The Universal Askey-Wilson Algebra and DAHA of Type $(C_1^{\vee },C_1)$

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 ${\Delta }_q$ 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 $\{t_i^{\pm 1}{\}}_{i=0}^3$ and relations (i) $t_i{t}_i^{-1}=t_i^{-1}{t}_i=1$; (ii) $t_i+t_i^{-1}$ is central; (iii) $t_0{t}_1{t}_2{t}_3=q^{-1}$. We display an injection of $\mathbb{F}$-algebras $\psi :{\Delta }_q\to {\hat{H}}_q$ that sends $A\mapsto t_1{t}_0+(t_1{t}_0{)}^{-1}$, $B\mapsto t_3{t}_0+(t_3{t}_0{)}^{-1}$, $C\mapsto t_2{t}_0+(t_2{t}_0{)}^{-1}$. 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.

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• Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity