Preface ........................................................ ix
Chapter 1 Preliminaries ........................................ 1
1.1 Flag manifolds, Grassmannians, Pliicker coordinates and
Plucker 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
Chapter 6 Pre-symplectic structures compatible with the
cluster algebra structure ..................................... 111
6.1 Cluster algebras of geometric type and pre-symplectic
structures ............................................... 111
6.2 Main example: Teichmiiller space ......................... 115
6.3 Restoring exchange relations ............................. 127
6.4 Summary .................................................. 130
Bibliographical notes ......................................... 130
Chapter 7 On the properties of the exchange graph ............ 133
7.1 Covering properties ...................................... 133
7.2 The vertices and the edges of the exchange graph ......... 135
7.3 Exchange graphs and exchange matrices .................... 138
7.4 Summary .................................................. 139
Bibliographical notes ......................................... 139
Chapter 8 Perfect planar networks in a disk and
Grassmannians ................................................. 141
8.1 Perfect planar networks and boundary measurements ........ 142
8.2 Poisson structures on the space of edge weights and
induced Poisson structures on Matk,m ..................... 147
8.3 Grassmannian boundary measurement map and induced
Poisson structures on Gk(n) .............................. 159
8.4 Face weights ............................................. 165
8.5 Summary .................................................. 171
Bibliographical notes ......................................... 172
Chapter 9 Perfect planar networks in an annulus and rational
loops in Grassmannians ........................................ 175
9.1 Perfect planar networks and boundary measurements ........ 175
9.2 Poisson properties of the boundary measurement map ....... 181
9.3 Poisson properties of the Grassmannian boundary
measurement map .......................................... 191
9.4 Summary .................................................. 196
Bibliographical notes ......................................... 197
Chapter 10 Generalized Backlund-Darboux transformations for
Coxeter-Toda flows from a cluster algebra perspective ......... 199
10.1 Introduction ............................................. 199
10.2 Coxeter double Bruhat cells .............................. 202
10.3 Inverse problem .......................................... 207
10.4 Cluster algebra .......................................... 214
10.5 Coxeter-Toda lattices .................................... 228
10.6 Summary .................................................. 237
Bibliographical notes ......................................... 237
Bibliography .................................................. 239
Index ......................................................... 243
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