Skew-symmetric cluster algebras of finite mutation type

By Anna Felikson, Michael Shapiro, Pavel Tumarkin. 2008

159 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/48f3af0ec7d7801483592e58302ced5cf860f5bf

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Citations in the corpus

On cluster algebras arising from unpunctured surfaces II (2008, background): “Nevertheless, it has been shown recently [17] that in the skew-symmetric case, besides 11 exceptions all cluster algebras of finite mutation type and rank at least 3 are cluster algebras arising from surfaces.”
Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs (2014, background): “Froma classification point of view, this class is very important, since it has been shown in [6] that almost all (skew-symmetric) mutation finite cluster algebras are of surface type.”
Cluster algebras and derived categories (2012, unknown): “Then the correspondence x 7→ ϕx, c 7→ ϕc extends to an algebra isomorphism ϕ : AQ̃ ⊗Q C ∼→ C[X ] so that C[X ] carries a cluster algebra structure of type Q̃ with initial seed ϕxi , 1 ≤ i ≤ m. 4.2.”