Cluster algebras, deformation theory and beyond

Abstract: The purpose of this talk is to explain a fruitful interaction of ideas/constructions coming from the theory of cluster algebras, representation theory of quivers and deformation theory.

The representation theory of quivers is a well developed branch of mathematics that has been very active for nearly 50 years. The theory of cluster algebras is much younger, it was initiated by Fomin and Zelevinsky in 2001. Various important developments in these theories have emerged in the last 15 years thanks to the deep relation that exists in between them. After a gentle introduction to this circle of ideas I will recall the construction of a simplicial complex K(A) -- the tau-tilting complex-- associated to a finite dimensional path algebra A. Then I will report on one aspect of work-in-progress with Nathan Ilten and Hipólito Treffinger. We show that if K(A) is a cluster complex of finite type then the associated cluster algebra with universal coefficients is equal to a canonically identified subfamily of the semiuniversal family for the Stanley-Reisner ring of K(A). Time permitting, and depending on the audience's preference, I will elaborate either on some aspects of the "non-cluster" case (namely, when K(A) is not a cluster complex) or on the interpretation of these results from the point of view of tropical geometry.

algebraic geometrycombinatorics

Audience: researchers in the topic

(LAGARTOS) Latin American Real and Tropical Geometry Seminar

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