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∈C such that q4≠1. Define a C-algebra Δq by generators and relations in the following way. The generators are A, B, C. The relations assert that each ofA+qBC−q−1CBq2−q−2,B+qCA−q−1ACq2−q−2,C+qAB−q−1BAq2−q−2 is central in Δq. The algebra Δq is called the universal Askey–Wilson algebra. Let Γ denote a distance-regular graph that has q-Racah type. Fix a vertex x of Γ and let T=T(x) denote the corresponding subconstituent algebra. In this paper we discuss a relationship between Δq and T. Assuming that every irreducible T-module is thin, we display a surjective C-algebra homomorphism Δq→T. This gives a Δq action on the standard module of T.