The Universal Askey–Wilson Algebra

Let $\mathbb F$ denote a field, and fix a nonzero $q\in\mathbb F$ such that $q^4\not=1$. We define an associative $\mathbb F$-algebra $\Delta=\Delta_q$ by generators and relations in the following way. The generators are $A$, $B$, $C$. The relations assert that each of
\begin{gather*} A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}},\qquad B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}},\qquad C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}} \end{gather*}
is central in $\Delta$. We call $\Delta$ the universal Askey–Wilson algebra. We discuss how $\Delta$ is related to the original Askey–Wilson algebra AW(3) introduced by A. Zhedanov. Multiply each of the above central elements by $q+q^{-1}$ to obtain $\alpha$, $\beta$, $\gamma$. We give an alternate presentation for $\Delta$ by generators and relations; the generators are $A$, $B$, $\gamma$. We give a faithful action of the modular group ${\rm {PSL}}_2(\mathbb Z)$ on $\Delta$ as a group of automorphisms; one generator sends $(A,B,C)\mapsto (B,C,A)$ and another generator sends $(A,B,\gamma)\mapsto (B,A,\gamma)$. We show that $\lbrace A^iB^jC^k \alpha^r\beta^s\gamma^t| i,j,k,r,s,t\geq 0\rbrace$ is a basis for the $\mathbb F$-vector space $\Delta$. We show that the center $Z(\Delta)$ contains the element
\begin{gather*} \Omega=qABC+q^2A^2+q^{-2}B^2+q^2C^2-qA\alpha-q^{-1}B\beta -q C\gamma. \end{gather*}
Under the assumption that $q$ is not a root of unity, we show that $Z(\Delta)$ is generated by $\Omega$, $\alpha$, $\beta$, $\gamma$ and that $Z(\Delta)$ is isomorphic to a polynomial algebra in 4 variables. Using the alternate presentation we relate $\Delta$ to the $q$-Onsager algebra. We describe the 2-sided ideal $\Delta\lbrack \Delta,\Delta\rbrack \Delta$ from several points of view. Our main result here is that $\Delta\lbrack \Delta,\Delta \rbrack \Delta + \mathbb F 1$ is equal to the intersection of $(i)$ the subalgebra of $\Delta$ generated by $A$, $B$; $(ii)$ the subalgebra of $\Delta$ generated by $B$, $C$; $(iii)$ the subalgebra of $\Delta $ generated by $C$, $A$.