Origins of Cluster Algebras

Cluster algebras where conceived around 2001 by Fomin and Zelevinsky [55] as a tool to study questions concerning dual canonical bases and total positivity. They are constructively defined commutative algebras with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of fixed cardinality. They have the unusual feature that both the generators and the relations among them are not given from the outset, but are produced by an elementary iterative process of seed mutation. This procedure appears at a first glance not intuitive, but it seems to encode a somehow universal phenomenon which might explain the explosive development of this topic. Interestingly, the same mutation rule came up recently in work on Seiberg-Witten dualities in string theory. The theory of cluster algebras was further developed in the subsequent papers [56,57,14,15,59,44]. Remarkably, in the last paper of this series superpotentials borrowed from mathematical physics play a prominent role. In the next few sections we outline the rapid development in various directions of research which where heavily influenced by the notions around cluster algebras.