Berkeley Lectures on Lie Groups and Quantum Groups

Description

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