Contents
Preface ix
Chapter 1. Preliminaries 1
1.1. Flag manifolds, Grassmannians, Pl¨ ucker coordinates and Pl¨ucker
relations 1
1.2. Simple Lie algebras and groups 3
1.3. Poisson-Lie groups 8
Bibliographical notes 13
Chapter 2. Basic examples: Rings of functions on Schubert varieties 15
2.1. The homogeneous coordinate ring of G2(m) 15
2.2. Rings of regular functions on reduced double Bruhat cells 24
Bibliographical notes 36
Chapter 3. Cluster algebras 37
3.1. Basic definitions and examples 37
3.2. Laurent phenomenon and upper cluster algebras 43
3.3. Cluster algebras of finite type 49
3.4. Cluster algebras and rings of regular functions 60
3.5. Conjectures on cluster algebras 63
3.6. Summary 64
Bibliographical notes 65
Chapter 4. Poisson structures compatible with the cluster algebra structure 67
4.1. Cluster algebras of geometric type and Poisson brackets 67
4.2. Poisson and cluster algebra structures on Grassmannians 73
4.3. Poisson and cluster algebra structures on double Bruhat cells 92
4.4. Summary 98
Bibliographical notes 99
Chapter 5. The cluster manifold 101
5.1. Definition of the cluster manifold 101
5.2. Toric action on the cluster algebra 102
5.3. Connected components of the regular locus of the toric action 104
5.4. Cluster manifolds and Poisson brackets 107
5.5. The number of connected components of refined Schubert cells in real
Grassmannians 109
5.6. Summary 110
Bibliographical notes 110
vii
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