Introduction to Quantum Groups

by George Lusztig

Overview

According to Drinfeld, a quantum group is the same as a Hopf algebra. This includes as special cases, the algebra of regular functions on an algebraic group and the enveloping algebra of a semisimple Lie algebra. The qu- tum groups discussed in this book are the quantized enveloping algebras introduced by Drinfeld and Jimbo in 1985, or variations thereof. Although such quantum groups appeared in connection with problems in statistical mechanics and are closely related to conformal field theory and knot theory, we will regard them purely as a new development in Lie theory. Their place in Lie theory is as follows. Among Lie groups and Lie algebras (whose theory was initiated by Lie more than a hundred years ago) the most important and interesting ones are the semisimple ones. They were classified by E. Cartan and Killing around 1890 and are quite central in today's mathematics. The work of Chevalley in the 1950s showed that semisimple groups can be defined over arbitrary fields (including finite ones) and even over integers. Although semisimple Lie algebras cannot be deformed in a non-trivial way, the work of Drinfeld and Jimbo showed that their enveloping (Hopf) algebras admit a rather interesting deformation depending on a parameter v. These are the quantized enveloping algebras of Drinfeld and Jimbo. The classical enveloping algebras could be obtained from them for v —» 1.

Product Details

ISBN-13:	9780817647162
Publisher:	Birkhäuser Boston
Publication date:	11/02/2010
Series:	Modern Birkhäuser Classics
Edition description:	1st ed. 1993. Corr. 2nd printing 1994. 3rd. printing 2010
Pages:	352
Product dimensions:	6.00(w) x 9.20(h) x 0.80(d)

Table of Contents

Part I The Drinfeld-Jimbo Algebra U 1

Chapter 1 The Algebra f 2

1.1 Cartan Datum 2

1.2 The Algebras ′f and f 2

1.3 Preliminaries on Gaussian Binomial Coefficients 9

1.4 The Quantum Serre Relations 10

Chapter 2 Weyl Group, Root Datum 14

2.1 The Weyl Group 14

2.2 Root Datum 15

2.3 Coroots 18

Chapter 3 The Algebra U 19

3.1 The Algebras ′U and U 19

3.2 Triangular Decomposition for ′U and U 25

3.3 Antipode 28

3.4 The Category C 30

3.5 Integrable Objects of C 31

Chapter 4 The Quasi-R-Matrix 34

4.1 The Element θ 34

4.2 Some Identities for θ 37

Chapter 5 The Symmetries T′_i,e, T″_i,e of an Integrable U-Module 40

5.1 The Category C′_i 40

5.2 First Properties of T′_i,e,, T″_i,e 42

5.3 The Operators L′_i, L″_i 45

Chapter 6 Complete Reducibility Theorems 48

6.1 The Quantum Casimir Operator 48

6.2 Complete Reducibility in C^hi∩^C′ 51

6.3 Affine or Finite Cartan Data 53

Chapter 7 Higher Order Quantum Serre Relations 55

Notes on Part I 59

Part II Geometric Realization of f 61

Chapter 8 Review of the Theory of Perverse Sheaves 63

Chapter 9 Quivers and Perverse Sheaves 68

9.1 The Complexes L_ν 68

9.2 The Functors Ind and Res 71

9.3 The Categories P_V;I′;≥γ and P_V;I′;γ 77

Chapter 10 Fourier-Deligne Transform 81

10.1 Fourier-Deligne Transform and Restriction 81

10.2 Fourier-Deligne Transform and Induction 84

10.3 A Key Inductive Step 87

Chapter 11 Periodic Functors 89

Chapter 12 Quivers with Automorphisms 92

12.1 The Group K(Qv) 92

12.2 Inner Product 95

12.3 Properties of L_ν 95

12.4 Verdier Duality 99

12.5 Self-Dual Elements 100

12.6 L_ν as Additive Generators 102

Chapter 13 The Algebras o′k and k 106

13.1 The Algebra o′k 106

13.2 The Algebra k 110

Chapter 14 The Signed Basis of f 113

14.1 Cartan Data and Graphs with Automorphisms 113

14.2 The Signed Basis B 118

14.3 The Subsets B_i;n of B 120

14.4 The Canonical Basis B of f 122

14.5 Examples 125

Notes on Part II 127

Part III Kashiwara's Operators and Applications 129

Chapter 15 The Algebra U 130

Chapter 16 Kashiwara's Operators in Rank 1 132

16.1 Definition of the Operators φ_i, ε_i and F_i, E_i 132

16.2 Admissible Forms 133

16.3 Adapted Bases 139

Chapter 17 Applications 142

17.1 First Application to Tensor Products 142

17.2 Second Application to Tensor Products 148

17.3 The Operators φ_i, ε_i : f → f 150

Chapter 18 Study of the Operators F_i, E_i on Λ_λ 152

18.1 Preliminaries 152

18.2 A General Hypothesis and Some Consequences 154

18.3 Further Consequences of the General Hypothesis 159

Chapter 19 Inner Product on Λ 164

19.1 First Properties of the Inner Product 164

19.2 Normalization of Signs 167

19.3 Further Properties of the Inner Product 170

Chapter 20 Bases at ∞ 173

20.1 The Basis at ∞ of Λ_λ 173

20.2 Basis at ∞ in a Tensor Product 175

Chapter 21 Cartan Data of Finite Type 177

Chapter 22 Positivity of the Action of F_i, E_i in the symmetric case 179

Notes on Part III 182

Part IV Canonical Basis of U 183

Chapter 23 The Algebra U 185

23.1 Definition and First Properties of U 185

23.2 Triangular Decomposition, A-form for U 188

23.3 U and Tensor Products 189

Chapter 24 Canonical Bases in Certain Tensor Products 192

24.1 Integrality Properties of the Quasi-R-Matrix 192

24.2 A Lemma on Systems of (Semi)-Linear Equations 194

24.3 The Canonical Basis of ^ωΛ_λ ⊗ Λ_λ 195

Chapter 25 The Canonical Basis B of U 198

25.1 Stability Properties 198

25.2 Definition of the Basis B of U 202

25.3 Example (Rank 1) 205

25.4 Structure Constants 206

Chapter 26 Inner Product on U 208

26.1 First Definition of the Inner Product 208

26.2 Definition of the Inner Product as a Limit 211

26.3 A Characterization of B ($$$) (-B) 212

Chapter 27 Based Modules 214

27.1 Isotypical Components 214

27.2 The Subsets B[λ] 217

27.3 Tensor Product of Based Modules 219

Chapter 28 Bases for Coinvariants and Cyclic Permutations 224

28.1 Monomials 224

28.2 The Isomorphism P 226

Chapter 29 A Refinement of the Peter-Weyl Theorem 230

29.1 The Subsets B [λ] of B 230

29.2 The Finite Dimensional Algebras U/U[P]p231

29.3 The Refined Peter-Weyl Theorem 233

29.4 Cells 235

29.5 The Quantum Coordinate Algebra 237

Chapter 30 The Canonical Topological Basis of (U- ⊗ U+)ˆ 238

30.1 The Definition of the Canonical Topological Basis 238

30.2 On the Coefficients Pb₁, b′₁; b₂, b′₂ 240

Notes on Part IV 242

Part V Change of Rings 244

Chapter 31 The Algebra _RU 245

31.1 Definition of _RU 245

31.2 ntegrable _RU-Modules 249

31.3 Highest Weight Modules 251

Chapter 32 Commutativity isomorphism 252

32.1 The Isomorphism _ƒR_M,M′ 252

32.2 The Hexagon Property 255

Chapter 33 Relation with Kac-Moody Lie Algebras 258

33.1 The Specialization υ = 1 258

33.2 The Quasi-Classical Case 260

Chapter 34 Gaussian Binomial Coefficients at Roots of 1 265

Chapter 35 The Quantum Frobenius Homomorphism 269

35.1 Statements of Results 269

35.2 Proof of Theorem 35.1.8 271

35.3 The Structure of Certain Highest Weight Modules of _RU 273

35.4 A Tensor Product Decomposition of _Rf 276

35.5 Proof of Theorem 35.1.7 278

Chapter 36 The Algebras _Rƒ, _Ru 280

36.1 The Algebra _Rƒ 280

36.2 The algebras _Ru, _Ru 282

Notes on Part V 285

Part VI Braid Group Action 286

Chapter 37 The Symmetries T′_i,e, T″_i,e of U 287

37.1 Definition of the Symmetries 287

37.2 Calculations in Rank 2 288

37.3 Relation of the Symmetries with Comultiplication 293

Chapter 38 Symmetries and Inner Product on f 294

38.1 The Algebras f[i], σf[i] 294

38.2 A Computation of Inner Products 300

Chapter 39 Braid group relations 304

39.1 Preparatory Results 304

39.2 Braid Group Relations for U in Rank 2 305

39.3 The Quantum Verma Identities 311

39.4 Proof of the Braid Group Relations 314

Chapter 40 Symmetries and U⁺ 318

40.1 Preparatory Results 318

40.2 The Subspace U⁺ (w, e) of U⁺ 321

Chapter 41 Integrality Properties of the Symmetries 324

41.1 Braid Group Action on U 324

41.2 Braid Group Action on Integrable _RU-Modules 326

Chapter 42 The ADE Case 328

42.1 Combinatorial Description of the Left Colored Graph 328

42.2 Remarks on the Piecewise Linear Bijections R^h′_h : Nⁿ ≅ Nⁿ 334

Notes on Part VI 338

Index of Notation 339

Index of Terminology 341

Editorial Reviews

From the reviews:

"There is no doubt that this volume is a very remarkable piece of work...Its appearance represents a landmark in the mathematical literature."

—Bulletin of the London Mathematical Society

"This book is an important contribution to the field and can be recommended especially to mathematicians working in the field."

—EMS Newsletter

"The present book gives a very efficient presentation of an important part of quantum group theory. It is a valuable contribution to the literature."

—Mededelingen van het Wiskundig

"Lusztig's book is very well written and seems to be flawless...Obviously, this will be the standard reference book for the material presented and anyone interested in the Drinfeld–Jimbo algebras will have to study it very carefully."

—ZAA

"[T]his book is much more than an 'introduction to quantum groups.' It contains a wealth of material. In addition to the many important results (of which several are new–at least in the generality presented here), there are plenty of useful calculations (commutator formulas, generalized quantum Serre relations, etc.)."

—Zentralblatt MATH

“George Lusztig lays out the large scale structure of the discussion that follows in the 348 pages of his Introduction to Quantum Groups. … A significant and important work. … it’s terrific stuff, elegant and deep, and Lusztig presents it very well indeed, of course.” (Michael Berg, The Mathematical Association of America, January, 2011)

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