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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.
pdf (546 kb)
tex (30 kb)
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