The Universal Askey-Wilson Algebra

[摘要] ( this is shortened html-version of the paper's 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 PSL 2 ( 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.

[关键词] Askey–Wilson relations;Leonard pair;modular group;q-Onsager algebra