viii CONTENTS
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: Teichm¨ uller 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 acklund–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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