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This article is cited in 36 scientific papers (total in 36 papers)
The Universal Askey–Wilson Algebra and the Equitable Presentation of $U_q(\mathfrak{sl}_2)$
Paul Terwilliger Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA
Abstract:
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.
Keywords:
Askey–Wilson relations, Leonard pair, Casimir element.
Received: July 19, 2011; in final form October 10, 2011; Published online October 25, 2011
Citation:
Paul Terwilliger, “The Universal Askey–Wilson Algebra and the Equitable Presentation of $U_q(\mathfrak{sl}_2)$”, SIGMA, 7 (2011), 099, 26 pp.
Linking options:
https://www.mathnet.ru/eng/sigma657 https://www.mathnet.ru/eng/sigma/v7/p99
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