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

Abstract

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 $ {\textbf{ Condition A: }} \; abc, a^{-1}bc, ab^{-1}c, abc^{-1} \notin \left \{q^{n-2i+1}| 1 \leq i \leq n\right \}. $ 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 Condition A.

Keywords

• universal askey-wilson algebra
• irreducible module
• verma module