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

Wanxia Wang; Shilin Yang; Wanxia Wang; Shilin Yang

doi:10.3934/math.2023964

AIMS Mathematics

2023, Volume 8, Issue 8: 18930-18947. doi: 10.3934/math.2023964

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Research article

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

Wanxia Wang ,
Shilin Yang ^,

Faculty of Science, Beijing University of Technology, Beijing, China

Received: 24 March 2023 Revised: 23 May 2023 Accepted: 25 May 2023 Published: 05 June 2023
MSC : 20G42, 33D45, 33D80

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.
- universal Askey-Wilson algebra,
- irreducible module,
- Verma module
Citation: Wanxia Wang, Shilin Yang. On finite-dimensional irreducible modules for the universal Askey-Wilson algebra[J]. AIMS Mathematics, 2023, 8(8): 18930-18947. doi: 10.3934/math.2023964

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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.

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