Recognizing cluster algebras of finite typeAhmet SevenNortheastern University
April 9,

ABSTRACT


One of the most striking results in the theory of cluster algebras due to S.Fomin and A.Zelevinsky is the classification of cluster algebras of finite type, which turns out to be identical to the CartanKilling classification. This result can be stated purely combinatorially in terms of certain transformations ("mutations") of certain edgeweighted directed graphs ("diagrams"). Fomin and Zelevinsky proved that a diagram is of finite type if and only if it is mutationequivalent to a Dynkin diagram. We will discuss a complete solution of the following natural recognition problem: how to recognize whether a given diagram is of finite type without having to perform an unspecified number of mutations. We solve this problem by providing the list of all minimal infinite type diagrams. The list contains all extended Dynkin diagrams but also has 6 more infinite series, and a substantial number of exceptional diagrams with at most 9 vertices. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

