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Mutation Classes of 3 x 3 Generalized Cartan Matrices

One of the recent developments in representation theory has been the introduction of cluster algebras by Fomin and Zelevinsky. It is now well known that these algebras are closely related with different areas of mathematics. A particular analogy exists between combinatorial aspects of cluster algebras and Kac-Moody algebras: roughly speaking, cluster algebras are associated with skew-symmetrizable matrices, while Kac-Moody algebras correspond to (symmetrizable) generalized Caftan matrices. In this paper, we describe an interplay between these two classes of matrices in size 3. In particular, we give a characterization of the mutation classes associated with the generalized Cartan matrices of size 3, generalizing results of Beineke-Bruestle-Hille.