Skew-symmetric cluster algebras of finite mutation type

TL;DR

This paper classifies all skew-symmetric cluster algebras of finite mutation type, identifying 11 exceptional cases beyond known classes, and provides criteria to determine such finiteness and discusses their growth rates.

Contribution

It provides a complete classification of skew-symmetric cluster algebras of finite mutation type, including 11 exceptional cases, and introduces a criterion for finiteness and growth rate analysis.

Findings

Identified 11 exceptional skew-symmetric cluster algebras of finite mutation type.

Classified all such algebras beyond surface triangulations and rank 2 cases.

Presented a criterion to determine if a skew-symmetric cluster algebra has finite mutation type.

Abstract

In 2003, 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 $E_6$, $E_7$, $E_8$, $\widetilde E_6$, $\widetilde E_7$, $\widetilde E_8$, $E_6^{(1,1)}$, $E_7^{(1,1)}$, $E_8^{(1,1)}$; two remaining were recently found by Derksen and Owen. 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.