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 $d^{\rm \, th}$ root of unity with $d\not=1,2,4$. The universal Askey--Wilson algebra $\triangle_q$ 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 $\triangle_q$-module is of dimension less than or equal to $$ \left{ \begin{array}{ll} d \quad &\hbox{if $d$ is odd}; \newline d/2 \quad &\hbox{if $d$ is even}. \end{array} \right. $$ Moreover we provide an example to show that the bound is tight.