155 Graduate Texts in Mathematics Editorial Board I.H. Ewing F.W. Gehring P.R. Halmos Graduate Texts in Mathematics TAKEUTIIZARING. Introduction to Axiomatic 33 HIRSCH. Differential Topology. Set Theory. 2nd ed. 34 SPITZER. Principles of Random Walk. 2nd ed. 2 OXTOBY. Measure and Category. 2nd ed. 35 WERMER. Banach Algebras and Several 3 SCHAEFFER. Topological Vector Spaces. Complex Variables. 2nd ed. 4 HILTON/STAMMBACH. A Course in 36 KELLEy/NAMIOKA et al. Linear Topological Homological Algebra. Spaces. 5 MAC LANE. Categories for the Working 37 MONK. Mathematical Logic. Mathematician. 38 GRAUERT/FRITZSCHE. Several Complex 6 HUGHES/PIPER. Projective Planes. Variables. 7 SERRE. A Course in Arithmetic. 39 ARVESON. An Invitation to C*-Algebras. 8 TAKEUTI/ZARING. Axiomatic Set Theory. 40 KEMENy/SNELL/KNAPP. Denumerable Markov 9 HUMPHREYS. Introduction to Lie Algebra~ Chains. 2nd ed. and Representation Theory. 41 ApOSTOL. Modular Functions and Dirichlet 10 COHEN. A Course in Simple Homotopy Series in Number Theory. 2nd ed. Theory. 42 SERRE. Linear Representations of Finite 11 CONWAY. Functions of One Complex Groups. Variable. 2nd ed. 43 GILLMAN/JERISON. Rings of Continuous 12 BEALS. Advanced Mathematical Analysis. Functions. I3 ANDERSON/FULLER. Rings and Categories of 44 KENDIG. Elementary Algebraic Geometry. Modules. 2nd ed. 45 LoEVE. Probability Theory I. 4th ed. 14 GOLUBITSKy/GUILEMIN. Stable Mappings and 46 LoEvE. Probability Theory II. 4th ed. Their Singularities. 47 MOISE. Geometric Topology in Dimensions 2 15 BERBERIAN. Lectures in Functional Analysis and 3. and Operator Theory. 48 SACHSlWu. General Relativity for 16 WINTER. The Structure of Fields. Mathematicians. 17 ROSENBLATT. Random Processes. 2nd ed. 49 GRUENBERGIWEIR. Linear Geometry. 2nd ed. 18 HALMos. Measure Theory. 50 EDWARDS. Fermat's Last Theorem. 19 HALMos. A Hilbert Space Problem Book. 51 KLINGENBERG. A Course in Differential 2nd ed. Geometry. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 52 HARTSHORNE. Algebraic Geometry. 21 HUMPHREYS. Linear Algebraic Groups. 53 MANIN. A Course in Mathematical Logic. 22 BARNES/MACK. An Algebraic Introduction to 54 GRAVERIW ATKINS. Combinatorics with Mathematical Logic. Emphasis on the Theory of Graphs. 23 GREUB. Linear Algebra. 4th ed. 55 BROWN/PEARCY. Introduction to Operator 24 HOLMES. Geometric Functional Analysis and Theory I: Elements of Functional Analysis. Its Applications. 56 MASSEY. Algebraic Topology: An 25 HEWITT/STROMBERG. Real and Abstract Introduction. Analysis. 57 CROWELL/Fox. Introduction to Knot Theory. 26 MANES. Algebraic Theories. 58 KOBUTZ. p-adic Numbers, p-adic Analysis, 27 KELLEY. General Topology. and zeta-Functions. 2nd ed. 28 ZARISKIISAMUEL. Commutative Algebra. 59 LANG. Cyclotomic Fields. Vol.l. 60 ARNOLD. Mathematical Methods in Classical 29 ZARISKIISAMUEL. Commutative Algebra. Mechanics. 2nd ed. Vol.lI. 61 WHITEHEAD. Elements of Homotopy Theory. 30 JACOBSON. Lectures in Abstract Algebra I. 62 KARGAPOLOVIMERLZJAKOv. Fundamentals of Basic Concepts. the. Theory of Groups. 31 JACOBSON. Lectures in Abstract Algebra II. 63 BOLLOBAS. Graph Theory. Linear Algebra. 64 EDWARDS. Fourier Series. Vol. I. 2nd ed. 32 JACOBSON. Lectures in Abstract Algebra III. Theory of Fields and Galois Theory. continued after index Christian Kassel Quantum Groups With 88 Illustrations Springer-Science+Business Media, LLC Christian Kassel Institut de Recherche Mathematique Avancee Universite Louis Pasteur-C.N.R.S. 67084 Strasbourg France Editorial Board J.H. Ewing F. W. Gehring P.R. Halmos Department of Department of Department of Mathematics Mathematics Mathematics Indiana University University of Michigan Santa Clara University Bloomington, IN 47405 Ann Arbor, MI 48109 Santa Clara, CA 95053 USA USA USA Mathematics Subject Classification (1991): Primary-17B37, 18DlO, 57M25, 81R50; Secondary-16W30, 17B20, 17B35, 18D99, 20F36 Library of Congress Cataloging-in-Publication Data Kassel, Christian. Quantum groups/Christian Kassel. p. cm. - (Graduate texts in mathematics; voI. 155) Includes bibliographical references and index. ISBN 978-1-4612-6900-7 ISBN 978-1-4612-0783-2 (eBook) DOI 10.1007/978-1-4612-0783-2 1. Quantum groups. 2. Hopf algebras. 3. Topology. 4. Mathematical physics. 1. Title. II. Series: Graduate texts in mathematics; 155. QC20.7.G76K37 1995 512'.55-dc20 94-31760 Printed on acid-free paper. © 1995 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 1995 Softcover reprint of the hardcover 1s t edition 1995 All rights reserved. This work may not be translated or copied in whole or in part without the written permission ofthe publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly anaJysis. Use in connection with any form of information storage and retrievaJ, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely byanyone. Production managed by Francine McNeill; manufacturing supervised by Genieve Shaw. Photocomposed pages prepared using Patrick D.F. Ion's TeX files. 987654321 ISBN 978-1-4612-6900-7 Preface {( Eh bien, Monsieur, que pensez-vous des x et des y ?» Je lui ai repondu : {( C'est bas de plafond. » V. Hugo [Hug51] The term "quantum groups" was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley (1986). It stands for certain special Hopf algebras which are nontrivial deformations of the enveloping Hopf algebras of semisimple Lie algebras or of the algebras of regular functions on the corresponding algebraic groups. As was soon ob served, quantum groups have close connections with varied, a priori remote, areas of mathematics and physics. The aim of this book is to provide an introduction to the algebra behind the words "quantum groups" with emphasis on the fascinating and spec tacular connections with low-dimensional topology. Despite the complexity of the subject, we have tried to make this exposition accessible to a large audience. We assume a standard knowledge of linear algebra and some rudiments of topology (and of the theory of linear differential equations as far as Chapter XIX is concerned). We divided the book into four parts we now briefly describe. In Part I we introduce the language of Hopf algebras and we illustrate it with the Hopf algebras SLq(2) and Uq(.s((2)) associated with the classical group 8L2. These are the simplest examples of quantum groups, and actually the only ones we treat in detail. Part II focuses on two classes of Hopf algebras that provide solutions of the Yang-Baxter equation in a systematic way. We review a method due to Faddeev, Reshetikhin, and Takhtadjian as well as Drinfeld's quantum double construction, both designed to produce quan tum groups. Parts I and II may form the core of a one-year introductory course on the subject. Parts III and IV are devoted to some of the spectacular connections alluded to before. The avowed objective of Part III is the construction of isotopy invariants of knots and links in R3, including the Jones polynomial, Preface VI from certain solutions of the Yang-Baxter equation. To this end, we intro duce various classes of tensor categories that are responsible for the close relationship between quantum groups and knot theory. Part IV presents more advanced material: it is an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations. Our aim is to highlight Drinfeld's deep result expressing the braided tensor category of modules over a quantum enveloping algebra in terms of the corresponding semisimple Lie algebra. We conclude the book with the construction of a "universal knot invariant". This is a nice, far-reaching application of the algebraic techniques developed in the preceding chapters. I wish to acknowledge the inspiration I drew during the composition of this text from [Dri87] [Dri89a] [Dri89b] [Dri90] by Drinfeld, [JS93] by Joyal and Street, [Tur89] [RT90] by Reshetikhin and Turaev. After having become acquainted with quantum groups, the reader is encouraged to return to these original sources. Further references are given in the notes at the end of each chapter. Lusztig's and Turaev's monographs [Lus93] [Tur94] may complement our exposition advantageously. This book grew out of two graduate courses I taught at the Department of Mathematics of the Universite Louis Pasteur in Strasbourg during the years 1990-92. Part I is the expanded English translation of [Kas92]. It is a pleasure to express my thanks to C. Bennis, R. Berger, C. Mitschi, P. Nuss, C. Reutenauer, M. Rosso, V. Turaev, M. Wambst for valuable discussions and comments, and to Raymond Seroul who coded the figures. lowe special thanks to Patrick Ion for his marvellous job in preparing the book for printing, with his attention to mathematical, English, typographical, and computer details. Christian Kassel March 1994, Strasbourg Notation. - Throughout the text, k is a field and the words "vector space", "linear map" mean respectively "k-vector space" and "k-linear map". The boldface letters N, Z, Q, R, and C stand successively for the nonnegative integers, all integers, the field of rational, real, and complex numbers. The Kronecker symbol l5ij is defined by l5ij = 1 if i = j and is zero otherwise. We denote the symmetric group on n letters by Sri' The sign of a permutation u is indicated by c(u). The symbol 0 indicates the end of a proof. Roman figures refer to the numbering of the chapters. Contents Preface v Part One Quantum 8L(2) 1 I Preliminaries 3 1 Algebras and Modules . 3 2 Free Algebras ...... 7 3 The Affine Line and Plane 8 4 Matrix Multiplication ... 10 5 Determinants and Invertible Matrices 10 6 Graded and Filtered Algebras 12 7 Ore Extensions . . 14 8 Noetherian Rings 18 9 Exercises 20 10 Notes ....... 22 II Tensor Products 23 1 Tensor Products of Vector Spaces 23 2 Tensor Products of Linear Maps 26 3 Duality and Traces ........ 29 4 Tensor Products of Algebras .. 32 5 Tensor and Symmetric Algebras 34 6 Exercises 36 7 Notes ............... 38 viii Contents III The Language of Hopf Algebras 39 1 Coalgebras.. 39 2 Bialgebras ............ 45 3 Hopf Algebras . . . . . . . . . . 49 4 Relationship with Chapter I. The Hopf Algebras GL(2) and SL(2). . . . . . . . . . . 57 5 Modules over a Hopf Algebra. . . . . . . . . . . . . . . 57 6 Comodules......................... 61 7 Comodule-Algebras. Coaction of SL(2) on the Affine Plane 64 8 Exercises 66 9 Notes.............................. 70 IV The Quantum Plane and Its Symmetries 72 1 The Quantum Plane . . . . . . . . . . . . . 72 2 Gauss Polynomials and the q-Binomial Formula 74 3 The Algebra Mq(2) ......... . 77 4 Ring-Theoretical Properties of Mq(2) . 81 5 Bialgebra Structure on Mq(2) . .... . 82 6 The Hopf Algebras GLq(2) and SLq(2) 83 7 Coaction on the Quantum Plane 85 8 Hopf *-Algebras 86 9 Exercises 88 10 Notes ..... . 90 V The Lie Algebra of SL(2) 93 1 Lie Algebras . . . . . 93 2 Enveloping Algebras . . 94 3 The Lie Algebra .5[(2) . 99 4 Representations of .5[(2) 101 5 The Clebsch-Gordan Formula. 105 6 Module-Algebra over a Bialgebra. Action of .5[(2) on the Affine Plane ....................... 107 7 Duality between the Hopf Algebras U(.5[(2)) and SL(2) 109 8 Exercises 11 7 9 Notes............................ 119 VI The Quantum Enveloping Algebra of .5[(2) 121 1 The Algebra Uq(.5[(2)) . . . . . . . . . . . . . 121 2 Relationship with the Enveloping Algebra of .5[(2) 125 3 Representations of Uq . . . . . . . . . . . . . . . . 127 4 The Harish-Chandra Homomorphism and the Centre of U 130 q Contents ix 5 Case when q is a Root of Unity. 134 6 Exercises 138 7 Notes .............. . 138 VII A Hopf Algebra Structure on Uis[(2)) 140 1 Comultiplication............. 140 2 Semi simplicity . . . . . . . . . . . . . . 143 3 Action of Uq(.s[(2)) on the Quantum Plane 146 4 Duality between the Hopf Algebras Uq(.s[(2)) and SLq(2) 150 5 Duality between U (.s[(2))-Modules and SLq(2)-Comodules 154 q 6 Scalar Products on U (.s[(2))-Modules 155 q 7 Quantum Clebsch-Gordan . 157 8 Exercises 162 9 Notes............ 163 Part Two Universal R-Matrices 165 VIII The Yang-Baxter Equation and (Co)Braided Bialgebras 167 1 The Yang-Baxter Equation . . . . . . . . . . . . 167 2 Braided Bialgebras. . . . . . . . . . . . . . . . . . . . . 172 3 How a Braided Bialgebra Generates R-Matrices .... 178 4 The Square of the Antipode in a Braided Hopf Algebra 179 5 A Dual Concept: Cobraided Bialgebras 184 6 The FRT Construction . . . . . . 188 7 Application to GLq(2) and SLq(2) 194 8 Exercises 196 9 Notes................ 198 IX Drinfeld's Quantum Double 199 1 Bicrossed Products of Groups ..... . 199 2 Bicrossed Products of Bialgebras . . . . . 202 3 Variations on the Adjoint Representation 207 4 Drinfeld's Quantum Double . . . . . . . . 213 5 Representation-Theoretic Interpretation of the Quantum Double . . . . 220 6 Application to Uq(.s[(2)) . 223 7 R-Matrices for U q 230 8 Exercises 236 9 Notes ...... . 238