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

Let $\mathbb F$ denote a field, and fix a nonzero $q\in\mathbb F$ such that $q^4\ne=1$. The universal Askey–Wilson algebra is the associative $\mathbb F$-algebra $\Delta=\Delta_q$ defined by generators and relations in the following way. The generators are $A$, $B$, $C$. The relations assert that each of
$$ A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}, \qquad B + \frac{qCA-q^{-1}AC}{q^2-q^{-2}}, \qquad C + \frac{qAB-q^{-1}BA}{q^2-q^{-2}} $$
is central in $\Delta$. In this paper we discuss a connection between $\Delta$ and the $\mathbb F$-algebra $U=U_q(\mathfrak{sl}_2)$. To summarize the connection, let $a$, $b$, $c$ denote mutually commuting indeterminates and let $\mathbb F \lbrack a^{\pm 1}, b^{\pm 1}, c^{\pm 1}\rbrack$ denote the $\mathbb F$-algebra of Laurent polynomials in $a$, $b$, $c$ that have all coefficients in $\mathbb F$. We display an injection of $\mathbb F$-algebras $ \Delta\to U \otimes_\mathbb F \mathbb F \lbrack a^{\pm 1}, b^{\pm 1}, c^{\pm 1}\rbrack$. For this injection we give the image of $A$, $B$, $C$ and the above three central elements, in terms of the equitable generators for $U$. The algebra $\Delta $ has another central element of interest, called the Casimir element $\Omega$. One significance of $\Omega$ is the following. It is known that the center of $\Delta$ is generated by $\Omega$ and the above three central elements, provided that $q$ is not a root of unity. For the above injection we give the image of $\Omega$ in terms of the equitable generators for $U$. We also use the injection to show that $\Delta$ contains no zero divisors.