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

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

Let $\F$ denote a field and fix a nonzero $q\in \F$ such that $q^4 ot=1$. The universal Askey-Wilson algebra is an associative unital $\F$-algebra $\Delta=\Delta q$ defined by generators and relations. The generators are $A,$ $B,$ $C$ and the relations assert that each of A+ \frac{qBC-q^{-1}CB}{q^2-q^{-2}}, \qquad B+ \frac{qCA-q^{-1}AC}{q^2-q^{-2}}, \qquad C+ \frac{qAB-q^{-1}BA}{q^2-q^{-2}} is central in $\Delta.$ We classify up to isomorphism the finite-dimensional irreducible $\Delta$-modules, provided $\F$ is algebraically closed and $q$ is not a root of unity.

Author: **Hau-wen Huang**

Source: https://archive.org/