Finite-Dimensional Irreducible Modules of the Universal Askey–Wilson Algebra

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 ▵q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\triangle_q}$$\end{document} of the Askey–Wilson algebras was recently studied. In this paper, we consider a family of infinite-dimensional ▵q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\triangle_q}$$\end{document}-modules. By the universal property of these ▵q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\triangle_q}$$\end{document}-modules, we classify the finite-dimensional irreducible ▵q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\triangle_q}$$\end{document}-modules when q is not a root of unity.