MIT Combinatorics Seminar

Cluster algebras of finite type and positive symmetrizable matrices

Andrei Zelevinsky  (Northeastern University, visiting MIT)

Wednesday, November 10, 2004    4:15 pm    Room 2-338


This is an account of a joint work with M.Barot and C.Geiss (UNAM, Mexico). Although motivated by the theory of cluster algebras, the talk will be purely combinatorial, so no knowledge of algebraic properties of cluster algebras (including their definition) will be assumed or needed. One should just bear in mind 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 discuss an interplay between the two classes of matrices. In particular, we establish a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type.