Cluster Algebras and Poisson Geometry
About this Title
Michael Gekhtman, University of Notre Dame, Notre Dame, IN, Michael Shapiro, Michigan State University, East Lansing, MI and Alek Vainshtein, University of Haifa, Haifa, Mount Carmel, Israel
Publication: Mathematical Surveys and Monographs
Publication Year 2010: Volume 167
ISBNs: 978-0-8218-4972-9 (print); 978-1-4704-1394-1 (online)
MathSciNet review: MR2683456
MSC (2000): Primary 13F60; Secondary 37K20, 37K25, 53D17
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.
Research mathematicians interested in cluster algebras and applications to geometry.
Table of Contents
- 1. Preliminaries
- 2. Basic examples: Rings of functions on Schubert varieties
- 3. Cluster algebras
- 4. Poisson structures compatible with the cluster algebra structure
- 5. The cluster manifold
- 6. Pre-symplectic structures compatible with the cluster algebra structure
- 7. On the properties of the exchange graph
- 8. Perfect planar networks in a disk and Grassmannians
- 9. Perfect planar networks in an annulus and rational loops in Grassmannians
- 10. Generalized Bäcklund-Darboux transforms for Coxeter-Toda flows from a cluster algebra perspective