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