Quantum loop algebras, quiver varieties, and cluster algebras

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. This takes the form of a general conjecture depending on a level $\ell \in \mathbb{N}$. The conjecture for $\ell =1$ is now proved thanks to some joint work with Hernandez, and a subsequent paper of Nakajima. The general case is still open.