MIT Combinatorics Seminar

Cluster Algebras of Finite Mutation

Michael Shapiro  (Michigan State University)

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


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.