Preface ......................................................... v
Chapter 1. Geometric Constructions of the Irreducible
Representations of GLn
Joel Kamnitzer ....................................... 1
Introduction .................................................... 1
1.1 Representation theory of GLn ............................... 2
1.2 Borel-Weil theory .......................................... 6
1.3 Ginzburg construction ...................................... 9
1.4 Geometric Satake correspondence ........................... 12
1.5 Geometric skew Howe duality ............................... 15
Chapter 2. Introduction to Crystal Bases
Seok-Jin Kang ....................................... 19
Introduction ................................................... 19
2.1 Lie algebras .............................................. 20
2.2 Kac-Moody algebras ........................................ 23
2.3 Quantum groups ............................................ 26
2.4 Crystal bases ............................................. 28
2.5 Abstract crystals ......................................... 31
2.6 Perfect crystals .......................................... 33
2.7 Combinatorics of Young walls .............................. 38
Chapter 3. Geometric Realizations of Crystals
Alistair Savage ..................................... 45
Introduction ................................................... 45
3.1 Motivating examples ....................................... 46
3.2 Quivers ................................................... 50
3.3 The Lusztig quiver variety ................................ 55
3.4 The lagrangian Nakajima quiver variety .................... 58
3.5 Connections to combinatorial realizations of crystal
graphs .................................................... 63
Chapter 4. Nilpotent Orbits and Finite W-Algebras
Weiqiang Wang ....................................... 71
Introduction ................................................... 71
4.1 Nilpotent orbits, Dynkin, and good  -gradings ............. 74
4.2 Definitions of W-algebras ................................. 78
4.3 Quantization of the Slodowy slices ........................ 82
4.4 An equivalence of categories .............................. 86
4.5 Good -gradings in type A ................................. 89
4.6 W-algebras and independence of good gradings .............. 92
4.7 Higher level Schur duality ................................ 96
4.8 W-(super)algebras in positive characteristic .............. 99
4.9 Further work and open problems ........................... 103
Chapter 5. Extended Affine Lie Algebras - An Introduction
to Their Structure Theory
Erhard Neher ....................................... 107
Introduction .................................................. 107
5.1 Affine Lie algebras and some generalizations ............. 109
5.2 Extended affine Lie algebras: Definition and first
examples ................................................. 120
5.3 The structure of the roots of an EALA .................... 129
5.4 The core and centreless core of an EALA .................. 144
5.5 The construction of all EALAs ............................ 156
Chapter 6. Representations of Affine and Toroidal Lie
Algebras
Vyjayanthi Chari ................................... 169
Introduction .................................................. 169
6.1 Simple Lie algebras ...................................... 170
6.2 Affine Lie algebras ...................................... 176
6.3 Affine Lie algebras integrable representations and
integral forms ........................................... 180
6.4 Finite-dimensional modules for loop algebras and their
generalizations .......................................... 183
6.5 Weyl modules, restricted Kirillov-Reshetikhin and
beyond ................................................... 187
6.6 Koszul algebras, quivers, and highest weight
categories ............................................... 191
Bibliography .................................................. 199
Index ......................................................... 209
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