The universal Askey-Wilson algebra and the equitable presentation of Uq(sl2)

Abstract

Let F denote a field, and fix a nonzero q ∈ F such that q4 ≠ 1. The universal Askey-Wilson algebra is the associative F-algebra Δ = Δq defined by generators and relations in the following way. The generators are A, B, C. The relations assert that each of is central in Δ. In this paper we discuss a connection between Δ and the F-algebra U = Uq(sl2). To summarize the connection, let a, b, c denote mutually commuting indeterminates and let F[a±1, b±1, c±1] denote the F-algebra of Laurent polynomials in a, b, c that have all coefficients in F. We display an injection of F-algebras Δ → U o×F F[a±1, b±1, c±1]. For this injection we give the image of A, B, C and the above three central elements, in terms of the equitable generators for U. The algebra Δ has another central element of interest, called the Casimir element Ω. One significance of Ω is the following. It is known that the center of Δ is generated by Ω and the above three central elements, provided that q is not a root of unity. For the above injection we give the image of Ω in terms of the equitable generators for U. We also use the injection to show that Δ contains no zero divisors.