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Cluster algebras of finite type and positive symmetrizable matrices

Michael Barot·Christof Geiss·Andrei Zelevinsky
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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 skew-symmetrizable matrix gives rise to a cluster algebra of finite type.

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Michael Barot, Christof Geiss, Andrei Zelevinsky. 2004-11-15. Cluster algebras of finite type and positive symmetrizable matrices. https://doi.org/10.1112/s0024610706022769

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