Quantum cluster algebra structure on the finite dimensional representations of $U_q(\widehat{sl_{2}})$

TL;DR

This paper establishes a quantum cluster algebra framework for the deformed Grothendieck ring of a subcategory of finite-dimensional representations of the quantum affine algebra U_q(sl2), linking representation theory with cluster algebra structures.

Contribution

It introduces a quantum cluster algebra structure on the Grothendieck ring of a specific subcategory of representations of U_q(sl2), expanding the understanding of algebraic structures in quantum groups.

Findings

Quantum cluster algebra structure is constructed on the Grothendieck ring.

The structure provides new insights into the representation theory of U_q(sl2).

Bridges the gap between quantum groups and cluster algebras.

Abstract

In this paper, we give a quantum cluster algebra structure on the deformed Grothendieck ring of $\CC_{n}$, where $\CC_{n}$ is a full subcategory of finite dimensional representations of $U_q(\widehat{sl_{2}})$ defined in section II.