**Mathematical Surveys and Monographs**

Volume: 167;
2010;
246 pp;

MSC: Primary 13; 53;
Secondary 05; 14; 16

**Electronic ISBN: 978-1-4704-1394-1
Product Code: SURV/167.E**

List Price: $87.00

AMS Member Price: $69.60

MAA Member Price: $78.30

#### Supplemental Materials

# Cluster Algebras and Poisson Geometry

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*Michael Gekhtman; Michael Shapiro; Alek Vainshtein*

Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are
commutative rings with unit and no zero divisors equipped with a
distinguished family of generators (cluster variables) grouped in
overlapping subsets (clusters) of the same cardinality (the rank of the
cluster algebra) connected by exchange relations. Examples of cluster
algebras include coordinate rings of many algebraic varieties that play a
prominent role in representation theory, invariant theory, the study of
total positivity, etc. The theory of cluster algebras has witnessed a
spectacular growth, first and foremost due to the many links to a wide
range of subjects including representation theory, discrete dynamical
systems, Teichmüller theory, and commutative and non-commutative
algebraic geometry.

This book is the first devoted to cluster algebras. After presenting the
necessary introductory material about Poisson geometry and Schubert
varieties in the first two chapters, the authors introduce cluster
algebras and prove their main properties in Chapter 3. This chapter can be
viewed as a primer on the theory of cluster algebras. In the remaining
chapters, the emphasis is made on geometric aspects of the cluster algebra
theory, in particular on its relations to Poisson geometry and to the
theory of integrable systems.

#### Reviews & Endorsements

[This book is] a rather complete, self-contained and concise introduction to the connections between cluster algebras and Poisson geometry...a good reference for researchers on the topic...It is also suitable for graduate students since it starts "from scratch"...One should also point out the successful pedagogical efforts that were made in order to render this book very clear and pleasant to read. ...The book is divided into ten chapters, ordered in an increasing level of difficulty, each of them starting with a clear introduction and ending with a summary and some bibliographical notes for further study.

-- Gregoire Dupont, Mathematical Reviews

#### Table of Contents

# Table of Contents

## Cluster Algebras and Poisson Geometry

- Contents vii8 free
- Preface ix10 free
- Chapter 1. Preliminaries 118 free
- Chapter 2. Basic examples: Rings of functions on Schubert varieties 1532
- Chapter 3. Cluster algebras 3754
- Chapter 4. Poisson structures compatible with the cluster algebra structure 6784
- Chapter 5. The cluster manifold 101118
- Chapter 6. Pre-symplectic structures compatible with the cluster algebra structure 111128
- Chapter 7. On the properties of the exchange graph 133150
- Chapter 8. Perfect planar networks in a disk and Grassmannians 141158
- Chapter 9. Perfect planar networks in an annulus and rational loops in Grassmannians 175192
- Chapter 10. Generalized B\"acklund--Darboux transformations for Coxeter--Toda flows from a cluster algebra perspective 199216
- Bibliography 239256
- Index 243260 free