Cluster Algebras of Finite Type and Positive Symmetrizable Matrices

Abstract

The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However, the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skewsymmetrizable matrix gives rise to a cluster algebra of finite type.

Cluster algebras and semipositive symmetrizable matrices

“…In this section, we recall some definitions and statements from [1,5,9]. Throughout the paper, a matrix always means a square integer matrix.…”

“…It follows that finite type skew-symmetrizable matrices can be characterized in terms of their diagrams as follows: B is of finite type if and only if its diagram Γ(B) is mutationequivalent to a Dynkin diagram ( Figure 2). Another characterization, which makes the relation to Cartan-Killing more explicit, was obtained in [1] using quasi-Cartan companions; in our setup it reads as follows: a skew-symmetrizable matrix B is of finite type if and only if it has an admissible quasi-Cartan companion which is positive [1, Theorem 1.2]. In particular, for a finite type skew-symmetrizable matrix, mutation of an admissible quasi-Cartan companion is also admissible.…”

“…quivers 3 ). More explicitly, we show that S is the mutation class of an extended Dynkin diagram corresponding to a skew-symmetric matrix if and only if every diagram in S has an admissible quasi-Cartan companion which is semipositive of 2 It exists, e.g., if all cycles in Γ(B) are cyclically oriented [1,Corollary 5.2]. 3 Replacing an edge from a vertex i to j with weight B 2 i,j by B i,j many arrows, the diagram of a skew-symmetric matrix B can be viewed as a quiver.…”

“…If the off-diagonal entries of a quasi-Cartan matrix are nonpositive, then it is a generalized Cartan matrix; these are the matrices that give rise to Kac-Moody algebras; see [11]. Motivated by the fact that cluster algebras and Kac-Moody algebras share the same classification of finite type objects [9], a notion of a quasi-Cartan companion was introduced in [1] to relate skew-symmetrizable and symmetrizable matrices: a quasi-Cartan companion of a skew-symmetrizable matrix B is a quasi-Cartan matrix A such that |A i,j | = |B i,j | for all i = j. In a slightly more general sense, we say that A is a quasi-Cartan companion of a diagram Γ if it is a quasi-Cartan companion of a skew-symmetrizable matrix whose diagram is equal to Γ.…”

Cluster algebras, quiver representations and triangulated categories

“…For a general quiver Q, a criterion for A Q to be cluster-finite in terms of quadratic forms was given in [14]. In practice, the quickest way to decide whether a concretely given quiver is clusterfinite and to determine its cluster-type is to compute its mutation-class using [88].…”