The Universal Askey-Wilson Algebra and the Equitable Presentation of $U_q(\mathfrak{sl}_2)$

TL;DR

This paper establishes a connection between the universal Askey-Wilson algebra and the quantum algebra U_q(sl_2) by constructing an algebra injection using the equitable presentation, revealing new structural insights.

Contribution

It introduces an algebra injection from the universal Askey-Wilson algebra into a relative of U_q(sl_2) using the equitable presentation, advancing understanding of their relationship.

Findings

Established an algebra injection from the universal Askey-Wilson algebra to a relative of U_q(sl_2)

Connected the universal Askey-Wilson algebra with the equitable presentation of U_q(sl_2)

Provided structural insights into the algebraic relationship between AW(3) and U_q(sl_2)

Abstract

Around 1992 A. Zhedanov introduced the Askey-Wilson algebra AW(3). Recently we introduced a central extension $\Delta$ of AW(3) called the universal Askey-Wilson algebra. In this paper we discuss a connection between $\Delta$ and the quantum algebra $U_q(\mathfrak{sl}_2)$. Our main result is an algebra injection from $\Delta$ into a relative of $U_q(\mathfrak{sl}_2)$; the relative is obtained from $U_q(\mathfrak{sl}_2)$ by adjoining three mutually commuting indeterminates. We describe the injection using the equitable presentation of $U_q(\mathfrak{sl}_2)$