Quantum Grothendieck rings as quantum cluster algebras

TL;DR

This paper constructs a quantum Grothendieck ring for a subcategory of quantum loop algebra representations, revealing its structure as a quantum cluster algebra and connecting it to known representation rings, especially in type A.

Contribution

It introduces a new quantum Grothendieck ring as a quantum cluster algebra for certain quantum loop algebra categories, extending previous understanding of their structure.

Findings

The quantum Grothendieck ring is a quantum cluster algebra.

In type A, it contains the quantum Grothendieck ring of finite-dimensional representations.

In type A1, notable relations are identified within the ring.

Abstract

We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category $\mathcal{O}$ of representations of the quantum loop algebra introduced by Hernandez-Jimbo. We use the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type $A$, we prove that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite-dimensional representations of the associated quantum affine algebra. In type $A_1$, we identify remarkable relations in this quantum Grothendieck ring.