CLUSTER ALGEBRAS OF FINITE TYPE AND POSITIVE SYMMETRIZABLE MATRICES

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.