Finite-dimensional irreducible modules of the universal Askey-Wilson algebra

TL;DR

This paper classifies finite-dimensional irreducible modules of a universal Askey-Wilson algebra variant, extending understanding of its representation theory for non-root of unity parameters.

Contribution

It provides a classification of finite-dimensional irreducible modules of the universal Askey-Wilson algebra, addressing an open problem in the field.

Findings

Classified finite-dimensional irreducible modules for non-root of unity q.

Introduced a family of infinite-dimensional modules for the algebra.

Utilized the universal property to achieve the classification.

Abstract

Since the introduction of Askey-Wilson algebras by Zhedanov in 1991, the classification of the finite-dimensional irreducible modules of Askey-Wilson algebras remains open. A universal analog $\triangle_q$ of the Askey-Wilson algebras was recently studied. In this paper, we consider a family of infinite-dimensional $\triangle_q$-modules. By the universal property of these $\triangle_q$-modules, we classify the finite-dimensional irreducible $\triangle_q$-modules when $q$ is not a root of unity.