ vi ∈ A⊗R A . A twist element for A is an invertible central element τ ∈ A such that ε(τ ) = 1 and  (ui τ vj ) ⊗ (vi τ uj ) . δ(τ ) = i,j

Diagrammatically the last equation becomes: R

γ ⊗τ ⊗τ ⊗γ

A⊗6

σ136245

A⊗6 µ⊗1µ⊗1

A⊗4

τ

µ⊗µ

A

δ

A⊗2 .

Twist elements τ for A are in bijection with twists θ for the braided tensor / M means it has the form category ModR (A) . Naturality of θM : M

76

Quantum Groups: A Path to Current Algebra

θM (m) = τ m for some τ ∈ A ; for θM to be an A-module morphism, τ needs to be central (meaning τ · a = a · τ for all a ∈ A); for θM to be an isomorphism, τ needs to be invertible; for θM = 1R , the condition ε(τ ) = 1 is needed; and of course the remaining twist conditions correspond. A balanced bialgebra is a braided bialgebra with a twist. Exercise 11.1 (a) In a braided tensor category V show that (ignoring the constraints a, l, r) cA,I = cI,A = 1A and A⊗C⊗B

c⊗1

C⊗A⊗B

1⊗c

1⊗c

A⊗B⊗C

C⊗B⊗A c⊗1

c⊗1

B⊗A⊗C

1⊗c

B⊗C⊗A

(b) For a braided bialgebra A and V = ModR (A), interpret the properties of c in (a) in terms of the braiding element γ ∈ A⊗R A. (c) Draw diagrams of braids which express the hexagonal diagram of (a) in the braid category B. Exercise 11.2 Define the centre Z V of a tensor category V to be the cat/ ⊗A egory whose objects are pairs (A, a) where A ∈ V and a : A⊗ is a natural isomorphism such that the following conditions hold: • aI = 1 (more precisely, aI is the composite of the canonical isomorphisms A⊗I ∼ =A∼ = I⊗A). • aX⊗Y = (1 ⊗ aY ) ◦ (aX ⊗ 1) for all X, Y ∈ V. / (B, b) in Z V is an arrow f : A An arrow f : (A, a) for all X ∈ V, we have bX ◦ (f ⊗ 1) = (1 ⊗ f ) ◦ aX .

/ B such that,

(a) Show that Z V becomes a tensor category with:   (A, a)⊗(B, b) = A⊗B, (a ⊗ 1) ◦ (1 ⊗ b) . (b) Show that the tensor category Z V is braided via c(A,a),(B,b) = aB : (A, a)⊗(B, b)

/ (B, b)⊗(A, a) .

Exercise 11.3 Show that the first dot-point condition on objects of Z V is redundant.

12 Internal homs and duals

Suppose V is a tensor category with A and B being objects of V. A (left) internal hom for A and B consists of an object [A , B ] of V together with an arrow /B (called evaluation) e : [A , B ]⊗A A

/ B , there exists a unique arrow fˆ :

such that, for all arrows f : C⊗A / [A , B ] with C

f

=



C⊗A

fˆ⊗1A

eA

[A , B ]⊗A

B



.

Thus we have a natural bijection V(C , [A , B ])

∼ =

V(C⊗A , B) eA ◦ (g ⊗ 1A ) .

g

A tensor category is called left-closed when each pair of objects has a left / D are arrows of V then, / A and g : B internal hom. If f : C provided the internal homs exist, there is a unique arrow [f , g ] : [A , B ]

/[C ,D ]

such that [f,g]⊗1C

[A , B ]⊗C

[ C , D ]⊗C

1[A,B] ⊗f

eC

[A , B ]⊗A

eA

B

g

D.

In this way, when V is left-closed the internal hom becomes a functor [ , ] : V op × V 77

/ V.

78

Quantum Groups: A Path to Current Algebra

From the universal property, the internal hom for A, B is unique up to isomorphism. An internal hom for I, A always exists; namely [I ,A] = A

with

/A .

eA = rA : A⊗I

If B, C and A⊗B, C have internal homs then so do A, [B , C ]; namely, /[B ,C ] .

[A , [ B , C ] ] = [A⊗B , C ] with eA = eˆA⊗B : [A⊗B , C ]⊗A

Usually the internal hom functor [ , ] is given a priori; in which case all we have are canonical isomorphisms [I ,A] ∼ =A

[A , [ B , C ] ] ∼ = [A⊗B , C ] .

and

There is a composition arrow / [A , C ]

[ B , C ]⊗[A , B ]

in V (whenever the internal homs exist) which corresponds, by using the universal property of [A , C ] , to the composite [B , C ]⊗[A , B ]⊗A

1⊗eA

[B , C ]⊗B

eB

C .

A right internal hom [A , B ]
for A, B comes equipped with an arrow eA
: A⊗[A , B ]


/B

which induces a bijection V(C , [A , B ]
) ∼ = V(A⊗C , B) for all objects C. If V is braided, each left internal hom [A , B ] gives a right internal hom via [A , B ]
= [A , B ] and eA
=



A⊗[A , B ]

cA,[A,B]

[A , B ]⊗A

eA

B



.

A tensor category is called closed when all left- and right-internal homs exist. (In the literature, “closed” is sometimes used for our left-closed, while “biclosed” is used for our closed.) When the internal homs exist, we have an arrow / [ [A , B ] , B ]
ωA
: A which corresponds to eA : [A , B ]⊗A ωA : A corresponding to eA
.

/ B via e
[A,B] . Similarly, we have

/ [ [A , B ]
, B ]

Internal homs and duals

79

Example 12.1 The symmetric tensor category ModR of modules over a commutative ring R is closed with internal homs given by [ M, N ] = [ M, N ]
= HomR (M, N ) . Example 12.2 Suppose H to be an R-Hopf algebra. Then the tensor category ModR (H) of H-modules (with tensor product ⊗R ) is left-closed with internal hom given by [ M, N ] = HomR (M, N ) ; see Proposition 10.1. Suppose the antipode ν for H is invertible. Using Proposition c(c),
(M, N ) for H
becomes a Hopf algebra having antipode ν −1 . Write HomR
HomR (M, N ) as a left H -module. Clearly, ModR (H) is right-closed with
[ M, N ]
= HomR (M, N ). Therefore ModR (H) becomes closed when ν is invertible. The forgetful functor ModR (H)

/ Mod R

preserves tensor product, and both left and right internal homs. Example 12.3 Let Ban denote the category of Banach spaces (over the / B are linear functions for complex numbers), where the arrows f : A which f (a) ≤ a . (The analysts in the audience will think these fairly uninteresting functions.) We make Ban into a symmetric tensor category by taking tensor products as vector spaces, completing in the obvious way. The internal hom [A , B ] exists for all Banach spaces A, B; it is the Banach space of bounded linear functions from A to B with the usual norm. (These functions are of more interest to the analyst.) Thus Ban is a closed symmetric tensor category. Suppose V is a tensor category. An object J is called left dualizing when, for all objects A, internal homs [A, J ], [A, J ]
exist, and the arrow ωA : A

/ [ [A , J ]
, J ]

is invertible. It follows that V is left-closed with [A , B ] = [A⊗[B , J ]
, J ] , since V(C , [A⊗[B, J ]
, J ] ) ∼ = ∼ =

V(C⊗A⊗[B, J ]
, J) V(C⊗A , [ [B, J ]
, J ]) ∼ =

V(C⊗A , B) .

80

Quantum Groups: A Path to Current Algebra

The concept of right dualizing object is defined in the same way, with ω replaced by ω
. A dualizing object is one which is both left and right dualizing. Example 12.4 The category of finite-dimensional vector spaces over a field has a dualizing object, namely the field itself. In this case, the dualizing object is the unit for tensor product.   Example 12.5 Fix a field k . A quadratic algebra is a pair V , R where V is a finite-dimensional vector R is a subspace of V ⊗V .  space  and / W , S  is a linear function A quadratic algebra morphism f : V , R / W for which f : V (f ⊗ f )(R) ⊆ S . Write  QA for the category of quadratic algebras. Each quadratic algebra V , R determines an actual algebra T (V )/R where T (V ) is the tensor algebra on V (see Example 6.2). The category QA has a symmetric tensor product given by       V , R ⊗ W , S = V ⊗W , σ1324 (R⊗S) .   The unit object is I = k , k ⊗k . We claim that J = k , 0 is a dualizing object. It is easy to see that     [ V , R , J ] = V ∗ , R⊥ where V ∗ = Hom k (V, k) and R⊥ is the kernel of the composite surjection V ∗ ⊗V ∗ with i : R

∼ =

(V ⊗V )∗

i∗

R∗

/ V ⊗V being inclusion. It follows that R is the kernel of

V ⊗V

ω ⊗ω

V ∗∗ ⊗V ∗∗

so we have an isomorphism   ω : V ,R

∼ =

(V ∗ ⊗V ∗ )∗

i∗

R⊥ ∗

/ V ∗∗ , R⊥⊥  .

The quadratic algebra which we identify with the quantum plane A2|0 q (recall Chapter 3) is   = k2 , (y ⊗ x − q x ⊗ y) A2|0 q

Internal homs and duals

81

where x = (1, 0), y = (0, 1) ∈ k2 . Let ξ , η ∈ k2 ∗ be the dual basis given by ξ(x) = η(y) = 1, ξ(y) = η(x) = 0. Then (y ⊗ x − q x ⊗ y)⊥ = ( ξ ⊗ ξ , η ⊗ η , ξ ⊗ η + q η ⊗ ξ ) as a subspace of k2 ∗⊗ k2 ∗ . Hence the quantum superplane arises as the quadratic algebra  2∗  A0|2 = k , ( ξ ⊗ξ , η ⊗ η , ξ ⊗ η + q η ⊗ ξ ) q [ A2|0 q ,J ] .

= Notice also that 2|0 A0|2 q ⊗ Aq

=



k2∗⊗ k2 , ( b ⊗ a − q a ⊗ b , d ⊗ c − q c ⊗ d ,

 q −1 b ⊗ c − q c ⊗ b + d ⊗ a − a ⊗ d )

(which should be compared with equations (∗∗) in the proof of Theorem 3.2) where a = ξ ⊗x , b = ξ ⊗ y , c = η ⊗ x , d = η ⊗y gives half the relations required for Mq (2). Let V be a tensor category. Write d , e : B A , or briefly B A , for objects A, B of V and arrows /I ,

e : B⊗A

/ A⊗B

d:I

when the following diagrams commute: A

d⊗1A

1A

B

A⊗B⊗A

1B ⊗d

1A⊗e

B⊗A⊗B e⊗1B

1B

A

B.

We call B a left dual for A, and we call A a right dual for B. We call e the counit and d the unit . Duals are uniquely determined up to isomorphism. The tensor category is called left autonomous when each object A has a left / B determines a unique arrow f ∗ : B ∗ / A∗ dual A∗. Each arrow f : A given by the composite B∗

1⊗d

B ∗ ⊗A⊗A∗

1⊗f ⊗1

B ∗ ⊗B⊗A∗

This makes left dual into a functor ( )∗ : V op

/ V.

e⊗1

A∗ .

82

Quantum Groups: A Path to Current Algebra

We also have (A⊗B)∗ ∼ = B ∗ ⊗A∗ and I ∗ ∼ = I. The tensor category is called autonomous when each object A has both a left dual A∗ and right dual A∨ . If V is a braided tensor category then each left dual A∗ is also a right dual with counit and unit, respectively A⊗A∗ I

cA,A∗

d

A∗⊗A A⊗A∗

e −1 cA,A ∗

I A∗⊗A .

This implies A∗∗ ∼ =A. A tortile tensor category is an autonomous balanced tensor category in which the twist is related to the dual via the condition θA∗ =

θA ∗ : A∗

/ A∗ .

A left dual A∗ for A gives a left internal hom [A , B ] = B⊗A∗ with “evaluation”: /B e = 1 ⊗ e : B⊗A∗⊗A A

B

for all objects B of V. A right dual A∨ for A gives a right internal hom [A , B ]
= A∨ ⊗B. Hence a left/right autonomous tensor category is left/right-closed. Example 12.6 For each commutative ring R, an object M of ModR has a (left) dual if and only if it is Cauchy (Theorem 5.2). Write PrfR for the full subcategory of ModR consisting of Cauchy R-modules. Since PrfR is closed under tensor, it is an autonomous (symmetric) tensor category. Example 12.7 Let H be an R-Hopf algebra. An object M of the tensor category ModR (H) has a left dual precisely when it is Cauchy as an Rmodule; in this case, M ∗ = HomR (M, R) (Proposition 10.1). If H has an
(M, R). invertible antipode then each such M has a right dual M ∨ = HomR Write PrfR (H) for the full subcategory of ModR (H) consisting of those H-modules M which are Cauchy when viewed as R-modules. For H with invertible antipode, PrfR (H) is an autonomous tensor category. Example 12.8 Tangles on strings. (This example was discovered by Freyd–Yetter [FY89].) Let P be a Euclidean plane. A geometric tangle T is a compact 1-dimensional oriented submanifold of [0, 1] × P which is tamely embedded and whose boundary ∂T is equal to T ∩∂([0, 1]×P ). Thus a geometric tangle T is a disjoint union of directed (topological) circles contained in (0, 1) × P and of directed paths connecting two points on the boundary ∂([0, 1] × P ). The target of T is the subset ∂T ∩ ({1} × P )

Internal homs and duals

83

as an oriented 0-dimensional manifold. The source of T is the subset ∂T ∩ ({0} × P ) but with orientation reversed. A geometric tangle can be pictured as follows: −

−

−

−

+

−

+

+

−

+

A tangle is an isotopy class of geometric tangles where isotopies keep the boundaries fixed. The source and target of a tangle are regarded as signed subsets of P . Let 1, 2, 3, . . . denote equally spaced collinear points on P . Now we can define the autonomous braided tensor category T of tangles. / {+, −} for n ≥ 0 , called The objects are functions A : {1, 2, . . . , n} signed sets. The arrows of T are the tangles which have these signed sets as sources and targets. Composition and tensor-product are as for braids. The braiding is illustrated below. +

−

+

+

+

−

+

−

+

−

+

+

∗

The left dual A of a signed set A is given by reversing the order and the sign of the points; that is, A∗(i) and A(n − i + 1) are opposite signs for 1 ≤ i ≤ n . The counit and unit are illustrated below for A = {− + −} .

∗

e : A ⊗A

d:I

+

−

+

−

+

−

−

+

−

+

−

+

/I

/ A ⊗A∗

84

Quantum Groups: A Path to Current Algebra

The next diagram proves one triangle for e and d; the other is similar. +

−

+

+

−

+

−

+

−

+

−

+

+

−

+

Example 12.9 Tangles on ribbons. (The full details of this example appear in the thesis of Shum [Shu94].) The category T of tangles on ribbons is obtained from T just as we obtained B from B in Example 11.5. The directed strings of tangles are thickened into (directed) ribbons. Ribbons obtained from strings with boundary may be twisted through complete turns. Those which are thickenings of closed strings may have twists, as long as they remain 2-sided 2-manifolds; the M¨ obius ribbon is not allowed. Again we obtain an autonomous tensor category. The counits and units look like this: +

−

−

+

+

− −

e

+

−

+

−

+

d

In fact, T is a tortile tensor category. The twist is as for B and the identity θA∗ = θA∗ can be seen from the following diagram.

=

13 Tensor functors and Yang–Baxter operators /V Suppose C and V are tensor categories. A tensor functor F : C / V (denoted by the same symbol) together consists of a functor F : C ∼ with a natural isomorphism φA,B : FA⊗FB = / F (A⊗B) and another iso∼ morphism φ0 : I = / FI , such that FA⊗FB⊗FC

φA,B ⊗1

F (A⊗B)⊗FC

1⊗φB,C

φA⊗B⊗C

FA⊗F (B⊗C)

φA,B⊗C

F (A⊗B⊗C)

and FA

1⊗φ0

1

φ0 ⊗1

FI⊗FA

φI,A

FA⊗FI φA,I

FA

(where we have suppressed the constraints a, r, l as usual). If the condition that φ, φ0 be invertible is dropped, we have a weak tensor functor. If in fact φ and φ0 are identities, then F is called a strict tensor functor. (Weak tensor functors are also called “monoidal functors” and tensor functors called “strong monoidal functors”.) If C and V are braided, we describe a tensor functor as braided when FA⊗FB

φA,B

F (cA,B )

cFA,FB

FB⊗FA

F (A⊗B)

φB,A

85

F (B⊗A) .

86

Quantum Groups: A Path to Current Algebra

If C and V are symmetric, we say F is symmetric instead of “braided”. If C and V are balanced, we say F is balanced, when it is braided and F (θA ) =

/ FA .

θFA : FA

/ V is a weak tensor functor and C and V are leftSuppose F : C closed tensor categories. Then the composite F [A , B ]⊗FA

φ[A,B],A

F eA

F ([A , B ]⊗A)

FB

corresponds, using the defining property of [FA , FB ] , to an arrow / [FA , FB ] . φ˜ : F [A , B ] A,B

We call F a left-closed tensor functor when each φ˜A,B is invertible; rightclosed and closed are now defined in the obvious way. (This differs from the notion of “closed functor” in the literature.) When it comes to duals, the situation is better: tensor functors preserve / V is a tensor functor and d , e : B A duals. More precisely, if F : C in C, then FB FA in V with unit I

φ0

FI

Fd

F (B⊗A)

−1 φ[A,B],A

FA⊗FB

and counit FB⊗FA

φ[B,A]

F (B⊗A)

Fe

FI

φ0−1

I .

Hence, if C is (left-) autonomous and V is (left-) closed, then each tensor / V is (left-) closed (since [A , B ] = B⊗A∗ in C). functor C Example 13.1 The category Set of (small) sets is a tensor category using cartesian product as tensor product. For each commutative ring R, the “free module functor” (see Chapter 4) FR : Set

/ Mod R

is a tensor functor. It is certainly not closed since we have HomR (FR X, FR Y ) ∼  YX. = (FR Y )X ∼ = / Set which takes each module to its underlying The functor | | : ModR set |M | is a good example of a weak tensor functor: we have functions φM,N : |M | × |N | (m , n) φ0 = η : which are not invertible.

1

/ |M ⊗N | m ⊗n / |R|

Tensor functors and Yang–Baxter operators

87

Example 13.2 Universal algebra. The “universal enveloping algebra” / Alg is a tensor functor (see Proposition 6.10). functor U : LieR R Example 13.3 Yang–Baxter operators. We want to examine what is / V from the braid category involved in giving a tensor functor F : B into an arbitrary tensor category V. A Yang–Baxter (YB) operator on an object A of V is an invertible / A⊗A such that the following hexagon commutes. arrow y : A⊗A A⊗A⊗A

1⊗y

A⊗A⊗A

y ⊗1

y ⊗1

A⊗A⊗A

A⊗A⊗A

1⊗y

1⊗y

A⊗A⊗A

y ⊗1

A⊗A⊗A

For example, the object 1 of B admits the following YB-operator: c1,1 = s1 : 1 + 1

/ 1+1

which is the element of the braid group B2 depicted by the next diagram. s1 : The YB-hexagon becomes the following simple identity.

=

/ V preserves tensor products “up to Since any tensor functor F : B coherent isomorphism”, we obtain a YB-operator y on F (1) = A ; namely, φ

y : A⊗A

1,1

F (1 + 1)

F s1

φ−1

F (1 + 1)

1,1

A⊗A .

Conversely, given a YB-operator y on an object A of V, we can determine / V such that F (1) = A and y is the above coma tensor functor F : B posite. In fact, F is unique up to isomorphism (arising from the different

88

Quantum Groups: A Path to Current Algebra

possible choices for the n-fold tensor product A⊗n ). Since F is to be a tensor functor, we are forced to have F (n) = F (1 + 1 + · · · + 1) ∼ = A⊗n . Each generator si of Bn can be written as si = 1i−1 ⊗ s1 ⊗ 1n−i−1 in B , so / A⊗n is forced. We just need to check that the definition of F si : A⊗n this is compatible with the braid relations (Example 11.1); but this follows from the YB-hexagon and the functoriality of tensor product. Details are left as an exercise (which is worth doing). Hence, up to the appropriate notion of isomorphism, tensor functors / V correspond to pairs (A, y) consisting of an object A of V and F :B a YB-operator y on A. We can express this by saying:   B , 1 , s1 is the free tensor category having an object equipped with a YB-operator. / V also deterExample 13.4 A tensor functor from ribbons F : B mines a YB-operator y on F 1 = A as in Example 13.3 (compare with / 1 in B gives an isomorphism [Tur88]). This time the twist θ1 : 1 / z = F θ1 : A A. Due to the equalities in B

,

=

=

this gives an example of the following concept. A YB-operator y on an object A of V is said to be balanced when it is / A such that these commute: equipped with an isomorphism z : A A⊗A

y

1⊗z

A⊗A

A⊗A z ⊗1

y

A⊗A

A⊗A ,

y

z ⊗1

A⊗A

A⊗A 1⊗z

y

A⊗A .

Each balanced YB-operator determines a unique (once the n-fold tensors / V from which A, y, z are recovA⊗n are chosen) tensor functor F : B ered as above. So we have that   B , 1 , c1,1 , θ1 is the free tensor category containing an object equipped with a balanced YB-operator.

Tensor functors and Yang–Baxter operators

89

The easier part of Example 13.3 can be obtained from two observations. • Tensor functors take YB-operators into YB-operators. / V is a tensor functor and if y is a YBMore precisely, if F : C / F (X⊗X) carries across operator on X in C, then F y : F (X⊗X) the isomorphism φX,X to a YB-operator y on F X in V. • A braiding on a tensor category gives, on each object X, a YB/ X⊗X . operator: cX,X : X⊗X Moreover, tensor functors take balanced YB-operators into balanced YBoperators. Furthermore, in a balanced tensor category there is a balanced YB-operator (cX,X , θX ) on each object X. (See Example 13.4 .) We now look at compatibility of YB-operators with duals. A YB-operator y on A is called (left-) dualizable when A has a left dual / A∗ given by the composites: A∗ and both the arrows u , v : A∗⊗A 1⊗1⊗d

A∗⊗A

A∗⊗A⊗A⊗A∗

1⊗y ⊗1 1⊗y

−1

A∗⊗A⊗A⊗A∗

⊗1

e⊗1⊗1

A⊗A∗

e⊗1⊗1

A∗⊗A∗

are invertible. It follows that the composite w given by 1⊗1⊗d

A∗⊗A∗

A∗⊗A∗⊗A⊗A∗

1⊗u⊗1

A∗⊗A⊗A∗⊗A∗

has inverse w−1 given by the composite: A∗⊗A∗

1⊗1⊗d

A∗⊗A∗⊗A⊗A∗

1⊗v ⊗1

A∗⊗A⊗A∗⊗A∗

e⊗1⊗1

A∗⊗A∗ .

Proposition 13.5 In a braided tensor category, if an object A has a dual, then y = cA,A is a dualizable YB-operator on A with −1 u = cA,A ∗ ,

−1 v = cA ∗,A ,

−1 w = cA ∗,A∗ .

Proof. To prove cA,A∗ ◦ u = 1A∗⊗A , it suffices (by the property of duals) to show that equality holds after applying A⊗ to both sides and composing with d ⊗ 1A . Thus the following diagram gives the first equation. A⊗A∗⊗A⊗A⊗A∗ 1⊗1⊗1⊗d

A⊗A∗⊗A d⊗1

A

1⊗1⊗cA,A ⊗1

d⊗1⊗1⊗1

A⊗A⊗A∗

A⊗A∗⊗A⊗A⊗A∗ d⊗1⊗1⊗1

cA,A ⊗1

A⊗A⊗A∗ cA,A⊗A∗

1⊗d

cA,I = 1

A

d ⊗1

1⊗e⊗1⊗1

1

A⊗A⊗A∗ 1⊗cA,A∗

A⊗A∗⊗A

90

Quantum Groups: A Path to Current Algebra

−1 The result for v follows by using the braiding cB,A in place of cA,B . For w,
consider A∗ in place of A.

/ V preserve dualizability. So if C is braided and Tensor functors F : C X has a dual in V, we obtain a dualizable YB-operator on F X in V. A YB-operator on A is called tortile when it is balanced, dualizable, and y ⊗1

A⊗A⊗A∗

1⊗v -1

A⊗A⊗A∗

A⊗A∗⊗A

1⊗d z

A

1⊗e

z

A

A.

Proposition 13.6 In a balanced tensor category, if an object A has a dual then the pair (cA,A , θA ) is a tortile YB-operator precisely when θA∗ = (θA )∗ . Proof. The following diagram proves the equation (θA∗ )∗ θA = (1 ⊗ e) ◦ (1 ⊗ v −1 ) ◦ (y ⊗ 1) ◦ (1 ⊗ d) .

θA

A

1⊗d

A

A⊗A⊗A∗

1⊗cA−1 ∗,A

A⊗A∗⊗A

θA⊗1⊗1

1⊗d 1⊗cA−1 ∗,A

A⊗A⊗A∗

A⊗A∗⊗A

cA−1 ∗,A

θA⊗θA∗⊗1

1⊗θA∗⊗1

A⊗A∗⊗A −1 cA,A ∗⊗1

cA−1 ∗,A⊗1

A∗⊗A⊗A y ⊗1

= cA,A⊗1

A∗⊗A⊗A

θA∗A⊗1 e⊗1

1⊗cA,A

e⊗1

A∗⊗A⊗A

A cA∗A,A

cA−1 ∗,AA

θI = I

A

cI,A = 1

1

cA∗,A⊗1

A⊗A⊗A∗

1⊗v −1 = 1⊗cA−1 ∗,A

A⊗A∗⊗A

1⊗e

A

So the balanced YB-operator (y, z) is tortile if and only if (θA∗ )∗ θA = z 2 ; that is, if and only if (θA∗ )∗ θA = (θA )2 ⇔ (θA∗ )∗ = θA ⇔ θA∗ = (θA )∗ .




Tensor functors and Yang–Baxter operators

91

It follows that, in a tortile tensor category, each object A is equipped with a tortile YB-operator (cA,A , θA ). Example 13.7 Since T is a tortile tensor category, we obtain a tortile YB-operator (c+,+ , θ+ ) on the object + of T . Thus, each tensor functor / V yields a tortile YB-operator on F (+) in V. F : T   In fact, T , + , c+,+ , θ+ is the free tensor category equipped with a tortile YB-operator ( [Shu94] together with [JS91c]). This means that, given a tortile YB-operator (y, z) on an object A in a tensor category V, there / V which exists a (unique up tensor functor F : T  to isomorphism)  takes + , c+,+ , θ+ to A , y , z . We do not intend to prove this here; after all, our geometric description of T was incomplete. We hope the result is believable. All we really need is that such a free T should exist; but the description of the realistic model is too pretty to omit. It should be clear how to define F in terms of A, y, z. For example, F (+ − − − + −) F (c+,− )

= =

A⊗A∗⊗A∗⊗A∗⊗A⊗A∗ u−1

F (c−,− ) F (θ+ )

= =

w z

F (θ− )

=

z∗

and so on. Any tangle of ribbons can be decomposed, using composition and tensor product in T , into single crossings (c+,+ , c+,− , c−,+ , c−,− , and their inverses), turnings ( e and d) and twists ( θ+ , θ− , and their inverses). So the value of F on the tangle is forced. The hard part, which we shall not include in these notes, is to show that this value is independent of the decomposition. It is instructive to see in this example what is meant by the equation z 2 = (1 ⊗ e) ◦ (1 ⊗ v −1 ) ◦ (y ⊗ 1) ◦ (1 ⊗ d). It is expressed by the following diagram (which can be tested by taking off your belt).

=

92

Bradshaw: “Barred Variation”, Sash Bradshaw Group, [Wal94, Plate 58].

14 A tortile Yang–Baxter operator for each finite-dimensional vector space

Let k be a field and let q ∈ k be a fixed non-zero element. Let V be a vector space over k with basis ε1 , ε2 , . . . , εn . Define a linear function y : V ⊗V

/ V ⊗V

on the basis elements εi⊗ εj of V ⊗V by    εj ⊗ εi y(εi⊗ εj ) = εj ⊗ εi + (q − q −1 ) εi⊗ εj   q εi⊗ εi

for i > j for i < j for i = j .

In order to check the YB-hexagon for y, we look at (y ⊗ 1)(1 ⊗ y)(y ⊗ 1), (1 ⊗ y)(y ⊗ 1)(1 ⊗ y) at each εi ⊗ εj ⊗ εk . There are thirteen of these cases to check to account for all possible relative positions of i, j and k. We shall only give three of these cases as an illustration: put ρ = q − q −1 and omit the ε and ⊗ symbols from the notation.

i < j < k : (ijk)

y ⊗1

(jik) + ρ (ijk)

1⊗y

(jki) + ρ (jik) + ρ (ikj) + ρ2 (ijk)

y ⊗1

(kji) + ρ (jki) + ρ (ijk) + ρ (kij) + ρ2 (ikj) + ρ2 (jik) + ρ3 (ijk)

(ijk)

1⊗y

(ikj) + ρ (ijk)

y ⊗1

(kij) + ρ (ikj) + ρ (jik) + ρ2 (ijk)

1⊗y

(kji) + ρ (kij) + ρ (ijk) + ρ (jki) + ρ2 (jik) + ρ2 (ijk) + ρ3 (ijk) ,

93

94

Quantum Groups: A Path to Current Algebra

j < i < k : (ijk)

(ijk)

y ⊗1 1⊗y

(jki) + ρ (jik)

y ⊗1

(kji) + ρ (jki) + ρ (ijk) + ρ2 (jik)

1⊗y

(ikj) + ρ (ijk)

y ⊗1 1⊗y

i = j < k : (iik)

(iik)

(jik)

y ⊗1

(kij) + ρ (ikj) + ρ (jik) (kji) + ρ (ijk) + ρ (jki) + ρ2 (jik) ,

q (iik)

1⊗y

q (iki) + ρq (iik)

y ⊗1

q (kii) + ρq (iki) + ρq 2 (iik)

1⊗y

(iki) + ρ (iik)

y ⊗1 1⊗y

(kii) + ρ (iki) + ρq (iik) q (kii) + ρ (iik) + ρq (iki) + ρ2q (iik) .

(Note that q 2 ρ = ρ + q ρ2 since ρ = q − q −1 .) Clearly y is invertible with inverse given by:    εj ⊗ εi −1 y (εi⊗ εj ) = εj ⊗ εi + (q −1 − q) εi⊗ εj   −1 q εi⊗ εi

for i < j for i > j for i = j .

Hence, y is a YB-operator on the object V of Modk . It is now possible to calculate the operators u, v, w and their inverses (see the definition of dualizable YB-operator in Chapter 13). For this, let ε1∗ , . . . , εn∗ ∈ V ∗ be the dual basis for ε1 , . . . , εn ∈ V ; this means εi∗ (εj ) = δij .

Tortile Yang–Baxter operator for vector spaces Recall that e : V ∗ ⊗V Now we obtain u(εi∗⊗ εj ) = =

/ k is the evaluation functor and d =  ε ⊗ ε∗ . k k k

(e ⊗ 1 ⊗ 1)(1 ⊗ y ⊗ 1)(1 ⊗ 1 ⊗ d)(εi∗⊗ εj )  (e ⊗ 1 ⊗ 1)(1 ⊗ y ⊗ 1)(εi∗⊗ εj ⊗ εk⊗ εk∗ ) k

=

95



  (e ⊗ 1 ⊗ 1) εi∗⊗ εk⊗ εj ⊗ εk∗ + (q − q −1 ) εi∗⊗ εj ⊗ εk⊗ εk∗

k>j

+ q (e ⊗ 1 ⊗ 1) (εi∗⊗ εj ⊗ εj ⊗ εj∗ ) +



(e ⊗ 1 ⊗ 1) (εi∗⊗ εk ⊗ εj ⊗ εk∗ )

k<j

=

  δik εj ⊗ εk∗ + (q − q −1 ) δij εk ⊗ εk∗ + q δij (εj ⊗ εj∗ ) k>j

+



δik εj ⊗ εk∗

k<j

=

 ∗   εj ⊗ εi εj ⊗ εi∗    −1 q εi⊗ εi∗ + (q − q −1 ) εk ⊗ εk∗

for j < i for i < j for i = j .

k>i

The other operators are calculated similarly. We record the results below:  for i = j εj ⊗ εi∗ ∗⊗ u(εi εj ) =  −1 ∗ ∗ q εi⊗ εi + (q − q ) εk ⊗ εk for i = j k>i  for i = j εj∗⊗ εi u−1 (εi⊗ εj∗ ) =  −1 (q − q)q −2(k−i) εk∗⊗ εk for i = j q −1 εi∗⊗ εi + k>i  ∗ ⊗ ε for i = j ε i j = v(εi∗⊗ εj )  −1 −1 ∗ ∗ q εi⊗ εi + (q − q) εk⊗ εk for i = j k j  εj ⊗ εi εj∗⊗ εi∗ + (q −1 − q) εi∗⊗ εj∗ for i < j w−1 (εi∗⊗ εj∗ ) =  −1 ∗ ∗ q εi ⊗ εi for i = j .

96

Quantum Groups: A Path to Current Algebra

Hence y is dualizable. It enriches to a balanced YB-operator on defining / V simply to be the homothety z: V z(x) = q n x . Proposition 14.1 The YB-operator (y, z) defined above is tortile. Proof. First observe that, for i = j , the value of e v −1 at εi ⊗ εj∗ is 0 ; while for i = j the value is  q + (q − q −1 )q 2(i−k) k

  = q + (q − q −1 ) q 2(i−1) + q 2(i−2) + · · · + q 2(i−(i−1)) (q 2 )i−1 − 1 q2 − 1 2 i−1 − q = q 2i−1 . = q + q(q ) = q + q(q 2 − 1)

Hence, we have the following remarkable calculation: (1 ⊗ e)(1 ⊗ v −1 )(y ⊗ 1)(1 ⊗ d)(εi )  (1 ⊗ (e v −1 ))(y ⊗ 1)(εi ⊗ εj ⊗ εj∗ ) = j

=

 j

+

(1 ⊗ (e v −1 ))(εj ⊗ εi⊗ εj∗ ) + (1 ⊗ (e v −1 )) q (εi⊗ εi⊗ εi∗ ) 

  (1 ⊗ (e v −1 )) (εj ⊗ εi⊗ εj∗ ) + (q − q −1 ) (εi ⊗ εj ⊗ εj∗ )

i<j

= 0 + q q 2i−1 εi + 0 +



(q − q −1 )q 2j−1 εi

i<j

= = = =

 2i  q + (q − q −1 )(q 2i+1 + q 2i+3 + · · · + q 2n−1 ) εi  2i  q + (q 2 − 1) q 2i (1 + q 2 + · · · + q 2(n−i−1) ) εi

2 n−i −1 2i 2 2i (q ) q + (q − 1) q εi q2 − 1  2i  q + q 2i (q 2n−2i − 1) εi

= q 2n εi   = z z(εi ) .


15 Monoids in tensor categories

A monoid in a tensor category V consists of an object A and arrows µ : A⊗A

/A

and

/A

η:I

which satisfy the usual identity and associativity conditions (see Chapters / B is an arrow in V which preserves 1 and 6). A monoid arrow f : A µ and η , in the diagrammatically expressed sense. It is also useful to consider “arrows between monoid arrows”. Suppose / B are monoid arrows. A 2-cell that f , g : A ξ : f ⇒g :A is defined to be an arrow ξ : I

/ B in V such that

ξ ⊗f

A

g ⊗ξ

/B

B ⊗B

µ

B.

The more 2-dimensional notation f

A

ξ

B

g

is also used. (In the case where V = Set , such a 2-cell amounts to an element ξ ∈ B for which ξf (a) = g(a)ξ for all a ∈ A .) There are two basic pasting operations for 2-cells. Given the situation f


A

h

A

ξ

B

k

B


g

where the arrows are monoid arrows and ξ is a 2-cell, there is a 2-cell kf h


A

kξ kgh

97

B


98

Quantum Groups: A Path to Current Algebra

obtained since we have the following factorization:

ξ ⊗(f h) ξ ⊗f

h

A


A

µ

B ⊗B

g ⊗ξ

(gh)⊗ξ

B

k⊗ k

k µ

B
⊗B


B


This is called whiskering ξ by h and k . The other basic pasting operation is vertical composition, which takes a pair of 2-cells ξ and ζ , as in the following situation f ξ

A

B

g

ζ h

to a 2-cell f

A

B

ζ∗ξ h

where ζ ∗ ξ =



I

ζ ⊗ξ

B ⊗B

 B .

µ

The identity 2-cell η : f ⇒ f : A vertical composition.

/ B is an identity for the operation of

When we write a diagram such as f2

f1

A

f3

B

ξ g1

g4 g2

g3

/ B . This allows us to it is intended that ξ : f3 f2 f1 ⇒ g4 g3 g2 g1 : A define a more general pasting operation, which assigns to a diagram like

Monoids in tensor categories

99

f3 f2

f4

f1

f5

f7

f8

f6

A

B g4

g3

g1 g2

/ B , obtained (quite a 2-cell f8 f7 f6 f5 f4 f3 f2 f1 ⇒ g4 f6 g3 g2 g1 : A possibly in several different ways) by first whiskering the 2-cells in the diagram to be of the form B

A

as well as being vertically composable, and then composing vertically. As an example of this pasting, consider the diagram f

A

ξ

g l

h

ζ

B

k

First whisker ξ and ζ appropriately, as in f

A

gl ξ

g

and

B

h

A

lh

ζ k

to obtain two vertically composable 2-cells gf glk

A

gξ

B. ζ kh

Then vertically compose to obtain a 2-cell gf

A

B kh

B

100

Quantum Groups: A Path to Current Algebra

which is called the pasted composite of the original diagram. In this case there was only one way of performing the pasting. As another example, consider the diagram f

A

h ξ

ζ

g

B.

k

On the one hand we can whisker ξ and ζ as in f

A

h ξ

h

B

,

A

g

ζ

g

B

k

and then vertically compose; while on the other hand we can whisker ξ and ζ as in

f

h

A

f ζ

B

,

A

ξ

k

B

g

k

and then vertically compose. The reader should verify that the resultant 2-cells of the form hf

A

B kg

are actually equal. It is a general fact that the result of pasting is independent of the way it is broken down into basic pasting operations. In fact, all ambiguities in the method can be traced back to instances of the last example. For the particular diagrams we shall use here, it is easily shown that they have a uniquely determined pasted composite. Write Mon(V) to denote the category of monoids in V ; the arrows are monoid arrows. With the extra structure of 2-cells, Mon(V) is an example of a “2-category” (see [KS74]). (For a commutative ring R , we have that Mon(ModR ) = AlgR and also Mon(ModRop ) = CogRop where ModR and ModRop have the same tensor product ⊗R . )

Monoids in tensor categories

101

• Weak tensor functors take monoids to monoids. / W is a weak tensor functor, each monoid A More precisely, if F : V in V gives a monoid F (A) in W with multiplication φA,A

F (A)⊗F (A)

F (µ)

F (A⊗A)

F (A)

and unit φ0

I

F (I)

F (η)

F (A) .

In fact, we obtain a functor Mon(F ) : Mon(V)

/ Mon(W)

which preserves the basic pasting operations of 2-cells (and so is an example of a “2-functor”). For each monoid A in V there is a category, called ModV (A) , of (left) A-modules. An A-module consists of an object M of V and an arrow / M , called the action, which satisfies all the usual defining µ : A⊗M /N diagrams for a module (see Chapter 9). An A-module arrow µ : M is an arrow in V such that A⊗M

1⊗u

A⊗N

µ

µ u

M

N .

/ V which simply forgets There is a “forgetful” functor UA : Mod V (A) the module action. / B determines a functor Each monoid arrow f : A Mod(f ) : Mod V (B)

/ Mod (A) V

given by “restriction of scalars” along f . That is, for a B-module M , we take Mod(f )(M ) to be M with A-action A⊗M

f ⊗1

µ

B⊗M

M .

Each B-module arrow becomes an A-module arrow, thereby giving the following commutative triangle of categories and functors: Mod V (B) UB

Mod(f )

V

Mod V (A) UA

102

Quantum Groups: A Path to Current Algebra

/ B between monoid arrows Furthermore, each 2-cell ξ : f ⇒ g : A f and g in V determines a natural transformation Mod(ξ) : Mod(f )

/ Mod(g)

whose component at the B-module M is the composite ξ ⊗1

M

B⊗M

µ

M

which is an A-module arrow, as can be seen from the diagram A⊗M

1⊗ξ ⊗1

A⊗B ⊗ M

f ⊗1

1⊗µ

A⊗M g ⊗1

g⊗1⊗1

A⊗M

ξ ⊗1⊗1

µ

B⊗B⊗M 1⊗µ

ξ ⊗1

M

1⊗µ

B⊗M µ

µ⊗1 µ

B⊗M

M

Naturality follows from the following diagram involving a B-module arrow /N. u: M M

ξ ⊗1

u

N

B⊗M

µ

M u

1⊗u ξ ⊗1

B⊗N

µ

N

The assignment ξ  / Mod(ξ) turns the two basic pasting operations into corresponding familiar operations on natural transformations. This gives an example of a “2-functor” ModV : Mon(V)op

/ Cat

where Cat is some appropriate “2-category” of categories. For our purposes, it is important to remember the forgetful functors / V . So, rather than Cat , we consider Cat/V, whose UA : Mon V (A) / V, whose arrows objects are functors F : C   /D ,G T : C ,F / D such that G T = F , and for which the 2-cells α : are functors T : C
/ T are arbitrary natural transformations from T to T
(that is, no T

Monoids in tensor categories

103

condition relating it to F and G ). Observe that ModV really lands in   Cat/V by taking A ∈ Mon(V)op to ModV (A) , UA . So we have that / Cat/V

ModV : Mon(V)op

is a 2-functor. There is an obvious candidate for a tensor product on Cat/V, namely 

    C ,F ⊗ D,G = C × D, C × D

F ×G

V×V

⊗

V



.

This tensor product respects all of the pasting operations for natural transformations. Ignoring the 2-cells, Cat/V becomes a tensor category with  / V . unit given by 1 , I : 1 Suppose now that V is braided. For monoids A and B in V, we enrich A⊗B with the multiplication 1⊗cB,A ⊗1

A⊗B⊗A⊗B

µ⊗µ

A⊗A⊗B⊗B

A⊗B

/ A⊗B ; the braiding properties imply (Exercise 15.1) and unit η ⊗ η : I that this makes A⊗B into a monoid. Thus Mon(V) becomes a tensor category such that the forgetful functor Mon(V)

/V

is a strict tensor functor. The tensor product on Mon(V) respects the basic pasting operations of 2-cells; see Exercise 15.2. We shall now see that ModV : Mon(V)op

/ Cat/V

is essentially a weak tensor functor. For this, observe that ModV(I) = V and we have arrows I

1

V 1V

I

Φ

ModV(A) × ModV(B)

ModV(A⊗B)

UA ⊗UB

V

V×V

UA⊗B ⊗

V

in Cat/V , where Φ(M , N ) = M ⊗N , with action A⊗B⊗M ⊗N

1⊗cB,M ⊗1

A⊗M ⊗B⊗N

µ⊗µ

M ⊗N .

104

Quantum Groups: A Path to Current Algebra

The reason for the word “essentially” is that the axioms for a weak tensor functor (see the beginning of Chapter 12) hold only up to isomorphism (instead of equality); in fact, the isomorphisms are precisely provided by the associativity and unit constraints a, r, l for the tensor product. Just as weak tensor functors take monoids to monoids, the 2-functor ModV takes tensor objects in Mon(V)op to tensor objects in Cat/V . • A tensor object is “essentially” a monoid. • A tensor object in Cat is precisely a tensor category; whereas a monoid in Cat is a strict tensor category.   • A tensor object in Cat/V is a pair C , F consisting of a tensor category / V. C and a strict tensor functor F : C • A tensor object in Mon(V)op will be called a quasi-bimonoid in V . This / A⊗A ,  : A /I, consists of a monoid A in V , monoid arrows δ : A and 2-cells: δ

A

A

A⊗2 1A

δ

1A

δ

δ ⊗1

α

ρ

λ ⊗2

A

⊗3

A

A

1⊗δ

⊗1

A⊗2

A

1⊗

which are invertible under vertical composition and satisfy the following equalities between pasted diagrams. δ

A

A⊗2

δ

A

δ

A

⊗3

A

1⊗α

1⊗δ

1⊗δ ⊗1

A A

δ

δ

A⊗2

1⊗δ

δ ⊗1

A⊗3

A⊗2

A δ

δ ⊗1

α

δ ⊗1⊗1

A⊗3

=

δ

A⊗2

1

A⊗2

1⊗1⊗δ

δ ⊗1

ρ⊗1

A⊗3

1⊗λ 1

1⊗⊗1

A⊗2

A⊗3

1⊗δ

A⊗4

1⊗1⊗δ

1⊗δ

α

A⊗2

δ ⊗1⊗1

⊗3

A⊗2

δ

=

α⊗1

1⊗δ

⊗2

δ ⊗1

α

A⊗3

δ ⊗1

α

A⊗2

δ

δ ⊗1

1

1⊗⊗1

A⊗2

A⊗4

Monoids in tensor categories

105

A bimonoid in V is a strict quasi-bimonoid; that is, one for which the 2-cells α, λ, ρ are actually identity 2-cells. (A bimonoid in ModR is an R-bialgebra; see Proposition 7.5.) Hence we have that a (quasi-) bimonoid A in V determines the structure of a tensor category on ModV(A) as well as a tensor functor structure / V (see Chapter 10 for those cases when V = Mod on UA : ModV(A) R op and when V = ModR ). A (quasi-) Hopf monoid in V is a (quasi-) bimonoid H together with / H , called the antipode, such that an arrow ν : H H

δ

ν ⊗1

H⊗H

1⊗ν

H⊗H



µ

H

η

I Drinfel
d [Dri89] obtained interesting examples of quasi-Hopf monoids in Modk . Proposition 15.1 Suppose V is a braided tensor category and H is a Hopf monoid in V . If M is an H-module which has a left dual M ∗ as an object of V then M ∗ becomes the left dual of M in ModV(H) via the action H⊗M ∗

ν ⊗1

H⊗M ∗

cH,M ∗

1⊗µ⊗1

M ∗ ⊗H

1⊗d

M ∗ ⊗M ⊗M ∗

M ∗ ⊗H⊗M ⊗M ∗ e⊗1

M∗ .

Proof. This is a matter of proving that e and d are module arrows. For this, compare with Propositions 10.1 and 10.5.
A braiding for a (quasi-) bimonoid A in V is a 2-cell A δ

δ γ

A⊗A

cA,A

A⊗A

which is invertible (with respect to vertical composition) and satisfies the two equalities below.

106

Quantum Groups: A Path to Current Algebra

A

δ δ

A⊗A

δ

A

δ

δ

A⊗A

A⊗A

α

γ α

α

A⊗A 1⊗δ

cA,A

δ ⊗1

A⊗A δ ⊗1

1⊗δ

=

A⊗A

δ

δ ⊗1

1⊗γ

1⊗δ

A⊗A⊗A

γ ⊗1

δ ⊗1

1⊗cA,A

1⊗δ

cA,A⊗1

A⊗A⊗A

A⊗A⊗A

cA⊗A,A

A

δ δ

A⊗A

A⊗A⊗A

δ

A

δ

δ

A⊗A

A⊗A

α-1

γ α-1

α-1

A⊗A δ ⊗1

cA,A

1⊗δ

δ

A⊗A 1⊗δ

δ ⊗1

=

A⊗A

A⊗A⊗A

cA⊗A,A

δ ⊗1

1⊗δ

γ ⊗1

A⊗A⊗A

1⊗γ

1⊗δ

cA,A⊗1

δ ⊗1

1⊗cA,A

A⊗A⊗A

A⊗A⊗A

cA⊗A,A

A⊗A⊗A

cA,A⊗A

A⊗A⊗A

Now suppose V is symmetric. A twist for A is a 2-cell 1A

A

A

τ 1A

which is invertible, with respect to ∗ , and satisfies δ

A 1

A

τ

A

δ

cA,A

1

A⊗A = δ

1

A⊗A γ

δ γ

A⊗A

τ ⊗τ

A⊗A

1 cA,A

1

A⊗A 

1

A

τ 1

A



I = A

1

I .



A (quasi-) bimonoid with a braiding and twist is called balanced.

Monoids in tensor categories

107

The reader should interpret the above pasting diagrams in the special case where V = Modk to see that these definitions agree with the definitions of braiding element and twist for bialgebras as in Examples 11.3 and 11.7 . We now have a conceptual version of the calculations in those examples. Proposition 15.2 Suppose V is a symmetric tensor category and A is a bimonoid in V. There is a bijection between braidings γ for A and braidings c for ModV(A) determined by cM,N as the composite: M ⊗N

γ ⊗1⊗1

1⊗cA⊗M,N

A⊗A⊗M ⊗N

A⊗N ⊗A⊗M

µ⊗µ

N ⊗M .

There is a bijection between twists τ for A and twists θ for ModV(A) determined by θM =



τ ⊗1

M

µ

A⊗M

M



.

Proof. Apply the 2-functor ModV to the triangle containing γ , and paste below it a square containing a natural isomorphism whose components are the symmetry of V (we omit the subscripts V on maps in the diagram): ModV(A) Mod(δ)

Mod(δ) Mod(γ)

ModV(A⊗A)

ModV(A⊗A)

Mod(cA,A )

Φ

Φ

c

ModV(A) × ModV(A)

σ

ModV(A) × ModV(A)

The result is a natural isomorphism, whose component at the pair (M , N ) ∈ ModV(A) × ModV(A) is the cM,N as stated in the proposition. The axioms on γ convert to the braiding axioms for c . Conversely, to recapture γ from c , take the composite γ =



I

η ⊗η

A⊗A

cA,A

A⊗A



,

where we regard A as an object of ModV(A) with action µ . The proof of the twist bijection is similar.


108

Quantum Groups: A Path to Current Algebra

A tortile bimonoid in V is a balanced Hopf monoid H such that I τ

H

τ ν

H .

Proposition 15.3 Suppose that V is a symmetric autonomous tensor category. If H is a tortile bimonoid in V then ModV(H) is a tortile tensor category. Proof. By Propositions 15.1 and 15.2, ModV(H) becomes an autonomous balanced tensor category. All that remains to see is that θM ∗ = (θM )∗ , which follows from ν ◦ τ = τ .
Consider the replacement of the tensor category V by its opposite tensor category V op . Monoids become comonoids, but bimonoids and Hopf / I on monoids are unchanged. For a bimonoid H in V , a cotwist τ : H op /I H in V is defined to be a twist on H in V ; a cobraiding γ : H⊗H op on H in V is defined to be a braiding on H in V . Thus we have the corresponding notions of cobraided, cobalanced and cotortile bimonoid in V . Our proposal for a definition of a quantum group over R is that it should be a cotortile bimonoid in ModR . In Chapter 18 we shall see that our main example, the quantum general linear group, does indeed give an instance of this concept.

Exercise 15.1 Check that the tensor product of monoids in a braided tensor category is a monoid. Exercise 15.2 Show that the natural tensor product on Mon(V) preserves whiskering and vertical composition of 2-cells in each variable. Exercise 15.3 Verify that braidings and twists in Modk , in the sense of this chapter, include those of Examples 11.3 and 11.7.

16 Tannaka duality

Given a compact group G, the set Rep G of isomorphism classes of appropriate representations admits various operations; for example direct sum and tensor product. Tannaka’s duality theorem (1939) provided a recipe for recovering a compact group Gp R from a structure R such as Rep G whereby Gp Rep G ∼ = G. For algebraic groups, Saavedra Rivano [SR72] considered the category of appropriate representations together with the tensor structure and the underlying functor into vector spaces. He gave criteria on a tensor functor into vector spaces under which it should be equivalent to such an underlying functor. A non-commutative generalization of this was given by Ulbrich [Ulb89]. We shall lead into this Hopf algebra version by examining the 2-functor Mod V of the previous chapter. For simplicity of exposition we suppose our tensor category V is strict. This loses no generality in fact since every tensor category is equivalent to a strict one (Mac Lane’s coherence theorem). We also suppose that V is symmetric (but we cannot suppose the tensor product is strictly commutative). A consequence of this simplification is that we really do have a weak tensor functor / Cat/V . Mod V : Monop V op

We are now interested in going back from Cat/V to Mon(V) . A possible way to do this is via a left adjoint to ModV , if it exists. Under reasonable   conditions, a left adjoint EF ∈ Mon(V) does exist at an object C , F of Cat/V. It is constructed as follows. If each internal hom [ FX , FX ] exists in V and if V is suitably complete, we put  [ FX , FX ] EF = X∈C

where the integral sign denotes the end (see Mac Lane [Mac71]) of the / V taking (X , Y ) to [ FX, F Y ]; it is the equalizer of functor C op × C the obvious pair of arrows (see Exercise 16.1)   [ FX , F Y ] . [ FX , FX ] f :X→Y

X

109

110

Quantum Groups: A Path to Current Algebra

There are projection arrows / [ FX , FX ]

πX : EF

for each object X ∈ C. These correspond, using the definition of internal hom, to arrows / FX . µX : EF ⊗FX The universal properties of end and internal hom show that there exists / E in V and natural families of a bijection between the arrows f : A F / FX , given by arrows θX : A⊗FX θX = µX ◦ (f ⊗ 1X ) . The natural families EF ⊗EF ⊗FX

1⊗µX

µX

EF ⊗FX

FX ,

FX

1FX

FX

induce, under such bijections, the monoid structure on EF : /E F

µ : EF ⊗EF

,

η:I

/E . F

Example 16.1 Take V = ModR for some commutative ring R . Then / V, the we have that Mon(V) = AlgR . Now for any functor F : C algebra EF has as elements the natural families θ = (θX )X∈G of R-linear / FX ; addition and multiplication by scalars are morphisms θX : FX done componentwise, while multiplication is componentwise composition. In particular, for any R-algebra A , if we have / Mod R

F = UA : ModR (A)

then there is a natural isomorphism of algebras EF ∼ = A. To see this, notice that each element m of an A-module M determines a / M in Mod (A) with m(1) ˆ = m ; so for a natural θ : unique m ˆ : A R / UA UA , we have A

θA

m ˆ

M

A m ˆ

θM

M

which implies θM (m) = θA (1) m ; so θ is determined by θA (1) ∈ A .

Tannaka duality

111

Suppose A is a monoid in V . The two axioms which are required for an / E to be in Mon(V) translate to the two conditions on arrow f : A F / FX which say that the corresponding family of arrows θX : A⊗FX each θX is an action of A on FX. This is precisely what is needed to lift / Mod (A) such that U T = F ; just put T X = F to a functor T : C V A   FX , θX . This gives a natural bijection between hom sets      , C , F , ModV (A) , UA Mon(V)(A , EF ) ∼ = Cat/V   op / Cat/V . whereby C , F  / EF is left adjoint to ModV : Mon(V) In fact, the above bijection becomes an isomorphism of categories, since it extends to 2-cells: T

C

f

A

ξ f

ModV (A) T

EF




F




UA

V   / E is left 2-adjoint to Mod . This is expressed by saying that C , F  F V Taking A = EF in the above bijection and looking at the image of the identity arrow, we obtain N

C F

ModV (EF ) UE

F

V   where N X = F X , µX . We obtain a (partial) 2-functor / (Mon(V))op : Cat/V   /  D , G  in Cat/V into the by taking the 2-cell α : T ⇒ T
: C , F / E in Mon(V) corresponding (under the 2-cell Eα : ET ⇒ ET
: EG F 2-adjunction) to the 2-cell in Cat/V :   /  Mod (E ) , U  . Nα : NT ⇒ NT
: C , F V G E E

G

/ V are Remark 16.2 The Formal Tannaka Duality criteria on F : C / ModV (EF ) should be faithful and that every “appropriate” that N : C EF -module should be isomorphic to some N X .

112

Quantum Groups: A Path to Current Algebra

We can equally well regard E E

as a 2-functor

: (Cat/V)op

/ Mon(V)

whereupon (for general reasons as an adjoint to ModV ) it is a weak tensor functor. It preserves the unit in the sense that EI ∼ = I, while we have a canonical arrow φ such that φF,G

EF ⊗FG

EF ⊗G πX⊗Y

πX⊗πY

[ FX , FX]⊗[ GY, GY ]

⊗

[ FX⊗GY, FX⊗GY ]

where the bottom arrow corresponds to the composite [FX, FX]⊗[ GY, GY ]⊗FX⊗GY

1⊗c⊗1

e⊗e

[FX, FX]⊗FX⊗[ GY, GY ]⊗GY FX⊗GY .

So E takes monoids in (Cat/V)op to monoids in Mon(V), the latter being the commutative monoids in V , but this is of no interest to us here. Our real interest is in to what extent E

: (Cat/V)

/ Mon(V)op

  takes monoids to monoids. This will be true of those monoids C , F in /E Cat/V for which φF,F : EF ⊗EF F ⊗F is invertible. Is this a reasonable condition? At first glance, invertibility of    / [ FX , FX ] ⊗ [ GY, GY ] [ FX⊗GY, FX⊗GY ] φF,G : X

Y

X,Y

looks unlikely. It would be implied by the two conditions: (a) each A⊗

: V

/ V preserves ends; and ⊗

(b) each [A , B ]⊗[ C , D ] [ A⊗C , B⊗D ] is invertible, for every A = FX and every C = GY . However, these look unlikely too, if we think in terms of Example 16.1 . We shall look at the conditions (a) and (b) more closely. If V is a closed tensor category then A⊗ preserves colimits (since it has a right adjoint

Tannaka duality

113

[A , ] ; see Mac Lane [Mac71, Chapter V §5]. But end is a limit, not a colimit. So (a) can be ensured by taking V to be the opposite of a complete closed tensor category. We need to be careful here since we still need the internal homs of the form [FX, FX ] in V , not in V op . Condition (b) is true, for example for finite-dimensional vector spaces. What is needed is that A and C should have duals; then we have canonical isomorphisms [ A , B ]⊗[ C , D ] ∼ = A∗⊗ B ⊗ C ∗ ⊗D ∼ = [ A⊗C , B⊗D ] . = (A⊗C)∗ ⊗ (B⊗D) ∼ Hence, conditions (a) and (b) are not unreasonable after all. They are satisfied when V is the opposite of a closed symmetric tensor category which is cocomplete enough for coends over C to exist, and when each FX and GY has a dual. Suppose then that V op is a closed symmetric (strict) tensor category which is (small) cocomplete. Suppose C is a left autonomous small (strict) / tensor category and F : C  V is a (strict) tensor functor. Then each FX ∗ has a dual FX . Since C , F is a monoid in Cat/V, we obtain a monoid EF in Mon(V)op ; that is, a bimonoid EF in V. This gives a factorization N

C

ModV (EF )

F

UEF

V of our tensor functor F into tensor functors N and UE . F

In fact, EF is a Hopf monoid. To see this, define F ∗ : C op ∗ F X = FX ∗ . We obtain a monoid arrow

/ V by

/ (C op , F ∗ )

( )∗ : (C, F )

in Cat/V. This induces a monoid arrow ν with ν

EF ∗ πX ∗

EF πX ∗

FX⊗FX ∗ . It is easy to see that EF ∗ = EF op as bimonoids in V (that is, EF ∗ is just EF with switched multiplication and switched comultiplication), and ν is an antipode for the bimonoid EF .

114

Quantum Groups: A Path to Current Algebra

Now suppose C is braided. The braiding can be regarded as an invertible 2-cell: 

   C ,F ⊗ C ,F



σ

   C ,F ⊗ C ,F

c ⊗

⊗



C ,F



in Cat/V. Applying E , we obtain an invertible 2-cell in Mon(V) : c

EF ⊗EF

EF ⊗EF . γ

δ

δ

EF The braiding arrows for c on C carry over precisely to those for γ on EF . / Mod (E ) becomes a braided tensor functor. Moreover, N : C V F Next suppose C is balanced. The twist on C can be regarded as being an invertible 2-cell: 

C ,F



1C θ



C ,F



1C

in Cat/V, and, applying E , we obtain a twist 1

EF

τ

EF

1

for the braided bimonoid EF . So EF becomes a balanced Hopf monoid and / Mod (E ) becomes a balanced tensor functor. N : C V F Finally, if C is a tortile tensor category, EF is a tortile bimonoid in V. To obtain Ulbrich’s [Ulb89] setting, we take V = Modop R for a commutative ring R . For each R-coalgebra C, we have Mod V (C)op = ComodR (C) . We use the notation ComodR (C)c to denote the full subcategory consisting of C-comodules M for which the underlying R-module UC M is Cauchy.

Tannaka duality Consider a small category C and a functor F : C values FX are Cauchy R-modules. The coend  X FX ⊗R (FX)∗ EF
=

115 / Mod whose R

becomes an R-coalgebra and we have N

C F

ComodR (EF
) U

ModR (since we can apply our previous theory to F regarded as going from C op
to V = Modop R ). Notice that N actually lands in ComodR (EF )c . If C is a tensor category and F is a tensor functor then EF
becomes an R-bialgebra and N becomes a tensor functor. If C is left autonomous then EF
becomes a Hopf algebra with invertible antipode. If C is a tortile tensor category then EF
becomes a cotortile R-bialgebra (quantum group!) and N becomes a balanced tensor functor. An important case of Tannaka duality is the characterization of those / Mod for some / Mod equivalent to U : Comod (H) F :C c R H R R Hopf algebra H. This can be investigated by looking at when the functor / Comod (E
) is an equivalence. N : C R F c The question arises here as to whether EF
∼ = C when the equality / Mod holds for a coalgebra C . We cannot F = UC : Comod(C)c R use the technique of Example 16.1 since, although C is a C-comodule, it is generally not Cauchy as an R-module. Proposition 16.3 If C is a coalgebra over a field R and U denotes the / Mod , then there is a coalgebra isoforgetful functor U : Comod(C)c R morphism EU
∼ = C. Proof. We need to show that C has the universal property of EU
; that / (f ⊗ 1) ◦ δ determines a natural bijection between is, the assignment f  / X and families of R-module morphisms R-module morphisms F : C / X⊗ U (M ) natural in M ∈ Comod(C) . θM : U (M ) c We need to apply the fundamental theorem on coalgebras (see Sweedler [Swe69, p.46]): (when R is a field) “the sub-coalgebra generated by an element of C is Cauchy”. Given a family θM , we must define f (c) for each c ∈ C. Let M be any subcoalgebra of C which contains c and is finite dimensional. Such M exist

116

Quantum Groups: A Path to Current Algebra

by the above fundamental theorem, and can be regarded as C-comodules. Put f (c) = (1 ⊗ ε )θM (c). This is independent of the choice of M since θM / (f ⊗ 1) ◦ δ is is natural in M . The proof that this gives the inverse to f  now easy.
Exercise 16.1 Give a precise definition of the “obvious pair of arrows” in the defining equalizer for EF . Exercise 16.2 Complete the proof of Proposition 16.3.

Bradshaw: Clothes Peg Figure Period, [Wal94, Plate 67].

17 Adjoining an antipode to a bialgebra

Tannaka duality allows the possibility of taking an R-bialgebra A , applying some categorical construction to ComodR (A)c , and asking whether the result again has the form ComodR (B)c for some R-bialgebra B . An example of an appropriate categorical construction is adjoining leftdual objects to a tensor category. To each tensor category C , there is a / A (C) left autonomous tensor category A (C) and a tensor functor C  which induces a natural equivalence between the category of tensor func/ D and the category of tensor functors C / D for all left tors A (C) autonomous tensor categories D . (See [JS91a], [JSV96] and [KSW02].) / Mod is a tensor functor whose values FX Suppose that F : C R are Cauchy R-modules. Then we obtain a corresponding tensor functor / Mod . Fˆ : A (C) R Proposition 17.1 EFˆ
is the reflection of the R-bialgebra EF
into the category of Hopf R-algebras. Proof. Let H be a Hopf R-algebra. Then we have that ComodR (H)c is a left autonomous tensor category. Thus we have an equivalence between / Comod (H) , over Mod and tensor functors tensor functors A (C) c R R / C ComodR (H)c over ModR . By the left adjoint property of E
, / H correspond to bialgebra it follows that bialgebra morphisms EFˆ

/ morphisms EF H, as required.
This gives a construction for adjoining an antipode to a bialgebra over a field R ; that is, a construction for a left adjoint to the inclusion of the category HopfR of Hopf algebras in the category BigR of bialgebras. Given / Mod . By Proposition a bialgebra A , put F = UA : ComodR (A)c R
. By Proposition 17.1, the Hopf algebra H = EFˆ
is 16.3, we have A ∼ E = F the required reflection. If we require the adjoined antipode to be invertible, we must replace (C) in the above by A(C) which is the free autonomous tensor category A on the tensor category C . And so on. 117

118

Bradshaw: Sash Bradshaw Group, [Wal94, Plate 54].

18 The quantum general linear group again

Let V be an n-dimensional vector space over a field k . Given an invertible q ∈ k , let (y , z) be the tortile Yang–Baxter operator on V defined in Chapter 14. By Example 13.1 and Example 13.2, there are strict tensor functors / Mod / Mod , G : T M : B k k     taking + , c+,+ , θ+ to be V , y , z (where we are identifying B with the subcategory of T whose objects are positively signed sets, with arrows being ribbons which do not bend around). Applying Tannaka duality ideas (Chapter 16) to M and G, we obtain a cobalanced bialgebra EM
and a cotortile bialgebra EG
. Theorem 18.1 There are k-bialgebra isomorphisms (see Example 9.8): EM
∼ = Mq (n)

,

EG
∼ = GLq (n) .

Proof. Let A be the bialgebra EM
. It comes equipped with a universal / A⊗M Z for Z ∈ B . In particular, we have linear function δZ : M Z / / A⊗V ⊗V is the composite δ = δ+ : V A⊗V , and δ+,+ : V ⊗V V ⊗V

δ ⊗δ

while y : V ⊗V

A⊗V ⊗A⊗V

1⊗σ⊗1

A⊗A⊗V ⊗V

µ⊗1⊗1

A⊗V ⊗V

/ V ⊗V becomes a comodule morphism V ⊗V

δ+,+

y

V ⊗V

A⊗V ⊗V 1⊗y

δ+,+

A⊗V ⊗V

 Putting δ(εi ) = j xij ⊗εj , it is a straightforward, but tedious, matter to check that commutativity of the above square is equivalent to the elements 119

120

Quantum Groups: A Path to Current Algebra

X = {xij | i, j = 1, . . . , n} satisfying the defining relations for the quantum matrix monoid Mq (n) (see Example 9.8). We therefore have a bialgebra / A which can be seen to be invertible. morphism Mq (n) To introduce an antipode to the bialgebra A and thereby obtain the / A⊗V ∗ for δ : Hopf algebra H, we must introduce a left dual κ : V ∗ / A⊗V . Since T is the free autonomous tensor category on B , we have V H = EG
(as in Chapter 17). If we put  κ(εi∗ ) = wij ⊗εj∗ j

/ k and d : k / V ∗ ⊗V to and express what it means for e : V ∗ ⊗V be H-comodule morphisms, we obtain the conditions   wim xjm = δij and xmi wmj = δij , m

m

    which mean that the matrix wij is the inverse of the transpose of xij . The Hopf algebra H is therefore obtained from A by adjoining elements wij subject to the above two conditions. By means of a quantum Cramer’sRule (checked in Chapter 3 for the n = 2 case), we can take wij = t detq Xij (see Example 9.8) where t is an adjoined inverse for detq (X) . In this way we see that H ∼
= GLq (n) . Corollary 18.2 Mq (n) is a cobalanced bialgebra and GLq (n) is a cotortile bialgebra. / k , and cotwist given The cobraiding given by γ : GLq (n)⊗GLq (n) / k , satisfy the equations by τ : GLq (n)⊗GLq (n) y(εi⊗ εj )

=



γ(xim , xjr ) εm⊗ εr

m,r

q n εi

=



τ (xim ) εm .

m

This means:   

γ(xim , xjr ) τ (xij )

1 (q − q −1 ) =  q  0 = q n δij .

for i = j , m = j , r = i for i < j , m = i , r = j for i = j = m = r otherwise

19 Solutions to Exercises

Chapter 4 4.2

For a ∈ Z/(2), b ∈ Z/(5), put x = a ⊗ b ∈ Z/(2)⊗Z Z/(5). So we have: 5x = 5(a ⊗ b) = a ⊗ 5b = a ⊗ 0 = a ⊗ (0 × 0) = 0a ⊗ 0 = 0 ⊗ 0 = 0 , 2x = 2(a ⊗ b) = 2a ⊗b = 0 ⊗ b = 0 . Hence x = (5 − 2 × 2)x = 0 − 0 = 0. Elements of the form x generate, so Z/(2)⊗Z Z/(5) = {0}.

4.3

(a)

Let ⊗ denote ⊗Z . The multiplication and unit are given by (R⊗S)⊗(R⊗S) ∼ Z⊗Z η⊗η Z=

(b)



1⊗σ⊗1

(R⊗R)⊗(S⊗S)

µ⊗µ

R⊗S

R⊗S .

This makes it clear that µ is an abelian group morphism, so we automatically have distributivity. Now in terms of generating elements, the multiplication is (r ⊗ s)(r
⊗ s
) = (rr
) ⊗ (ss
) and the unit is 1 = 1 ⊗ 1 . Associativity and unit conditions only need to be checked on generators where they clearly follow from these conditions in R and S. / R ⊗S . Yes, ϕ(r) = r ⊗ 1 does define a ring morphism ϕ : R ϕ =

 R∼ = R⊗Z

1R ⊗η

R⊗S



is clearly an abelian group morphism. It remains to check that multiplication and unit are preserved: • ϕ(rr
) = (rr
) ⊗ 1 = (r ⊗ 1)(r
⊗ 1) = ϕ(r) ϕ(r
) ; • ϕ(1) = 1 ⊗ 1 = 1 using the definition in (a). (c)

ϕ / ψ R⊗Z S o S , with ϕ(r) = r ⊗ 1 and ψ(s) = 1 ⊗ s ; Let R these are ring morphisms as in (b). These give our “coprojecf / g T o S with f, g ring morphisms tions”. Now given R

121

122

Quantum Groups: A Path to Current Algebra and T commutative, we must show there is a unique ring mor/ T with h ◦ ϕ = f , h ◦ ψ = g . These phism h : R⊗S   last equations force us to define h(r ⊗ s) = h (r ⊗ 1)(1 ⊗ s) = /T , h(r ⊗ 1)h(1 ⊗ s) = f (r) g(s) . It is easily checked that R×S / f (r) g(s) is bilinear. So h does give an abelian group (r, s)  morphism. It remains to show h preserves multiplication and unit.     h (r ⊗ s)(r
⊗ s
) = h rr
⊗ ss
= f (rr
) g(ss
) = f (r) f (r
) g(s) g(s
) = =

4.4

f (r) g(s) f (r
) g(s
) h(r ⊗ s) h(r
⊗ s
) .

So h is a ring morphism. Since the definition of h was forced, it is unique.  / A module M : R S is an abelian group with an abelian group / M , written µ(r ⊗ m ⊗ s) = rms , satisfying morphism µ : R⊗M ⊗S

r
(rms)s
= (r
r)m(ss
) , 1m1 = m . One can see that this agrees with the definition given in Chapter 4 (given left R- , right S-scalar multiplications satisfying (rm)s = r(ms) we define rms = (rm)s; all the distributive laws precisely summarize to trilinearity (over Z); conversely, given µ, define the two scalar multiplications by rm = rm1 and ms = 1ms). Now a left R⊗S op -module is an abelian group / M , written M with an abelian group morphism µ : (R⊗S)⊗M

⊗ ⊗ ⊗ ⊗ µ(r s m) = (r  s)m , satisfying (1 1)m = m and (r r ⊗ ss )m = (r
⊗ s
) (r ⊗ s)m . Clearly to give the abelian group morphisms µ, µ are the “same thing” via the diagram: R⊗M ⊗S

µ

∼ = 1⊗σ

R⊗S⊗M

4.5

M . µ

Moreover the conditions on µ directly translate to those on µ.  /M  /N  /L Suppose R S T U . As an abelian group we have: M ⊗S N ⊗T L = B where B is the subset of the abelian group FZ (M × N × L) consisting of all elements of the forms: (m + m
, n, l) − (m, n, l) − (m
, n, l) , (m, n + n
, l) − (m, n, l) − (m, n
, l) , (m, n, l + l
) − (m, n, l) − (m, n, l
) , (ms, n, l) − (m, sn, l) , (m, nt, l) − (m, n, tl) .

Solutions to Exercises

123

The equivalence class of (m, n, l) is denoted by m ⊗ n ⊗ l. We now define  / U . Then r(m ⊗ n ⊗ l)µ = (rm) ⊗ n ⊗ (lµ) yielding M ⊗S N ⊗T L : R ∼ HomU R (M ⊗S N ⊗T L, K) = Mult (M, N, L; K) . Chapter 5 5.2

We use the Fundamental Theorem of Morita Theory. Suppose M is finitely generated and projective. By Theorem 5.2, we have the / R satisfying / M ∗⊗ M , e : M ⊗ M ∗ morphisms d : R R R 

M∗

 1⊗e M ∗ ⊗R M ⊗R M ∗ M ∗ = 1M ∗ and (e ⊗ 1) ◦ (1 ⊗ d) = 1M .

d⊗1

   d / σ / σ / M ∗ ⊗R M M ⊗R M ∗ , e
= M ∗ ⊗R M Put d
= R  e / R . We can now apply Theorem 5.2(iii) (replacing M , M ⊗R M ∗ N , e, d with M ∗ , M , e
, d
respectively) and by (iv) M ∗ is finitely generated and projective. 5.3

∗ / Hom (M, L) is “natural” in M (and Notice that ρM L : M ⊗R L R in L too for that matter), meaning that for any module morphism f : / N : R  / S , the following “f -square” commutes: M

N ∗ ⊗R L

ρN L

HomR (N, L)

f ∗ ⊗1

◦f

M ∗ ⊗R L

ρM L

HomR (M, L) .

Suppose now that M is a retract of a Cauchy module N ; so we have / M , r ◦ i = 1 and ρN invertible. We can /N, r : N i: M L M show that the composite Hom (M, L)

◦r

Hom (n, L)

−1 (ρN L)

N ∗ ⊗L

i∗ ⊗1

M ∗ ⊗L

is an inverse for ρM L . This is seen as follows: ∗ N −1 N −1 ρM ◦ ( ◦ r) = ( ◦ i) ◦ ρN ◦ ( ◦ r) L ◦ (i ⊗ 1) ◦ (ρL ) L ◦ (ρL )

(by the i-square) = ( ◦ i) ◦ 1Hom(N,L) ◦ ( ◦ r) = ( ◦ i) ◦ ( ◦ r) =

◦ (ri) =

◦ 1M = 1Hom(M,L)

124

Quantum Groups: A Path to Current Algebra and −1 ∗ N −1 ∗ (i∗ ⊗ 1) ◦ (ρN ◦ ( ◦ r) ◦ ρM ◦ ρN L) L = (i ⊗ 1) ◦ (ρL ) L ◦ (r ⊗ 1) (by the r-square)

= (i∗ ⊗ 1) ◦ (r∗ ⊗ 1) = (ri)∗ ⊗ 1 = 1M ∗ ⊗ 1L = 1M ∗ ⊗L establish the inverses, as required. Chapter 6 6.1

(a)

(b)

(c)

(d)

/ End (M ⊗N ) , whereby g  / m ⊗ n  / We have that G R   (gm)⊗ (gn) , is a monoid morphism since 1  / 1M⊗N and gh  / m ⊗n  / (hm) ⊗ (hn)  / (ghm) ⊗ (ghn) . It extends to a / End (M ⊗ N ) . So M ⊗ N unique R-algebra morphism R(G) R R R is an R(G)-module. / Hom (M, L) be the R-module morLet µ ˆ(g) : HomR (M, L) R phisms given by µ ˆ(g)(µ)(m) = gµ(g −1 m). Then we have that µ ˆ(1)(µ)(m) = µ(m), ˆ(gh)(µ)(m) = ghµ(h−1 g −1 m) =  and µ  −1 gµ ˆ(h)(µ)(g m) = µ ˆ(g) ◦ µ ˆ(h) (µ)(m). / End Hom (M, L) is a monoid morphism. Hence So µ ˆ:G R R HomR (M, L) becomes an R(G)-module.   evM g ◦ (m ⊗ µ) = evM (gm ⊗ gµ) = (gµ)(gm) = g µ(g −1 gm) = g µ(m) = g evM (m ⊗ µ), so evM preserves the R(G)-action. / Hom (M, M ⊗ N ) , n  We also need d : N R R to be an R(G)-module morphism. We have

/ (m 

/ m ⊗ n)

  d(gn)(m) = m ⊗ gn = g (g −1 m) ⊗ n   = gd(n)(g −1 m) = gd(n) (m) , so d(gn) = gd(n). The required isomorphism is the restriction of   HomR N, HomR (M, L) f (g ◦ ) ◦ d

o



∼ = 

/

HomR (M ⊗R N, L) e ◦ (1M ⊗ f ) g

to R(G)-module morphisms f and g; since e and d are such, so then are e ◦ (1M ⊗ f ) and (g ◦ ) ◦ d when f and g are. (This will be generalized from R(G) to an arbitrary Hopf algebra in Chapter 9.)

Solutions to Exercises

125

Chapter 7 7.1

There is a little abuse of notation here since the four parts to part (a), for example, are elements of C⊗R C, R⊗R C⊗R C, C⊗R C⊗R R, C⊗R R⊗R C respectively. But these modules are canonically isomorphic, and so “=” really means “corresponds under the canonical isomorphism to”. (a)

To prove the first part: σ

C⊗C

ε⊗1

C⊗C

R⊗C 1⊗δ

δ ∼ =

C

σ

1⊗ε

C⊗R

δ ⊗1

C⊗C⊗R

∼ =

R⊗C⊗C

∼ =

δ

C⊗C (b)

Similarly for the second: C⊗C⊗C δ

C

δ

δ ⊗1

C⊗C

1⊗δ

C⊗C⊗C

ε⊗1⊗1 ∼ =

ε⊗1⊗1

R⊗C⊗C

1⊗σ

R⊗C⊗C

∼ =

σ

C⊗C (c)

For the third: C⊗C⊗C δ δ ⊗1

C

δ

C⊗C

∼ =

1⊗σ

ε⊗1⊗1

R⊗C⊗C

C⊗C⊗C ε⊗1⊗1

1⊗σ

R⊗C⊗C

1⊗1⊗ε

1⊗ε

∼ =

C⊗R ∼ =

R⊗C⊗R

ε⊗ε⊗1 1⊗ε⊗1 1⊗σ

R⊗R⊗C

126

Quantum Groups: A Path to Current Algebra

Chapter 9 9.1

(a)

Suppose I is a coideal of C, that is, a submodule satisfying δ(I) ⊆ I⊗C + C⊗I and ε(I) = 0. The composite morphism C

δ

C⊗C

ρ⊗ρ

C/I ⊗ C/I

maps I to 0 since (ρ ⊗ ρ)δ(I) ⊆ (ρ ⊗ ρ)(I⊗C + C⊗I) = ρ(I) ⊗ ρ(C) + ρ(C) ⊗ ρ(I) = 0 ⊗ C/I + C/I ⊗ 0 = 0. So there exists a unique / C/I ⊗ C/I such that δ ◦ ρ = module morphism δ : C/I (ρ ⊗ ρ) ◦ δ. Similarly, ε(I) = 0 implies that there exists a unique / R with ε ◦ ρ = ε. These properties of δ, ε will mean ε : C/I / C/I is a coalgebra morphism once we know C/I is a ρ: C / C/I ⊗ C/I , coalgebra. To prove coassociativity of δ : C/I take the coassociativity diagram for C/I and precompose with / C/I : ρ: C C

ρ

C/I

δ

C/I ⊗ C/I

ρ⊗ρ

δ ⊗1 1⊗δ

C/I ⊗ C/I ⊗ C/I

ρ⊗ρ⊗ρ

δ

C⊗C

δ ⊗1 1⊗δ

C⊗C⊗C

The result commutes by coassociativity of C. But ρ is surjective; so the coassociativity diagram for C/I commutes. Similarly we / R is a counit. If C is a bialgebra and can prove ε : C/I I is also an ideal, certainly C/I becomes an algebra. All that remains to check are the extra bialgebra conditions (see Proposition 7.5). The main one, showing that δ preserves multiplication, is obtained by precomposing the diagram with ρ ⊗ ρ and using the corresponding condition for C. This gives the result since ρ ⊗ ρ is surjective. Since ρ is a bialgebra morphism, the only possible way C/I can become a Hopf algebra is to have ν ◦ ρ = ρ ◦ ν. This forces us to ask whether ν(I) ⊆ I for the antipode of C. (b)

Put B = R x, y, z. The given equations define algebra mor/ B ⊗B, ε : B / R since B is free as an algephisms δ : B bra. By Proposition 7.5, it remains to see that these morphisms make B a coalgebra. First look at the coassociativity: (δ ⊗ 1)δ(x) = (δ ⊗ 1)(x ⊗ x) = x ⊗ x ⊗ x = (1 ⊗ δ)(x ⊗ x) = (1 ⊗ δ)δ(x) .

Solutions to Exercises

127

Similarly for y: (δ ⊗ 1) δ(z) = (δ ⊗ 1)(1 ⊗ z + z ⊗ x) = 1 ⊗ 1 ⊗ z + (1 ⊗ z + z ⊗ x) ⊗ x = 1 ⊗ (1 ⊗ z + z ⊗ x) + z ⊗ x ⊗ x = (1 ⊗ δ)(1 ⊗ z + z ⊗ x) = (1 ⊗ δ)δ(z) . Then check the counit conditions: (ε ⊗ 1) δ(x) = (ε ⊗ 1)(x ⊗ x) = ε(x)x = x = (1 ⊗ ε)(x ⊗ x) = (1 ⊗ ε)δ(x) , and similarly for y: (ε ⊗ 1)δ(z) = (ε ⊗ 1)(1 ⊗ z + z ⊗ x) = z + 0x = z = 0 + z = (1 ⊗ ε)(1 ⊗ z + z ⊗ x) = (1 ⊗ ε) δ(z) . (c)

We have: δ(xy − 1) (x ⊗ x)(y ⊗ y) − 1 ⊗ 1 = xy ⊗ xy − 1 ⊗ 1 = (xy − 1) ⊗ xy + 1 ⊗ xy − 1 ⊗ 1 = (xy − 1) ⊗ xy + 1 ⊗ (xy − 1) ⊆ I⊗B + B⊗I .

(d)

We must check: H

δ

ν ⊗1

H⊗H

1⊗ν

ε

H⊗H

µ

H

η

R µ(ν ⊗ 1) δ(x) = µ(ν ⊗ 1)(x ⊗ x) = µ(y ⊗ x) = yx ≡ 1 = νε(x) ≡ xy = µ(x ⊗ y) = µ(1 ⊗ ν)(x ⊗ x) = µ(1 ⊗ ν) δ(x) . Similarly for y: µ(ν ⊗ 1) δ(z) = µ(ν ⊗ 1)(1 ⊗ z + z ⊗ x) = µ(1 ⊗ z + (−zy) ⊗ x) = z − zyx ≡ 0 = ηε(z) = 0 = −zy + zy = µ(1 ⊗ (−zy) + z ⊗ y) = µ(1 ⊗ ν)(1 ⊗ z + z ⊗ x) = µ(1 ⊗ ν) δ(z) . So H is a Hopf algebra. By Proposition 9.1, ν reverses both multiplication and comultiplication. The formulas for ν r (z) are trivial for r = 0 or 1. Also ν 2n (z) = xn zy n implies: ν 2n+1 (z) = ν(y)n ν(z)ν(x)n = xn (−zy)y n = −xn zy n+1

128

Quantum Groups: A Path to Current Algebra which gives: ν 2n+2 (z) = −ν(y)n+1 ν(z)ν(x)n = −xn+1 (−zy)y n = xn+1 zy n+1 . So the formulas follow by induction. If ν had finite order we would have either xn z = zxn , or xn z = −zxn+1 , which are both false in H.

(e)(i)

Evaluate each of ν, δ and ε at (xn zy n − z) : ν(xn zy n − z) = ν(y)n ν(z)ν(x)n − ν(z) = xn (−zy)y n + zy = −xn zy n+1 + zy = (xn zy n − z)(−y) ∈ In , δ(x zy − z) = (xn ⊗ xn )(1 ⊗ z + z ⊗ x)(y n ⊗ y n ) − 1 ⊗ z − z ⊗ x n

n

= (xn ⊗ xn z + xn z ⊗ xn+1 )(y n ⊗ y n ) − 1 ⊗ z − z ⊗ x = xn y n ⊗ xn zy n + xn zy n ⊗ xn+1 y n − 1 ⊗ z − z ⊗ x = 1 ⊗ xn zy n + xn zy n ⊗ x − 1 ⊗ z − z ⊗ x = 1 ⊗ (xn zy n − z) + (xn zy n − z) ⊗ x ∈ In ⊗H + H⊗In , ε(xn zy n − z) = 0 . So In is a Hopf ideal in H. Also compute: ν(xn − 1) = y n − 1 ≡ −y n (xn − 1) ∈ Jn , δ(xn − 1) = xn ⊗ xn − 1 ⊗1 = (xn − 1) ⊗ xn + 1 ⊗ (xn − 1) ∈ Jn ⊗H + H⊗Jn , ε(x − 1) = 1n − 1 = 0 . n

So Jn is a Hopf ideal in H. (e)(ii)

ν 2n (x) = x and ν 2n (y) = y since ν just switches x and y. ν 2n (z) = xn zy n ≡ z ν

2n

(mod In )

(z) = x zy ≡ |z| = z n

n

(mod Jn ) .

Chapter 10 10.1

Let M be a Cauchy R-module. The diagrams (coassociativity and counit) showing E to be a coalgebra are: MM∗

1⊗d⊗1

M M ∗M M ∗

1⊗1⊗1⊗d⊗1 1⊗d⊗1⊗1⊗1

M M ∗M M ∗M M ∗

Solutions to Exercises MM∗

1⊗d⊗1

129 e⊗1⊗1

M M ∗M M ∗

MM∗ .

1⊗1⊗e

These follow from functionality of ⊗R and Theorem 5.2. / δˆ characterizes a bijection between R-linear functions Certainly δ  / C . The / C⊗ M and R-linear functions ω : M ⊗ M ∗ δ: M R R / ∨ ω is given by: inverse assignment ω    1⊗d ω ⊗1 ∨ ω = M M M ∗M CM . (That these assignments are mutually inverse follows from the properties of e and d in Theorem 5.2.) It remains to see that coaction axioms on δ translate precisely to coalgebra morphisms on ω. We shall do the translation for the coaction axiom: δ ⊗1

δ

M

CM

CCM

1⊗δ

(where δ is the comultiplication of C). This is equivalent to: δ ⊗1

MM∗

CM M ∗

δ ⊗1⊗1

CCM M ∗

1⊗δ ⊗1

1⊗1⊗e

Using: δ ⊗1

MM∗

δ ⊗1⊗1

CM M ∗

CCM M ∗

1⊗e

1⊗1⊗e

δˆ

C

CC

δ

and 1⊗d⊗1

MM∗

M M ∗M M ∗ δ ⊗1⊗1

CM M ∗ M M ∗

δ ⊗1

δˆ⊗1

1⊗1⊗δ ⊗1 1⊗e⊗1⊗1

CM M

∗

CM M ∗

1

we see that the axiom becomes: δˆ

MM∗

C δ

1⊗d⊗1

M M ∗M M ∗

δˆ⊗1

CM M ∗

ˆ which is a coalgebra morphism axiom on δ.

1⊗δˆ

CC

CC .

130

Quantum Groups: A Path to Current Algebra

Chapter 11 11.1 (a)

cA,I = cA,I ◦ cA,I from the triangle below. Since cA,I is invertible, cA,I = 1A . Similarly cI,A = 1A . A

cA,I⊗I

cA,I ⊗1I

A

1I ⊗cA,I

A The hexagon can be subdivided into commutative regions thus: ACB

c ⊗1

CAB

1⊗c

1⊗c cAB,C

ABC

CBA . cBA,C c ⊗1

c ⊗1

BAC (b)

Put γ =



1⊗c

BCA

µi ⊗ vi ∈ A⊗A so that cM,N (m ⊗ n) =

i



(µi n) ⊗ (vi n):

i



m ⊗ µi l ⊗ vi n

c⊗1



µj µi l ⊗ vj m ⊗ vi n

i,j

i 1⊗c

1⊗c

X.

m ⊗ n ⊗l c⊗1

c ⊗1





µi n ⊗ vi m ⊗ l 1⊗c

i

µi n ⊗ µj l ⊗ vj vi m

i,j

The hexagon gives us the condition:   X= µj µi l ⊗ µk vi n ⊗ vk vj m = µk µj l ⊗ vk µi n ⊗ vj vi m i,j,k

i,j,k

in A⊗A⊗A . Diagrammatically this becomes: R

γ ⊗γ ⊗γ

A⊗6

σ315264 σ536142

A⊗6

µ⊗µ⊗µ

A⊗3 .

Solutions to Exercises

131

(c) A

B

C

A

B

C

C

B

A

=

C

B

A

Bradshaw: Elegant Action Figure Group, [Wal94, Plate 79].

132

Bradshaw: “Broad Clothes Peg Figure Group”, Clothes Peg Figure Period, [Wal94, Plate 71].

References

[Abe80] Eiichi Abe. Hopf Algebras, volume 74 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1980. MR594432 [Dri87] Vladimir G. Drinfel
d. Quantum groups. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pp. 798–820, Amer. Math. Soc., Providence, RI, 1987. MR934283

Vladimir G. Drinfel
d. Quasi-Hopf algebras and Knizhnik–Zamolodchikov equations. In Problems of Modern Quantum Field Theory (Alushta, 1989), Res. Rep. Phys., pp. 1–13. Springer, Berlin, 1989. Previously: Acad. Sci. Ukr. (Preprint ITP-89-43E, 1989). MR1091757 [EK66] Samuel Eilenberg and G. Max Kelly. Closed categories. In Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), pp. 421–562. MR0225841 Springer, New York, 1966.
[FRT88] Ludwig D. Fadde ev, Nikolai Yu. Reshetikhin, and Leon A. Takhtajan. Quantization of Lie groups and Lie algebras. In Algebraic Analysis, Vol. I, pp. 129–139. Academic Press, Boston, MA, 1988.

[Dri89]

MR992450

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[JS91c] [JS93] [JS95]

Peter J. Freyd and David N. Yetter. Braided compact closed categories with applications to low-dimensional topology. Adv. Math., 77(2):156–182, 1989. MR1020583 Andr´e Joyal and Ross Street. The geometry of tensor calculus. I. Adv. Math., 88(1):55–112, 1991. MR1113284 Andr´e Joyal and Ross Street. An introduction to Tannaka duality and quantum groups. In Category Theory (Como, 1990), volume 1488 of Lecture Notes in Mathematics, pp. 413–492. Springer, Berlin, MR1173027 1991. Andr´e Joyal and Ross Street. Tortile Yang–Baxter operators in tensor categories. J. Pure Appl. Algebra, 71(1):43–51, 1991. MR1107651 Andr´e Joyal and Ross Street. Braided tensor categories. Adv. Math., MR1250465 102(1):20–78, 1993. Andr´e Joyal and Ross Street. The category of representations of the general linear groups over a finite field. J. Algebra, 176(3):908–946, MR1351369 1995. 133

134 [JSV96] Andr´e Joyal, Ross Street, and Dominic Verity. Traced monoidal categories. Math. Proc. Cambridge Philos. Soc., 119(3):447–468, 1996. MR1357057

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[Maj95] [Man88]

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[Wal94]

[Yet01]

Christian Kassel. Quantum Groups, volume 155 of Graduate Texts in Mathematics. Springer, New York, 1995. MR1321145 G. Max Kelly and Ross Street. Review of the elements of 2-categories. In Category Seminar (Proc. Sem., Sydney, 1972/1973), volume 420 of Lecture Notes in Mathematics, pp. 75–103, Springer, Berlin, MR0357542 1974. Piergiulio Katis, Nicoletta Sabadini, and Robert F. C. Walters. Feedback, trace and fixed-point semantics. Theor. Inform. Appl., MR1948768 36(2):181–194, 2002. Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics, Springer, New York, MR0354798 1971. Shahn Majid. Foundations of Quantum Group Theory. Cambridge University Press, Cambridge, 1995, (paperback, 2000). MR1381692 Yuri I. Manin. Quantum groups and noncommutative geometry. Universit´e de Montr´eal Centre de Recherches Math´ematiques, Montreal, QC, 1988. MR1016381 Shum Mei Chee. Tortile tensor categories. J. Pure Appl. Algebra, MR1268782 93(1):57–110, 1994. Neantro Saavedra Rivano. Cat´egories Tannakiennes, volume 265 of MR0338002 Lecture Notes in Mathematics Springer, Berlin, 1972. Steven Shnider and Shlomo Sternberg. Quantum Groups: From CoAlgebras to Drinfel
d Algebras. Graduate Texts in Mathematical Physics, II. International Press, Cambridge, MA, 1993. MR1287162 Moss E. Sweedler. Hopf Algebras. Mathematics Lecture Note Series. MR0252485 W. A. Benjamin, Inc., New York, 1969. V. G. Turaev. The Yang–Baxter equation and invariants of links. MR939474 Invent. Math., 92(3):527–553, 1988. Karl-Heinz Ulbrich. Tannakian categories for non-commutative Hopf algebras. In International Conference on Hopf Algebras, Beer Sheva, January 1989. Grahame L. Walsh. Bradshaws: Ancient Rock Paintings of Northwestern Australia. Edition Limit´ee, 1994. Privately commissioned by the Bradshaw Foundation. David N. Yetter. Functorial Knot Theory, volume 26 of Series on Knots and Everything. World Scientific Publishing Co. Inc., River Edge, NJ, 2001. MR1834675

Index

2-category, 100, 102 2-cell, 97 2-category, 100 2-functor, 101, 102 identity, 98 pasting, 97 2-functor, 101, 102 A-module, 101 action, 101 arrow, 101 left, 101 A-point, 55 action, 59, 101, 111 of an A-module, 101 adjoining an antipode, 117 an inverse for the determinant, 56 left-dual objects, 117 adjoint left, 111 left 2-adjoint, 111 Lie algebra, 33 algebra exterior, 31 Hopf, 51 morphism, 27 of endomorphisms, 28 of formal power series, 42 opposite, 30 over R, 27 polynomial algebra, 45 polynomial R-algebra, 46 R-algebra, 27 symmetric, 45 tensor, 28 universal, 87 universal enveloping, 40 algebraic geometry, 5 algebraic varieties, 5 antipode, 7, 51, 105, 117

appropriate representation, 109 arrow between monoid arrows, 97 composition, 78 monoid, 97 of category, 6 associativity, 2, 27, 67 autonomous, 82 left, 81 left/right, 82 balanced, 106 YB-operator, 88 bialgebra, 76 Banach spaces, 79 example of closed symmetric tensor category, 79 bialgebra balanced, 76 braided, 72 commutative over k, 7 matrix bialgebra, 46 morphism, 44 quasitriangular, 72 R-bialgebra, 42 symmetric, 72 triangular, 72 biclosed, 78 bijection canonical, 1 bilinear R-bilinear, 15 universal bilinear function, 15 bimodule left R-/right S-bimodule, 16 bimonoid, 105 balanced, 106 cobalanced, 108 cobraided, 108 cotortile, 108

135

136 in ModR , 105 in V , 105 strict quasi-bimonoid, 105 tortile, 108, 114 with a braiding, 106 with a twist, 106 Bradshaws, xvii Age of String, xviii Bradshaw Foundation, xvii Edition Limit´ ee, xvii Grahame Walsh, xvii silhouette, viii, 8, 20, 26, 36, 46, 58, 66, 92, 116, 118, 131, 132, 141 braid category, 69 composition, 69 group, of Artin, 69 tensor product, 69 braided, 114 bialgebra, 72 tensor category, 68 braiding, 68, 83, 105 element, 72 for a bimonoid, 105 for a tensor category, 68 Brown, Ronald, xviii Byrne, Simon, xvi C ∗ -algebra commutative, 5 Cartan, Eli, 33 cartesian product, x, 1 of no sets, 1 category, xi dual, xii functor category, xiv functors, xiii monoidal, 67 morphisms, xi objects, xi of braids, 69 of monoids in V , 100 opposite, xii strict tensor, 104 tensor, 67 Cauchy H-comodule, 65 module, 21 closed functor, 86 left-closed, 77 tensor category, 78 co-element co-T -elements, x coaction, 62

Index coalgebra morphism, 38 over a ring, 37 primitive element, 41 set-like element, 40 cobraiding, 108 coCauchy, 47 cocommutative coalgebra, 38 codomain target, xii coend, 115 coherence theorem of Mac Lane, 68, 109 commutative, 3, 30 Lie algebra, 35 monoid, 3 R-algebra, 30 rig, 4 commutativity, 3 commutator defines a Lie bracket, 32 commute with diagonals, 45 comodule, 59, 62 left, 62 comodule morphism, 62 composable morphisms, xii composite pasted, 100 composite morphism, xii composition arrow, 78 of braids, 69 vertical, 98 comultiplication, 7, 8, 37 constraint associativity, 67 left unit, 67 right unit, 67 convergent, 24 convolution product, 39 structure, 38 coordinate algebra, 9 quantum matrices, 9 coordinate k-algebra, 8 of general linear group, 8 coproduct, xiii coprojection, 8 coset, 14 H-coset, 14 coterminal object, xiii cotwist, 108 counit, 7, 37, 81

Index Cramer Rule quantum, 120 derivation, 32 diagonal, 1 formal, 37 in category X, 7 ternary, 2 direct sum of Lie algebras, 34 of modules, 24 disjoint union X +Y, x distributive, 4 domain source, xii Drinfeld, Vladimir G., 105 International Congress of Mathematicians, 1986, ix dual, 21 category, xii left dual, 21, 81 opposite, xii right dual, 81 duality principle, x dualizing object, 80 left, 79 right, 80 end, 109, 110 is a limit, 113 endomorphism algebra, 28 enrich, 18 abelian group with module structure, 18 equalizer, 109 essentially, 104 evaluation, 77 functor, 95 morphism, 19 exponential series, 30 exterior algebra, 31 Faddeev, Ludwig D., 10 field, 4 finitely generated, 13 forgetful functor, 31, 101, 102, 115 formal diagonal, 37 power series, 42 Formal Tannaka Duality, 111 free, 14 constructions, 31 module, 14 free module

137 from R to S, generated by X, 16 functor, 86 Freyd, Peter, 82 function complex-valued, 5 continuous, 5 diagonal, 1 identity, 1 functor, xiii essentially weak tensor, 103 evaluation, 95 forgetful, 101 natural transformation, xiv representable, xiv represented by T , xiii self-adjoint, 54 tensor, 85 weak tensor functor, 101 functor category, xiv fundamental theorem on coalgebras, 115 funny superscripts, 10 Gelfand duality, 5 general linear group, 8 commutative Hopf algebra, 8 coordinate k-algebra, 8 generalised co-element, x generate, 13 generic point, 10 geometric series, 28 Grassmannian algebra, 10 group, 2, 7 affine over k, 7 diagrammatic definition, 7 Lie group, 7 R-algebra, 29 topological group, 7 homomorphism generalised co-element, x homothety, 96 Hopf algebra, 7, 51 commutative, 7, 53 Hopf monoid, 113 quasi, 105 Hurwitz polynomials, 45 ideal, 29 in an algebra, 29 identity, 1, 2, 27, 84 2-cell, 98 identity morphism, xii incidence

138 projective plane, xi indeterminate, 42 initial object, xiii injective, 24 morphism, 24 internal hom, 77, 110 left, 77 right, 78 inverse left- and right-inverses agree, xiii invertibility, 2 invertible, 2 isomorphism, xiii morphism, xiii isomorphic, xiii isomorphism, xiii, 4 invertible, xiii isomorphic, xiii natural, xiv Jacobi identity, 32 Joyal, Andr´e, xvi, 91, 117 k-algebra, 4 coordinate algebra, 8 morphism, 4 Katis, Piergiulio, 117 Kobyzev, Yu., 10 Kronecker delta, 57 left adjoint, 111 left dual, 81 as a functor, 81 of a module, 21 of a signed set, 83 left inverse, xiii left R-linear, 14 left 2-adjoint, 111 left-closed, 77 Leibniz rule, 32 Lie, Sophus, 33 Lie algebra, 32, 33 adjoint, 33 commutative, 35 direct sum, 34 Lie bracket, 32 morphism, 33 universal enveloping, 33 Lie bracket, 32, 33 Lie group, 33 linearly independent, 14 Mac Lane, Saunders, 109, 113 Mac Lane

Index coherence theorem, 68, 109 Manin, Yuri I., 10, 64, 65 McCrudden, Paddy, xvi module Cauchy, 21 finitely generated, 13 from R to S, 16 left R-module, 13 morphism, 14 projective, 21 right R-module, 13 monoid, 2, 7, 97 affine over k, 7 arrow, 97 category, 100 commutative, 3 diagrammatic definition, 7 homomorphism, 3 morphism, 3 morphisms preserve invertibility, 4 quasi-Hopf, 105 R-algebra, 29 monoidal category tensor category, 67 monoidal functor tensor functor, 85 Moore, Ross LATEX2HTML, xvii XY-pic, xvi Morita Theory, 62 fundamental theorem, 22 morphism algebra, 5 comodule, 62 composable, xii composite, xii evaluation, 19 identity, xii invertible, xiii left inverse, xiii map of varieties, 6 module morphism, 14 of bialgebras, 44 of C ∗ -algebras, 5 of coalgebras, 38 of k-algebras, 4, 6 of Lie algebras, 33 of monoids, 3 of R-algebras, 27 of rigs, 4 retraction, 21 right inverse, xiii morphisms, xi multilinear function, 17

Index multiplication opposite, 13 scalar, 13 multiplicative matrices, 64 natural family, 68 natural isomorphism, xiv natural numbers, 4 example of a rig, 4 natural transformation, xiv, 102 naturality, xiv non-commutating indeterminates, 9 object coterminal, xiii initial, xiii of category, 6 representing, xiv terminal, xiii, 7 unit, 67 objects, xi opposite algebra, 30 category, xii dual, xii multiplication, 13 pasted composite, 100 pasting, 99 2-cells, 97 Planck constant, 9 Poincar´ e–Birkhoff–Witt, 35 point, 5, 6 as algebra morphism, 5 B-point, 6 B-point of a k-algebra, 6 of a k-algebra, 6 primitive element in a coalgebra, 41 product in category X, 6 of modules, 24 of objects, xii tensor, 67 tensor product, 7 projection, xii, 24 universal property, xii projective, 21 projective plane, xi axioms, xi incidence, xi reverse, P rev , xi QIST, 10

139 quadratic algebra, 80 category QA, 80 morphism, 80 quantization, 9 deforming commutative algebras to non-commutative ones, 9 quantum Cramer Rule, 120 deformation, 57 determinant, 57 general linear group, 12, 57 group, 108, 115 group over R, 108 inverse scattering transform, 10 matrices, 9 plane, 10, 80 spaces, 9 special linear group, 12 superplane, 10, 81 quantum group cotortile bimonoid in ModR , 108 quantum spaces correspond to k-algebras, 9 quasi-bimonoid, 104 in V , 104 quasitriangular bialgebra, 72 R-algebra, 27 commutative, 30 group, 29 monoid, 29 skew commutative, 31 symmetric, 30 R-coalgebra, 37 representable functor, xiv representation, 29 appropriate, 109 of group on monoid, 29 representing object, xiv restriction of scalars, 59, 101 retract, 21 retraction morphism, 21 reverse-arrow universal property, 63 ribbons tangles, 84 YB-operator, 88 rig, 4 commutative, 4 morphism, 4 natural numbers, 4 right inverse, xiii ring, 4

140 with opposite multiplication, 13 Rivano, see Saavedra Rivano R-Lie algebra, 32 R-module derivation, 32 two-sided, 29 Saavedra Rivano, Neantro, 109 Sabadini, Nicoletta, 117 scalar multiplication, 4 self-adjoint, 54 set-like element in a coalgebra, 40 Shum Mei Chee, 84, 91 signed sets, 83 skew commutative, 31 small sets, 86 source of a tangle, 83 space seen from the other side of your brain, 5 span, 13 strict, 104 strong monoidal functor tensor functor, 85 submodule, 14 generated by a subset, 15 supergeometry, 10 Sweedler, Moss E., 115 switch, 3, 70 symmetric R-algebra, 30 tensor category, 69, 79 symmetry for a tensor category, 68 taking off your belt, 91 Talent, Ross, xvi tangle, 83 autonomous braided category, 83 geometric, 82 source, 83 tangles on ribbons, 84 tangles on strings, 82 target, 82 Tannaka duality, 115, 117 duality theorem, 109 target codomain, xii of a tangle, 82, 83 tensor algebra, 28

Index functor, 85 object, 104 product, 67 tensor category, 67, 104 autonomous, 84 balanced, 73 braided, 68 closed, 78 free autonomous, 117 left autonomous, 117 opposite, 68 strict, 68, 70, 104 symmetric, 69 tortile, 82, 84 tensor functor balanced, 86, 114, 115 braided, 85, 114 closed, 86 left-closed, 86 monoidal functor, 85 preserves dualizability, 90 preserves duals, 86 preserves products, 87 right-closed, 86 strict, 85 strong monoidal functor, 85 symmetric, 86 takes YB-operator into YB-operator, 89 weak, 85 tensor product, 67 as composition of modules, 16 multiple, 16 of R-modules, 29 of braids, 69 represent bilinear function as module morphism, 18 tensor–hom, 61 terminal object, xiii, 7 twist, 73, 84, 106 element, 75 two-sided R-module, 29 Ulbrich, Karl-Heinz, 114 unit, 81 left, 67 object, 67 right, 67 universal algebra, 87 property, 115 universal enveloping algebra, 33, 40, 46, 87 is a cocommutative bialgebra, 45

141 universal property, x end, 110 for internal hom, 78 internal hom, 110 reverse-arrow, 63 up to coherent isomorphism, 87 Vaughan, Elaine, xvi vector space over R, 13 Verity, Dominic R., 117 Walsh, Grahame, see Badshaws, xvii Walters, Robert F. C., 117 weak tensor functor essentially, 103 takes monoids to monoids, 101 whisker, 98–100 Williams, Sam, xvi XY-pic Ross Moore, xvi Kristoffer Rose, xvi Yang–Baxter, 87 hexagon, 87 YB-operator, 87 YB-hexagon, 87, 93 YB-operator balanced, 88, 89 compatability with duals, 89 dualizable, 89 given by braiding, 89 in braided tensor category, 89 left-dualizable, 89 tortile, 90, 91, 96 under tensor functors, 89 Yetter, David, 82 Yoneda Lemma, xiv

Bradshaw: Schematized Bradshaw Group, [Wal94, Plate 45].