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

SIGMA 3 (2007), 063, 15 pages      math.QA/0612730      https://doi.org/10.3842/SIGMA.2007.063
Contribution to the Vadim Kuznetsov Memorial Issue

The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case

Tom H. Koornwinder
Korteweg-de Vries Institute, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands

Received December 22, 2006, in final form April 23, 2007; Published online April 27, 2007
A slight error in formula (2.8) for Q0 is corrected November 07, 2007

Abstract
Zhedanov's algebra AW(3) is considered with explicit structure constants such that, in the basic representation, the first generator becomes the second order q-difference operator for the Askey-Wilson polynomials. It is proved that this representation is faithful for a certain quotient of AW(3) such that the Casimir operator is equal to a special constant. Some explicit aspects of the double affine Hecke algebra (DAHA) related to symmetric and non-symmetric Askey-Wilson polynomials are presented and proved without requiring knowledge of general DAHA theory. Finally a central extension of this quotient of AW(3) is introduced which can be embedded in the DAHA by means of the faithful basic representations of both algebras.

Key words: Zhedanov's algebra AW(3); double affine Hecke algebra in rank one; Askey-Wilson polynomials; non-symmetric Askey-Wilson polynomials.

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