Distance-regular graphs of $q$-Racah type and the universal Askey-Wilson algebra

Abstract

Let $\C$ denote the field of complex numbers, and fix a nonzero $q \in \C$ such that $q^4 \ne 1$. Define a $\C$-algebra $\Delta_q$ by generators and relations in the following way. The generators are $A,B,C$. The relations assert that each of $A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}$, $B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}$, $C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}}$ is central in $\Delta_q$. The algebra $\Delta_q$ is called the universal Askey-Wilson algebra. Let $\Gamma$ denote a distance-regular graph that has $q$-Racah type. Fix a vertex $x$ of $\Gamma$ and let $T=T(x)$ denote the corresponding subconstituent algebra. In this paper we discuss a relationship between $\Delta_q$ and $T$. Assuming that every irreducible $T$-module is thin, we display a surjective $\C$-algebra homomorphism $\Delta_q \to T$. This gives a $\Delta_q$ action on the standard module of $T$.