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This article is cited in 33 scientific papers (total in 33 papers)
The Universal Askey–Wilson Algebra and DAHA of Type $(C_1^{\vee},C_1)$
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 $q^4\not=1$.
The universal Askey–Wilson algebra $\Delta_q$ is the associative $\mathbb F$-algebra defined 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 $\Delta_q$.
The universal DAHA $\hat H_q$ of type $(C_1^\vee,C_1)$ is the associative $\mathbb F$-algebra defined by
generators $\lbrace t^{\pm1}_i\rbrace_{i=0}^3$ and relations (i) $t_i t^{-1}_i=t^{-1}_i t_i=1$; (ii) $t_i+t^{-1}_i$ is central; (iii) $t_0t_1t_2t_3=q^{-1}$.
We display an injection of $\mathbb F$-algebras $\psi:\Delta_q\to\hat H_q$ that sends
\begin{gather*}
A\mapsto t_1t_0+(t_1t_0)^{-1},
\qquad
B\mapsto t_3t_0+(t_3t_0)^{-1},
\qquad
C\mapsto t_2t_0+(t_2t_0)^{-1}.
\end{gather*}
For the map $\psi$ we compute the image of the three central elements mentioned above.
The algebra $\Delta_q$ has another central element of interest, called the Casimir element $\Omega$.
We compute the image of $\Omega$ under $\psi$.
We describe how the Artin braid group $B_3$ acts on $\Delta_q$ and $\hat H_q$ as a group of automorphisms.
We show that $\psi$ commutes with these $B_3$ actions.
Some related results are obtained.
Keywords:
Askey–Wilson polynomials; Askey–Wilson relations; rank one DAHA.
Received: December 22, 2012; in final form July 7, 2013; Published online July 15, 2013
Citation:
Paul Terwilliger, “The Universal Askey–Wilson Algebra and DAHA of Type $(C_1^{\vee},C_1)$”, SIGMA, 9 (2013), 047, 40 pp.
Linking options:
https://www.mathnet.ru/eng/sigma830 https://www.mathnet.ru/eng/sigma/v9/p47
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