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

Abstract

Around 1992 A. Zhedanov introduced the Askey-Wilson algebra AW(3). Recently we introduced a central extension $Δ$ of AW(3) called the universal Askey-Wilson algebra. In this paper we discuss a connection between $Δ$ and the quantum algebra $U_q(\mathfrak{sl}_2)$. Our main result is an algebra injection from $Δ$ 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 $U_q(\mathfrak{sl}_2)$

misprint in Lemma 7.1 is corrected