Recognizing cluster algebras of finite type

Ahmet Seven

Northeastern University

April 9,
refreshments at 3:45pm


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 Cartan-Killing classification. This result can be stated purely combinatorially in terms of certain transformations ("mutations") of certain edge-weighted directed graphs ("diagrams"). Fomin and Zelevinsky proved that a diagram is of finite type if and only if it is mutation-equivalent 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.

Speaker's Contact Info: aseven(at-sign)

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