Cluster algebras and quantum affine algebras

Abstract

Let $C$ be the category of finite-dimensional representations of a quantum affine algebra $U_q(\stackrel{\wedge }{\mathfrak{g}})$ of simply laced type. We introduce certain monoidal subcategories $C_{\ell }(\ell \in \mathbb{N})$ of $C$, and we study their Grothendieck rings using cluster algebras.