Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 047, 40 pages      arXiv:1202.4673

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

Received December 22, 2012, in final form July 07, 2013; Published online July 15, 2013

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 $A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}$, $B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}$, $C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}}$ 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 $A\mapsto t_1t_0+(t_1t_0)^{-1}$, $B\mapsto t_3t_0+(t_3t_0)^{-1}$, $C\mapsto t_2t_0+(t_2t_0)^{-1}$. 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.

Key words: Askey-Wilson polynomials; Askey-Wilson relations; rank one DAHA.

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