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

Abstract

Inspired by a profound observation on the Racah-Wigner coefficients of Uq(sl2), 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 , C1) at roots of unity is obtained.

An algebraic interpretation of the intertwining operators associated with the discrete Fourier transform

“…Finally note that the Askey-Wilson algebra for roots of unity was considered by Huang in [20] and [21]. It would be of interest to clarify how the results of Huang are interrelated with those, discussed in this work.…”

Totally bipartite tridiagonal pairs

“…We will construct a TB tridiagonal system Φ over F that has eigenvalue array (68). Fix β ∈ F that satisfies (69). We will refer to Section 10 with this β.…”

“…A ↔ B). For more information on ∆ q , see [68][69][70]144,145,155]. The double affine Hecke algebra (DAHA) for a reduced root system was defined by Cherednik (see [34]), and the definition was extended to include nonreduced root systems by Sahi (see [127]).…”

Representations of Askey--Wilson algebra

“…(Only in the case when q is not a root of unity. The case when q is a root of unity is much more complicated and the classification problem is still open, see [5,6,8]. )…”