Graduate Texts in Mathematics 155. Editorial Board I.H. Ewing F.W. Gehring P.R. Halmos

Transcription

1 Graduate Texts in Mathematics 155 Editorial Board I.H. Ewing F.W. Gehring P.R. Halmos

2 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

3 Christian Kassel Quantum Groups With 88 Illustrations Springer-Science+Business Media, LLC

4 Christian Kassel Institut de Recherche Mathematique Avancee Universite Louis Pasteur-C.N.R.S Strasbourg France Editorial Board J.H. Ewing Department of Mathematics Indiana University Bloomington, IN USA F. W. Gehring Department of Mathematics University of Michigan Ann Arbor, MI USA P.R. Halmos Department of Mathematics Santa Clara University Santa Clara, CA 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 ISBN (ebook) DOI / Quantum groups. 2. Hopf algebras. 3. Topology. 4. Mathematical physics. 1. Title. II. Series: Graduate texts in mathematics; 155. QC20.7.G76K '.55-dc Printed on acid-free paper Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 1995 Softcover reprint of the hardcover 1 st 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 ISBN

5 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 observed, 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 spectacular 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 8L 2. 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 quantum 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 R 3, including the Jones polynomial,

6 VI Preface from certain solutions of the Yang-Baxter equation. To this end, we introduce 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 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 l5 ij is defined by l5 ij = 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.

7 Contents Preface v Part One Quantum 8L(2) 1 I II Preliminaries 3 1 Algebras and Modules. 3 2 Free Algebras The Affine Line and Plane 8 4 Matrix Multiplication Determinants and Invertible Matrices 10 6 Graded and Filtered Algebras 12 7 Ore Extensions Noetherian Rings 18 9 Exercises Notes Tensor Products 23 1 Tensor Products of Vector Spaces 23 2 Tensor Products of Linear Maps 26 3 Duality and Traces Tensor Products of Algebras Tensor and Symmetric Algebras 34 6 Exercises 36 7 Notes... 38

8 viii Contents III The Language of Hopf Algebras 39 1 Coalgebras Bialgebras Hopf Algebras Relationship with Chapter I. The Hopf Algebras GL(2) and SL(2) Modules over a Hopf Algebra Comodules Comodule-Algebras. Coaction of SL(2) on the Affine Plane 64 8 Exercises 66 9 Notes IV The Quantum Plane and Its Symmetries 1 The Quantum Plane Gauss Polynomials and the q-binomial Formula 3 The Algebra Mq(2) Ring-Theoretical Properties of Mq(2). 5 Bialgebra Structure on Mq(2) The Hopf Algebras GLq(2) and SLq(2) 7 Coaction on the Quantum Plane 8 Hopf *-Algebras 9 Exercises 10 Notes V The Lie Algebra of SL(2) 93 1 Lie Algebras Enveloping Algebras The Lie Algebra.5[(2) Representations of.5[(2) The Clebsch-Gordan Formula Module-Algebra over a Bialgebra. Action of.5[(2) on the Affine Plane Duality between the Hopf Algebras U(.5[(2)) and SL(2) Exercises Notes VI The Quantum Enveloping Algebra of.5[(2) The Algebra Uq(.5[(2)) Relationship with the Enveloping Algebra of.5[(2) Representations of U q The Harish-Chandra Homomorphism and the Centre of U q 130

9 Contents ix 5 Case when q is a Root of Unity. 6 Exercises 7 Notes VII A Hopf Algebra Structure on Uis[(2)) Comultiplication Semi simplicity Action of Uq(.s[(2)) on the Quantum Plane Duality between the Hopf Algebras Uq(.s[(2)) and SLq(2) Duality between U q (.s[(2))-modules and SLq(2)-Comodules Scalar Products on U q (.s[(2))-modules Quantum Clebsch-Gordan Exercises Notes Part Two Universal R-Matrices 165 VIII The Yang-Baxter Equation and (Co)Braided Bialgebras The Yang-Baxter Equation Braided Bialgebras How a Braided Bialgebra Generates R-Matrices The Square of the Antipode in a Braided Hopf Algebra A Dual Concept: Cobraided Bialgebras The FRT Construction Application to GLq(2) and SLq(2) Exercises Notes IX Drinfeld's Quantum Double 1 Bicrossed Products of Groups Bicrossed Products of Bialgebras Variations on the Adjoint Representation 4 Drinfeld's Quantum Double Representation-Theoretic Interpretation of the Quantum Double Application to Uq(.s[(2)). 7 R-Matrices for U q 8 Exercises 9 Notes

10 x Contents Part Three x XI Low-Dimensional Topology and Tensor Categories Knots, Links, Tangles, and Braids 1 Knots and Links Classification of Links up to Isotopy 3 Link Diagrams The Jones-Conway Polynomial 5 Tangles. 6 Braids.. 7 Exercises 8 Notes.. 9 Appendix. The Fundamental Group Tensor Categories 1 The Language of Categories and Functors. 2 Tensor Categories Examples of Tensor Categories Tensor Functors Turning Tensor Categories into Strict Ones 6 Exercises 7 Notes.... XII The Tangle Category 1 Presentation of a Strict Tensor Category 2 The Category of Tangles The Category of Tangle Diagrams Representations of the Category of Tangles 5 Existence Proof for Jones-Conway Polynomial 6 Exercises 7 Notes XIII Braidings Braided Tensor Categories The Braid Category Universality of the Braid Category The Centre Construction A Categorical Interpretation of the Quantum Double Exercises Notes

11 Contents xi XIV Duality in Tensor Categories 1 Representing Morphisms in a Tensor Category 2 Duality Ribbon Categories Quantum Trace and Dimension. 5 Examples of Ribbon Categories. 6 Ribbon Algebras 7 Exercises 8 Notes.... XV Quasi-Bialgebras 1 Quasi-Bialgebras... 2 Braided Quasi-Bialgebras 3 Gauge Transformations. 4 Braid Group Representations. 5 Quasi-Hopf Algebras. 6 Exercises 7 Notes Part Four Quantum Groups and Monodromy 383 XVI Generalities on Quantum Enveloping Algebras 1 The Ring of Formal Series and h-adic Topology 2 Topologically Free Modules. 3 Topological Tensor Product.. 4 Topological Algebras... 5 Quantum Enveloping Algebras 6 Symmetrizing the Universal R-Matrix 7 Exercises Notes Appendix. Inverse Limits XVII Drinfeld and Jimbo's Quantum Enveloping Algebras Semisimple Lie Algebras Drinfeld-Jimbo Algebras Quantum Group Invariants of Links The Case of s[(2) Exercises Notes

12 xii Contents XVIII Cohomology and Rigidity Theorems Cohomology of Lie Algebras Rigidity for Lie Algebras Vanishing Results for Semisimple Lie Algebras Application to Drinfeld-Jimbo Quantum Enveloping Algebras Cohomology of Coalgebras Action of a Semisimple Lie Algebra on the Cobar Complex 434 "1 Computations for Symmetric Coalgebras Uniqueness Theorem for Quantum Enveloping Algebras Exercises Notes Appendix. Complexes and Resolutions. 447 XIX Monodromy of the Knizhnik-Zamolodchikov Equations Connections Braid Group Representations from Monodromy The Knizhnik-Zamolodchikov Equations The Drinfeld-Kohno Theorem Equivalence of Uh(g) and Ag,t Drinfeld's Associator Construction of the Topological Braided Quasi-Bialgebra Ag,t Verification of the Axioms Exercises Notes Appendix. Iterated Integrals 480 XX Postlude. A Universal Knot Invariant Knot Invariants of Finite Type Chord Diagrams and Kontsevich's Theorem Algebra Structures on Chord Diagrams Infinitesimal Symmetric Categories A Universal Category for Infinitesimal Braidings Formal Integration of Infinitesimal Symmetric Categories Construction of Kontsevich's Universal Invariant Recovering Quantum Group Invariants 9 Exercises 10 Notes References Index