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Cluster Algebras and Poisson Geometry
Michael Gekhtman, University of Notre Dame, IN, Michael Shapiro, Michigan State University, East Lansing, MI, and Alek Vainshtein, University of Haifa, Mount Carmel, Israel

Mathematical Surveys and Monographs
2010; 246 pp; hardcover
Volume: 167
ISBN-10: 0-8218-4972-7
ISBN-13: 978-0-8218-4972-9
List Price: US$87
Member Price: US$69.60
Order Code: SURV/167
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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.


"[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

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