SPECIAL FEATURE: INTRODUCTION Cluster algebras

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, 48–53) and to the cluster algebras portal (www.math. lsa.umich.edu/~fomin/cluster.html) for various introductions to cluster algebras and their links with other subjects in mathematics (and physics). In brief, a cluster algebra A of rank k is a subring of an ambient field F of rational functions in k variables, say x1, . . ., xk. Unlike most commutative rings, a cluster algebra is not presented at the outset via a complete set of generators and relations. Instead, from the data of the initial seed—which includes the k initial cluster variables x1, . . ., xk, plus an exchange matrix—one uses an iterative procedure called “mutation” to produce the rest of the cluster variables. In particular, each new cluster variable is a rational expression in x1, . . ., xk. The cluster algebra is then defined to be the subring of F generated by all cluster variables. The set of all cluster variables has a remarkable combinatorial structure: It is a union of overlapping algebraically independent k subsets of F called “clusters,” which together have the structure of a simplicial complex called the “cluster complex.” The clusters are related to each other by birational transformations of the following kind: For every cluster x and every cluster variable x ∈ x, there is another cluster x′ = (x − {x}) ∪ {x′}, with the new cluster variable x′ determined by an exchange relation of the form