A quantum cluster algebra approach to representations of simply-laced quantum affine algebras

TL;DR

This paper introduces a quantum cluster algebra framework for the quantum Grothendieck ring of certain representations of simply-laced quantum affine algebras, providing new computational tools and confirming algebraic structures.

Contribution

It establishes a quantum cluster algebra structure on the quantum Grothendieck ring and introduces an algorithm for computing (q,t)-characters of irreducible representations.

Findings

Quantum cluster algebra structure on the Grothendieck ring

New algorithm for (q,t)-character computation

Validation of the algebraic structure in larger categories

Abstract

We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q,t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q,t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defined in a previous work as a quantum cluster algebra, contains indeed the well-known quantum Grothendieck ring of the category of finite-dimensional representations. Finally, we display our algorithm on a concrete example.