Berkeley Lectures on Lie Groups and Quantum Groups

Contents......Page 2
List of Theorems......Page 8
About this book......Page 13
I Lie Groups......Page 15
Group objects......Page 17
Lie algebra of a closed linear group......Page 19
Some analysis......Page 20
Classical compact Lie groups......Page 21
The classical groups......Page 22
Exercises......Page 23
Classical definition......Page 25
Manifold constructions......Page 26
Submanifolds......Page 27
Definition......Page 28
Integral curves......Page 29
Group actions......Page 30
Lie algebra of a Lie group......Page 31
Exercises......Page 32
The exponential map......Page 35
The Fundamental Theorem......Page 36
The definition......Page 39
Poincaré–Birkhoff–Witt theorem......Page 40
Ug is a bialgebra......Page 41
Geometry of the universal enveloping algebra......Page 42
The Baker–Campbell–Hausdorff Formula......Page 43
Relationship between Lie subgroups and Lie subalgebras......Page 44
Review of algebraic topology......Page 46
A dictionary between algebras and groups......Page 48
Basic examples: one- and two-dimensional Lie algebras......Page 49
Exercises......Page 50
Ug is a Hopf algebra......Page 55
Many definitions......Page 57
Nilpotency: Engel's theorem and corollaries......Page 58
Solvability: Lie's theorem and corollaries......Page 60
Jordan form......Page 61
Cartan's criteria......Page 63
The Casimir......Page 64
Review of Ext......Page 65
Complete reducibility......Page 66
Computing Exti(K,M)......Page 68
From Zassenhaus to Ado......Page 71
Exercises......Page 75
Reductive Lie algebras......Page 79
Guiding examples: sl(n) and sp(n) over C......Page 80
Representation theory of sl(2)......Page 86
Definition and existence......Page 88
More on the Jordan decomposition and Schur's lemma......Page 91
Precise description of Cartan subalgebras......Page 92
Motivation and a quick computation......Page 93
The definition......Page 95
Classification of rank-two root systems......Page 96
Positive roots......Page 98
Definitions......Page 99
Classification of finite-type Cartan matrices......Page 100
From Cartan matrix to Lie algebra......Page 104
Exercises......Page 109
Irreducible Lie-algebra representations......Page 113
Weyl Character Formula......Page 116
Some applications of the Weyl Character Formula......Page 121
Algebraic Lie groups......Page 123
Guiding example: SL(n) and PSL(n)......Page 124
Definition and general properties of algebraic groups......Page 125
Constructing G from g......Page 129
Exercises......Page 134
II Further Topics......Page 135
Lie groups in general......Page 137
Lie groups and Lie algebras......Page 139
Lie groups and finite groups......Page 140
Lie groups and real algebraic groups......Page 141
Important Lie groups......Page 142
Basic properties......Page 144
Unitary representations......Page 147
Clifford algebras......Page 149
Clifford groups, Spin groups, and Pin groups......Page 155
Examples of Spin and Pin groups and their representations......Page 159
Finite dimensional representations......Page 164
Background about infinite dimensional representations......Page 165
The unitary representations of SL(2,R)......Page 166
Exercises......Page 170
The E8 lattice......Page 171
The E8 Weyl group......Page 172
From lattice to Lie algebra......Page 178
Real forms......Page 183
Working with simple Lie groups......Page 186
Finishing the story......Page 188
Exercises......Page 194
General facts about algebraic groups......Page 197
Peter–Weyl theorem......Page 199
Homogeneous spaces......Page 202
Solvable groups......Page 203
Parabolic Lie algebras......Page 206
Flag manifolds for classical groups......Page 207
Bruhat decomposition......Page 208
Geometric induction......Page 211
Induction for the universal enveloping algebra......Page 213
The derived functor of induction......Page 214
Harish-Chandra's homomorphism......Page 215
Exponents of a semisimple Lie algebra......Page 219
The nilpotent cone......Page 222
The main theorem......Page 224
Differential operators and more on the nilpotent cone......Page 228
Twisted differential operators and Beilinson-Bernstein......Page 231
Kostant theorem......Page 236
Exercises......Page 238
III Poisson and Quantum Groups......Page 239
Lie bialgebras......Page 241
Lie bialgebras......Page 242
Poisson algebras......Page 243
Definition of Poisson Lie group......Page 245
Braid groups......Page 246
Quasitriangular Lie bialgebras......Page 248
Factorizable Lie bialgebras......Page 250
The Poisson bracket on SL(2,C)......Page 251
SL(2,C)*, a dual Lie group......Page 254
Classical doubles......Page 256
Kac–Moody algebras and their standard Lie bialgebra structure......Page 258
The Belavin–Drinfeld Classification......Page 261
Exercises......Page 264
Real forms of Lie bialgebras......Page 267
Bruhat decomposition......Page 269
Shubert cells......Page 271
Symplectic leaves......Page 273
Constructing the dressing action......Page 276
Orbits of the dressing action......Page 278
Symplectic leaves of K......Page 279
Symplectic leaves of G......Page 280
Exercises......Page 282
Quantum matrices......Page 285
The quantum determinant......Page 287
Uq sl(2)......Page 288
When q is not a root of unity......Page 289
When q is a root of unity......Page 291
Hopf structure......Page 292
Uq sl(2) and C(SLq(2)) are dual......Page 293
Hopf algebras and monoidal categories......Page 295
The Temperley–Lieb algebra......Page 301
Ribbon tangles......Page 304
Ribbon Hopf algebras......Page 306
Exercises......Page 309
Higher-rank quantum groups......Page 311
Deformation quantization......Page 313
The quantum Drinfeld double......Page 316
Quantizing sl(2)......Page 317
Quantizing the Serre relations......Page 321
The quantum Weyl group......Page 324
Highest weight theory for Uqg......Page 327
Z(Uqg)......Page 332
The quantum R-matrix......Page 336
Quantum Schur–Weyl duality......Page 340
Kashiwara's crystal bases......Page 341
Exercises......Page 350
Bibliography......Page 351
Index......Page 356