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

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 ${△}_q$ of the Askey--Wilson algebras was recently studied. For $q$ not a root of unity, it is known that $Z({△}_q)$ is isomorphic to the polynomial ring of four variables. A presentation for $Z({△}_q)$ 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.

References in corpus (5)

• The Universal Askey-Wilson Algebra
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
• Double affine Hecke algebras of rank 1 and affine cubic surfaces
• The quantum algebra $U_q(s{l}_2)$ and its equitable presentation

Cited by in corpus (4)

• An embedding of the Bannai-Ito algebra in $\mathcal{U}(\mathfrak{osp}(1,2))$ and $-1$ polynomials
• An algebraic interpretation of the intertwining operators associated with the discrete Fourier transform
• Ihara zeta function, coefficients of Maclaurin series, and Ramanujan graphs
• Representations of Askey--Wilson algebra