The Universal Askey–Wilson Algebra

Let F denote a field, and fix a nonzero q ∈ F such that q 6= 1. We define an associative F-algebra ∆ = ∆q by generators and relations in the following way. The generators are A, B, C. The relations assert that each of A+ qBC − q−1CB q2 − q−2 , B + qCA− q−1AC q2 − q−2 , C + qAB − q−1BA q2 − q−2 is central in ∆. We call ∆ the universal Askey–Wilson algebra. We discuss how ∆ is related to the original Askey–Wilson algebra AW(3) introduced by A. Zhedanov. Multiply each of the above central elements by q + q−1 to obtain α, β, γ. We give an alternate presentation for ∆ by generators and relations; the generators are A, B, γ. We give a faithful action of the modular group PSL2(Z) on ∆ as a group of automorphisms; one generator sends (A,B,C) 7→ (B,C,A) and another generator sends (A,B, γ) 7→ (B,A, γ). We show that {ABCαβγ|i, j, k, r, s, t ≥ 0} is a basis for the F-vector space ∆. We show that the center Z(∆) contains the element Ω = qABC + qA + q−2B2 + qC − qAα− q−1Bβ − qCγ. Under the assumption that q is not a root of unity, we show that Z(∆) is generated by Ω, α, β, γ and that Z(∆) is isomorphic to a polynomial algebra in 4 variables. Using the alternate presentation we relate ∆ to the q-Onsager algebra. We describe the 2-sided ideal ∆[∆,∆]∆ from several points of view. Our main result here is that ∆[∆,∆]∆ + F1 is equal to the intersection of (i) the subalgebra of ∆ generated by A, B; (ii) the subalgebra of ∆ generated by B, C; (iii) the subalgebra of ∆ generated by C, A.