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

Tom H. Koornwinder
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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.

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BibTeXRIS

Tom H. Koornwinder. 2007-11-07. The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. https://doi.org/10.3842/sigma.2007.063

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