Cluster algebras of finite type and positive symmetrizable matrices

Details Cluster algebras of finite type and positive symmetrizable matrices Cluster algebras of finite type and positive symmetrizable matrices

2004-11-15

arxiv.org/abs/math/0411341v5

Mathematics

Combinatorics

20 pages. In version 3, some new material is added in the end of section 2, discussing the classification and characterization

Scientific paper

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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