Cluster algebras

Abstract

What Is a Cluster Algebra?Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2-15; Poisson geometry (16-19); Teichmüller theory (20-24); string theory (25-31); discrete dynamical systems and integrability (6, 32-38); and combinatorics (39-47).Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36,(48)(49)(50)(51)(52)(53) and to the cluster algebras portal (www.math.