The Universal Askey-Wilson Algebra

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 {\it universal Askey-Wilson algebra}. We give a faithful action of the modular group ${\rm {PSL}}_2({\mathbb Z})$ 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.

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