Center of the universal Askey--Wilson algebra at roots of unity

TL;DR

This paper investigates the structure of the center of the universal Askey--Wilson algebra at roots of unity, providing a new presentation and applications to double affine Hecke algebras.

Contribution

It offers a presentation of the center of the universal Askey--Wilson algebra at roots of unity and applies this to the double affine Hecke algebra of type (C_1^∨, C_1).

Findings

Presented a new description of the center at roots of unity.

Connected the algebra's center to double affine Hecke algebras.

Extended known results from the non-root of unity case.

Abstract

Inspired by a profound observation on the Racah--Wigner coefficients of $U_q(\mathfrak{sl}_2)$, the Askey--Wilson algebras were introduced in the early 1990s. A universal analog $\triangle_q$ of the Askey--Wilson algebras was recently studied. For $q$ not a root of unity, it is known that $Z(\triangle_q)$ is isomorphic to the polynomial ring of four variables. A presentation for $Z(\triangle_q)$ at $q$ a root of unity is displayed in this paper. As an application, a presentation for the center of the double affine Hecke algebra of type $(C_1^\vee,C_1)$ at roots of unity is obtained.