Skew-symmetric cluster algebras of finite mutation type

Abstract

In the famous paper [FZ2] Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices. Besides cluster algebras of rank $2$ and cluster algebras associated with triangulations of surfaces there are exactly $11$ exceptional skew-symmetric cluster algebras of finite mutation type. More precisely, $9$ of them are associated with root systems $E_6$, $E_7$, $E_8$, ${\tilde{E}}_6$, ${\tilde{E}}_7$, ${\tilde{E}}_8$, $E_6^{(1,1)}$, $E_7^{(1,1)}$, $E_8^{(1,1)}$; two remaining were found by Derksen and Owen in [DO]. We also describe a criterion which determines if a skew-symmetric cluster algebra is of finite mutation type, and discuss growth rate of cluster algebras.