JournalsjemsVol. 14, No. 4pp. 1135–1180

Skew-symmetric cluster algebras of finite mutation type

  • Michael Shapiro

    Michigan State University, East Lansing, USA
  • Anna Felikson

    Independent University of Moscow, Russian Federation
  • Pavel Tumarkin

    Independent University of Moscow, Russian Federation
Skew-symmetric cluster algebras of finite mutation type cover
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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 22 and cluster algebras associated with triangulations of surfaces there are exactly 1111 exceptional skew-symmetric cluster algebras of finite mutation type. More precisely, 99 of them are associated with root systems E6E_6, E7E_7, E8E_8, E~6\widetilde E_6, E~7\widetilde E_7, E~8\widetilde E_8, E6(1,1)E_6^{(1,1)}, E7(1,1)E_7^{(1,1)}, E8(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.

Cite this article

Michael Shapiro, Anna Felikson, Pavel Tumarkin, Skew-symmetric cluster algebras of finite mutation type. J. Eur. Math. Soc. 14 (2012), no. 4, pp. 1135–1180

DOI 10.4171/JEMS/329