Finite dimensional simple modules of $(q, \mathbf{Q})$-current algebras

TL;DR

This paper classifies all finite-dimensional simple modules of a specific quantum current algebra related to the special linear Lie algebra, expanding understanding of its representation theory.

Contribution

It provides a complete classification of finite-dimensional simple modules for the $(q, extbf{Q})$-current algebra associated with $ ext{sl}_n$, a novel result in quantum algebra representation theory.

Findings

Classification of finite-dimensional simple modules achieved

New insights into the structure of $(q, extbf{Q})$-current algebras

Extension of representation theory for quantum current algebras

Abstract

The $(q, \mathbf{Q})$-current algebra associated with the general linear Lie algebra was introduced by the second author in the study of representation theory of cyclotomic $q$-Schur algebras. In this paper, we study the $(q, \mathbf{Q})$-current algebra $U_q(\mathfrak{sl}_n^{\langle \mathbf{Q} \rangle}[x])$ associated with the special linear Lie algebra $\mathfrak{sl}_n$. In particular, we classify finite dimensional simple $U_q(\mathfrak{sl}_n^{\langle \mathbf{Q} \rangle}[x])$-modules.