  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 skew-symmetrizable matrix gives rise to a cluster
algebra of finite type.
