The Universal Askey-Wilson Algebra

Abstract

In 1992 A. Zhedanov introduced the Askey-Wilson algebra AW=AW(3) and used it to describe the Askey-Wilson polynomials. In this paper we introduce a central extension Δ of AW, obtained from AW by reinterpreting certain parameters as central elements in the algebra. We call Δ the universal Askey-Wilson algebra. We give a faithful action of the modular group PSL₂(Z) on Δ as a group of automorphisms. We give a linear basis for Δ. We describe the center of Δ and the 2-sided ideal Δ[Δ,Δ]Δ. We discuss how Δ is related to the q-Onsager algebra.