The Universal Askey-Wilson Algebra

TL;DR

This paper introduces the universal Askey-Wilson algebra, a central extension of the original algebra, and explores its structure, automorphisms, and relations to other algebraic objects.

Contribution

It defines the universal Askey-Wilson algebra, provides a basis, describes its center, and establishes a faithful action of the modular group.

Findings

Introduces the universal Askey-Wilson algebra as a central extension.

Provides a linear basis for the algebra.

Describes the algebra's center and automorphism group.

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 $\Delta$ of AW, obtained from AW by reinterpreting certain parameters as central elements in the algebra. We call $\Delta$ the {\it universal Askey-Wilson algebra}. We give a faithful action of the modular group ${\rm {PSL}}_2({\mathbb Z})$ on $\Delta$ as a group of automorphisms. We give a linear basis for $\Delta$. We describe the center of $\Delta$ and the 2-sided ideal $\Delta[\Delta,\Delta]\Delta$. We discuss how $\Delta$ is related to the $q$-Onsager algebra.