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MIT Combinatorics Seminar

Cluster Algebras of Finite Mutation

Michael Shapiro  (Michigan State University)
mshapiro@math.msu.edu

Friday, March 4, 2005   4:15 pm    Room 2-338

ABSTRACT

Coordinate rings of some natural geometrical objects (for example, Grassmannians) possess distinguished sets of generators (Plucker coordinates). Cluster algebras are generalizations of such coordinate rings. Sets of generators of a cluster algebra are organized in a tree-like structure with simple transformation rules between neighboring sets of generators.

Fomin and Zelevinsky obtained a complete classification for cluster algebras of finite type (i.e., with finite number of generators). This classification coincides with the Cartan-Killing classification of simple Lie algebras. We try to describe cluster algebras with finitely many transformation rules, so called cluster algebras of finite mutation type. We prove that a special class of cluster algebras originating from triangulation of Riemann surfaces is of finite mutation type.

This is joint work with Sergey Fomin.