On finite-dimensional irreducible modules for the universal Askey-Wilson algebra

Abstract

<abstract><p>Let $ \Delta_q $ be the universal Askey-Wilson algebra. If $ q $ is not a root of unity, it is shown in the Huang's earlier paper that an $ (n+1) $-dimensional irreducible $ \Delta_q $-module is a quotient $ V_n(a, b, c) $ of a $ \Delta_q $-Verma module with</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ {\textbf{ Condition A: }} \; abc, a^{-1}bc, ab^{-1}c, abc^{-1} \notin \left \{q^{n-2i+1}| 1 \leq i \leq n\right \}. $\end{document} </tex-math></disp-formula></p> <p>The aim of this paper is to discuss the structures of $ (n+1) $-dimensional $ \Delta_q $-modules $ V_n(a, b, c) $ when the given triples $ (a, b, c) $ do not satisfy <bold>Condition A</bold>.</p></abstract>