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 alge-
braic 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.
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