Skew-symmetric cluster algebras of finite mutation type

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 E 6 , E 7 ,; 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.

Cluster algebras and semipositive symmetrizable matrices

“…In this paper we use more combinatorial methods for more general diagrams. Let us also mention that there are algorithms to check whether a given skew-symmetric matrix is of finite mutation type: one of them is realized in B. Keller's computer program (which is available at www.math.jussieu.fr/~keller/quivermutation); a polynomial-time algorithm is given in [7].…”

Tubular cluster algebras II: Exponential growth

“…These cluster algebras represent three of the 11 exceptional mutation finite cluster algebras with skew symmetric exchange matrix [8] and one is the surface algebra corresponding to the 4-punctured sphere. Figure 1 shows representatives of their exchange matrices in quiver form.…”

Untitled

“…We say that a matrix B (and the corresponding cluster algebra) is mutation finite (or is of finite-mutation type) if its mutation-equivalence class is finite, i.e., only finitely many matrices can be obtained from B by repeated matrix mutations. Felikson, Shapiro, and Tumarkin gave a classification of all skew-symmetric mutation-finite cluster algebras in [7]. They showed that these cluster algebras are the union of the following classes of cluster algebras:…”