Cluster algebras and tilting theory

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

The project focuses on the relation between cluster algebras and the representation theory of finite dimensional algebras. Finite dimensional algebras are often described as path algebras of quivers with relations. The modules of the algebra then correspond to the representations of the quiver. Cluster algebras are commutative algebras with a special combinatorial structure. The theory of cluster algebras is a fast developing field which is related to many areas of mathematics. One of its most active branches is its connection to the representation theory of finite dimensional algebras which has been established in the PI's joint work with Caldero and Chapoton and, independently, by Buan, Marsh, Reineke, Reiten and Todorov. Their construction of certain triangulated categories, the cluster categories, has provided unexpected new insights in tilting theory and has given rise to a whole new class of finite dimensional algebras, the cluster-tilted algebras. When cluster algebras were introduced by Fomin and Zelevinsky in 2002, their original motivation came from representation theory, which is a branch of modern algebra that is concerned with the study of symmetries of scientific models. Studying the symmetries of a model is often more fruitful than studying the model directly, and representation theory has found many applications in physics and chemistry, as well as in other mathematical fields. The cluster algebras provide a mathematical framework for fundamental patterns which occur throughout representation theory. Surprisingly, these patterns are also observed in various other branches of science which, a priori, are not related to representation theory. This motivates a further development of the theory of cluster algebras to which this project will contribute.