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An Introduction to the Theory of Quantum Groups Ryan W Eastern Washington University EWU Digital Commons EWU Masters Thesis Collection Student Research and Creative Works 2012 An introduction to the theory of quantum groups Ryan W. Downie Eastern Washington University Follow this and additional works at: http://dc.ewu.edu/theses Part of the Physical Sciences and Mathematics Commons Recommended Citation Downie, Ryan W., "An introduction to the theory of quantum groups" (2012). EWU Masters Thesis Collection. 36. http://dc.ewu.edu/theses/36 This Thesis is brought to you for free and open access by the Student Research and Creative Works at EWU Digital Commons. It has been accepted for inclusion in EWU Masters Thesis Collection by an authorized administrator of EWU Digital Commons. For more information, please contact [email protected]. EASTERN WASHINGTON UNIVERSITY An Introduction to the Theory of Quantum Groups by Ryan W. Downie A thesis submitted in partial fulfillment for the degree of Master of Science in Mathematics in the Department of Mathematics June 2012 THESIS OF RYAN W. DOWNIE APPROVED BY DATE: RON GENTLE, GRADUATE STUDY COMMITTEE DATE: DALE GARRAWAY, GRADUATE STUDY COMMITTEE EASTERN WASHINGTON UNIVERSITY Abstract Department of Mathematics Master of Science in Mathematics by Ryan W. Downie This thesis is meant to be an introduction to the theory of quantum groups, a new and exciting field having deep relevance to both pure and applied mathematics. Throughout the thesis, basic theory of requisite background material is developed within an overar- ching categorical framework. This background material includes vector spaces, algebras and coalgebras, bialgebras, Hopf algebras, and Lie algebras. The understanding gained from these subjects is then used to explore some of the more basic, albeit important, quantum groups. The thesis ends with an indication of how to proceed into the deeper areas of the theory. Acknowledgements First, and foremost, I would like to thank my wife, for her patience and encouragement throughout this process. A very deep and special thanks goes to Dr. Ron Gentle, my thesis advisor, for all his help, suggestions and insight. Similarly, I would like to thank Dr. Dale Garraway for his input and for serving, with Dr. Achin Sen, on my thesis committee. iii Contents Abstract ii Acknowledgements iii List of Figures vii 1 The Advent of Quantum Groups1 1.1 Introduction...................................1 1.1.1 Basic Description............................2 1.2 Background...................................3 1.2.1 The Mathematical Structure of Classical Mechanics........4 1.2.2 The Mathematical Structure of Quantum Mechanics........4 1.2.3 Quantum Groups Emerge.......................5 1.3 Overview of Approach.............................6 2 The Basics: Vector Spaces and Modules9 2.1 Vector Spaces..................................9 2.1.1 Direct Sums............................... 12 2.1.2 Quotient Spaces............................ 14 2.1.3 Tensor Products............................ 15 2.1.4 Duality................................. 30 2.2 Modules..................................... 32 2.2.1 Noetherian Rings and Noetherian Modules............. 34 2.2.2 Artinian Rings............................. 35 3 Algebras and Coalgebras 37 3.1 Algebras..................................... 37 3.1.1 Common Examples of Algebras.................... 39 3.1.2 Setting The Stage: A Preliminary Result.............. 41 3.1.3 Free Algebras.............................. 46 3.1.4 Tensor Products of Algebras..................... 48 3.1.4.1 Multilinear Maps and Iterated Tensor Products..... 52 3.1.4.2 Important Multilinear Maps................ 53 3.1.5 Graded Algebras............................ 55 3.1.6 The Tensor Algebra.......................... 56 iv Contents v 3.2 Coalgebras................................... 60 3.2.1 Sweedler's sigma notation....................... 69 3.2.2 Some Basic Coalgebra Theory..................... 71 3.2.3 The Tensor Product of Coalgebras.................. 79 3.2.4 The Algebra/Coalgebra Connection................. 81 3.2.5 Co-Semi-Simple Coalgebras...................... 97 4 Bialgebras and Hopf Algebras 100 4.1 Bialgebras.................................... 100 4.1.1 The Tensor Product of Bialgebras.................. 106 4.2 Hopf Algebras.................................. 108 4.2.1 The Tensor Product of Hopf Algebras................ 125 4.3 Comodules and Hopf Modules......................... 125 4.4 Actions and Coactions............................. 130 4.4.1 Actions................................. 131 4.5 The Group Algebra............................... 133 4.5.1 Grouplike Elements.......................... 140 4.6 M(2), GL(2) and SL(2)............................ 153 5 Lie Algebras 161 5.1 Background and Importance to the Theory of Quantum Groups...... 161 5.2 The Basics................................... 162 5.2.1 Introducing Lie Algebras....................... 162 5.2.2 Adjoints and the Commutator.................... 164 5.2.3 The General Linear Group and The General Linear Algebra.... 168 5.2.4 New Lie Algebras............................ 171 5.3 Enveloping Algebras.............................. 175 5.3.1 Representations of Lie Algebras.................... 176 5.3.2 The Universal Enveloping Algebra.................. 179 5.4 The Lie Algebra sl(2)............................. 188 5.5 Representations of sl(2)............................ 193 5.5.1 Weight Space.............................. 194 5.5.1.1 Constructing Irreducible sl(2)-modules.......... 199 5.5.2 The Universal Enveloping Algebra of sl(2).............. 202 5.5.3 Duality................................. 206 6 Deformation Quantization: The Quantum Plane and Other Deformed Spaces 220 6.1 Introduction................................... 220 6.2 The Affine Line and Plane........................... 220 6.2.1 The Affine Line............................. 221 6.2.2 The Affine Plane............................ 222 6.3 The Quantum Plane.............................. 233 6.3.1 Ore Extensions............................. 236 6.3.2 q-Analysis................................ 245 6.4 The Quantum Groups GLq(2) and SLq(2).................. 251 6.4.1 Mq(2).................................. 251 Contents vi 6.4.2 Quantum Determinant......................... 254 6.4.3 The Quantum Groups GLq(2) and SLq(2).............. 266 7 The Quantum Enveloping Algebra Uq(sl(2)) 270 7.1 Introduction................................... 270 7.2 Some Basic Properties of Uq(sl(2))...................... 270 7.2.1 q-Analysis Revisited.......................... 270 7.3 Motivating Uq(sl(2)).............................. 273 7.4 An Alternative Presentation of Uq(sl(2))................... 290 7.5 Representations of Uq(sl(2)).......................... 296 7.5.1 When q is not a Root of Unity.................... 297 7.5.1.1 Verma Modules....................... 301 7.5.2 When q is a Root of Unity...................... 304 7.5.3 Action on the Quantum Plane.................... 309 7.5.4 Duality between Uq(sl(2)) and SLq(2)................ 315 Bibliography 319 List of Figures 1.1..........................................8 3.1.......................................... 70 4.1 Revised diagram for f ? η" = f......................... 110 4.2 This diagram encodes the antipode axiom for a Hopf algebra........ 112 4.3 Hopf morphism diagram combined with antipode diagram.......... 113 4.4 Algebra diagram for κ[G]............................ 134 4.5 Unit diagram for κ[G]............................. 134 4.6 The coalgebra diagram for κ[G]........................ 139 4.7 The counit diagram for κ[G].......................... 139 vii For my wife Tara viii Chapter 1 The Advent of Quantum Groups 1.1 Introduction At the writing of this thesis the theory of quantum groups is a young and burgeoning area of study. The excitement surrounding the theory is due to its implications for both pure and applied mathematics. My particular interest in the subject was aroused on both accounts. I was actually introduced to the concept while reading a book on the mathematical structure of quantum mechanics. Given my affinity for both pure mathematics and mathematical physics, quantum groups was a very clear choice for me. And what aspiring mathematician wouldn't be thrilled to participate in a new and exciting area of research? This, however, is a double edged sword since although there is a lot of potential for one to contribute, brand new mathematical ideas are generally very involved, complicated, abstract and just plain difficult. They are built on and blossom from deep, as well as broad, mathematical ideas. One cannot hope to simply dive in, but needs a diverse wealth of mathematical background just to get started. Suffice it to say, this is what I very clearly discovered while writing this thesis and, to a large extent, is why it turned out so long. Even despite the length, I was only able to address the very basics of the theory. Nevertheless, the study was most worthwhile and enlightening, giving me occasion to greatly expand my mathematical knowledge and understanding as well as deepen my understanding of what I was taught in my course work. It is my hope that the reader will gain a similar benefit from exploring this thesis and will appreciate the beauty of this most fascinating subject. 1 Chapter 1. The Advent of Quantum Groups 2 1.1.1 Basic Description To some extent, quantum groups almost sound like science fiction, especially given the weirdness surrounding the discoveries of quantum physics. So, just what are these exciting new structures called quantum groups? It's always good to
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