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Skew-symmetric cluster algebras of finite mutation type

Felikson, A.; Shapiro, M.; Tumarkin, P.

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Authors

M. Shapiro



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 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.

Citation

Felikson, A., Shapiro, M., & Tumarkin, P. (2012). Skew-symmetric cluster algebras of finite mutation type. Journal of the European Mathematical Society, 14(4), 1135-1180. https://doi.org/10.4171/jems/329

Journal Article Type Article
Publication Date Apr 1, 2012
Deposit Date Mar 19, 2012
Publicly Available Date Mar 29, 2024
Journal Journal of the European Mathematical Society
Print ISSN 1435-9855
Electronic ISSN 1435-9863
Publisher EMS Press
Peer Reviewed Peer Reviewed
Volume 14
Issue 4
Pages 1135-1180
DOI https://doi.org/10.4171/jems/329
Keywords Cluster algebra, Finite mutation type, Triangulated surface, Growth rate.

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