Contents
Preface vii
Chapter 1. Introduction 1
1.1. How do you do ? 1
1.2. What are we interested in ? 1
1.3. Enveloping algebras 3
Chapter 2. The Serre relations 9
2.1. The Serre relations 9
2.2. The quantum algebra of type Ar−1 13
Chapter 3. Kac-Moody Lie algebras 15
3.1. Lie algebras by generators and relations 15
3.2. Kac-Moody Lie algebras 17
3.3. The quantum algebra of type Ar−1
(1)
21
Chapter 4. Crystal bases of Uv-modules 23
4.1. Integrable modules 23
4.2. The Kashiwara operators 24
4.3. Crystal bases 27
Chapter 5. The tensor product of crystals 29
5.1. Basics of crystal bases 29
5.2. Tensor products of crystal bases 34
Chapter 6. Crystal bases of Uv 37
6.1. Triangular decomposition of Uv 37
6.2. Integrable highest weight modules 39
6.3. Crystal bases of Uv

42
Chapter 7. The canonical basis 47
7.1. A review of Lusztig’s canonical basis 47
7.2. The lattice of the canonical basis 51
Chapter 8. Existence and uniqueness (part I) 55
8.1. Preparatory lemmas 55
8.2. The first main theorem 59
Chapter 9. Existence and uniqueness (part II) 61
9.1. Preparatory results 61
9.2. The second main theorem 65
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