
SIGMA 3 (2007), 063, 15 pages math.QA/0612730
http://dx.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
Kortewegde 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 Q_{0} 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
qdifference operator for the AskeyWilson 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 nonsymmetric AskeyWilson
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; AskeyWilson polynomials; nonsymmetric AskeyWilson polynomials.
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