Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity

Abstract

Let $\mathbb{F}$ denote an algebraically closed field and assume that $q\in \mathbb{F}$ is a primitive $d^{\mathrm{th}}$ root of unity with $d\ne 1,2,4$. The universal Askey--Wilson algebra ${△}_q$ is a unital associative $\mathbb{F}$-algebra defined by generators and relations. The generators are $A,B,C$ and the relations assert that each of \begin{gather*} 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}} \qquad \end{gather*} commutes with $A,B,C$. We show that every finite-dimensional irreducible ${△}_q$-module is of dimension less than or equal to $$\{\begin{array}{ll}d& ifdis\; odd;\\ d/2& ifdis\; even.\end{array}$$ Moreover we provide an example to show that the bound is tight.

14 pages

References in corpus (7)

• The Universal Askey-Wilson Algebra
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Cited by in corpus (1)

• Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz