Quantum loop algebras, quiver varieties, and cluster algebras

TL;DR

This paper explores the connections between quantum loop algebras, their representations, quiver varieties, and cluster algebras, highlighting geometric and algebraic structures relevant to representation theory.

Contribution

It introduces Nakajima's geometric approach to q-characters and discusses recent developments linking tensor structures to cluster algebras.

Findings

Geometric description of irreducible q-characters via graded quiver varieties

Insights into the tensor structure of finite-dimensional representations

Connections established between quantum algebras and cluster algebra frameworks

Abstract

These notes reflect the contents of three lectures given at the workshop of the 14th International Conference on Representations of Algebras (ICRA XIV), held in August 2010 in Tokyo. We first provide an introduction to quantum loop algebras and their finite-dimensional representations. We explain in particular Nakajima's geometric description of the irreducible q-characters in terms of graded quiver varieties. We then present a recent attempt to understand the tensor structure of the category of finite- dimensional representations by means of cluster algebras.