Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 069, 24 pages      arXiv:1104.2813      https://doi.org/10.3842/SIGMA.2011.069
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

The Universal Askey-Wilson Algebra

Paul Terwilliger
Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA

Received April 17, 2011, in final form July 09, 2011; Published online July 15, 2011

Abstract    (this is shortened html-version of the paper's 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 universal Askey-Wilson algebra. We give a faithful action of the modular group PSL2(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.

Key words: Askey-Wilson relations; Leonard pair; modular group; q-Onsager algebra.

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