Cluster algebras and tilting theory II

University Of Connecticut, Storrs CT

Investigators

Abstract

The project focuses on cluster algebras and their relation to the representation theory of finite dimensional algebras. 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. The PI will continue his study of the special type of cluster algebras associated to Riemann surfaces; he will introduce generalized cluster variables and use them to construct canonical bases for these cluster algebras, and he will develop combinatorial models for explicit computations inside the cluster algebra. The PI will also investigate cluster-tilted algebras, which are certain finite dimensional algebras, whose modules correspond to elements of the cluster algebra. Cluster-tilted algebras provide a new point of view on tilting theory over hereditary algebras, and their module categories carry interesting information about the cluster algebra. 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.