Publication details for Pavel TumarkinFelikson, A., Shapiro, M. & Tumarkin, P. (2012). Skew-symmetric cluster algebras of finite mutation type. Journal of the European Mathematical Society 14(4): 1135-1180.
- Publication type: Journal Article
- ISSN/ISBN: 1435-9855, 1435-9863
- DOI: 10.4171/JEMS/329
- Keywords: Cluster algebra, Finite mutation type, Triangulated surface, Growth rate.
- Further publication details on publisher web site
- Durham Research Online (DRO) - may include full text
Author(s) from Durham
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 E6, E7, E8, E˜6, E˜7, E˜8, E(1,1)6, E(1,1)7, E(1,1)8; 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.