The Universal Askey–Wilson Algebra and the Equitable Presentation of Uq(sl2)

Let F denote a field, and fix a nonzero q ∈ F such that q 6= 1. The universal Askey–Wilson algebra is the associative F-algebra ∆ = ∆q defined 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 ∆. In this paper we discuss a connection between ∆ 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 ∆→ U ⊗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 ∆ 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 ∆ 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 ∆ contains no zero divisors.