Cluster algebras and discrete integrability

Abstract

Cluster algebras are a class of commutative algebras whose generators are defined by a recursive process called mutation. We give a brief introduction to cluster algebras, and explain how discrete integrable systems can appear in the context of cluster mutation. In particular, we give examples of birational maps that are integrable in the Liouville sense and arise from cluster algebras with periodicity, as well as examples of discrete Painlev\'e equations that are derived from Y-systems.