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

Let C denote the field of complex numbers, and fix a nonzero q ∈ C such that q 4 ≠ 1 . Define a C -algebra Δ q by generators and relations in the following way. The generators are A, B, C. The relations assert that each of A + q B C − q − 1 C B q 2 − q − 2 , B + q C A − q − 1 A C q 2 − q − 2 , C + q A B − q − 1 B A q 2 − q − 2 is central in Δ q . The algebra Δ q 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 Δ q and T. Assuming that every irreducible T-module is thin, we display a surjective C -algebra homomorphism Δ q → T . This gives a Δ q action on the standard module of T.