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 is an element of C such that q(4) not equal 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 + qBC - q(-1)CB/q(2) - q(-2), B + qCA - q(-1)AC/q(2) - q(-2), C + 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) -> T. This gives a Delta(q) action on the standard module of T. (C) 2014 Elsevier Inc. All rights reserved.