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

SIGMA 7 (2011), 099, 26 pages      arXiv:1107.3544      https://doi.org/10.3842/SIGMA.2011.099

The Universal Askey-Wilson Algebra and the Equitable Presentation of Uq(sl2)

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

Received July 19, 2011, in final form October 10, 2011; Published online October 25, 2011; Misprint in Lemma 7.1 is corrected March 16, 2012

Abstract    (this is shortened html-version of the paper's abstract)
Around 1992 A. Zhedanov introduced the Askey-Wilson algebra AW(3). Recently we introduced a central extension Δ of AW(3) called the universal Askey-Wilson algebra. In this paper we discuss a connection between Δ and the quantum algebra Uq(sl2). Our main result is an algebra injection from Δ into a relative of Uq(sl2); the relative is obtained from Uq(sl2) by adjoining three mutually commuting indeterminates. We describe the injection using the equitable presentation of Uq(sl2).

Key words: Askey-Wilson relations; Leonard pair; Casimir element.

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