Hugh Thomas (New Brunswick), n-representation finiteness and the search for n-cluster algebras

Abstract:

Basic tilting modules for the linearly oriented A_n path algebra correspond to triangulations of the (n+2)-gon. They can further be considered as providing a combinatorial scaffolding for the corresponding cluster algebra. It is natural to investigate n-representation finite algebras in the hope of finding similar structures, which might give some hints as to what an n-cluster algebra might be. We will consider replacing linearly oriented A_n by its higher Auslander algebras. We find that they possess a natural class of tilting modules which correspond to triangulations of cyclic polytopes. Some aspects of the theory of cluster algebras generalize, but so far only on the tropical level. This is joint work with Steffen Oppermann. The talk will be based on our joint paper arXiv:1001.5437 together with subsequent work.