The Universal Askey-Wilson Algebra

2011/04/30 by Paul Terwilliger · 2 citations
Mathematics · #math.RA #math.QA #msc:33D80

paper · pdf · doi:10.3842/sigma.2011.069

published as SIGMA 7 (2011), 069 · 24 pages

arxiv created 2011/07/15 · arxiv updated 2011/07/18

Abstract

In 1992 A. Zhedanov introduced the Askey-Wilson algebra AW=AW(3) and used it to describe the Askey-Wilson polynomials. In this paper we introduce a central extension Δ of AW, obtained from AW by reinterpreting certain parameters as central elements in the algebra. We call Δ the \it universal Askey-Wilson algebra. We give a faithful action of the modular group \rm PSL₂(\mathbb Z) on Δ as a group of automorphisms. We give a linear basis for Δ. We describe the center of Δ and the 2-sided ideal Δ[Δ,Δ]Δ. We discuss how Δ is related to the q-Onsager algebra.

Cited by

• Verma modules and finite-dimensional irreducible modules of the universal Askey–Wilson algebra at roots of unity
• Using the quantum torus to investigate the q-Onsager algebra

Related

• The Universal Askey-Wilson Algebra and DAHA of Type (C₁^\vee,C₁)
• The Universal Askey-Wilson Algebra and the Equitable Presentation of U q (sl 2 )
• Finite-dimensional irreducible modules of the universal Askey-Wilson algebra
• Representations of Askey--Wilson algebra
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra