The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
| Authors | |
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| Publication date | 2007 |
| Journal | Symmetry, Integrability and Geometry : Methods and Applications (SIGMA) |
| Article number | 063 |
| Volume | Issue number | 3 |
| Number of pages | 15 |
| Organisations |
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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.
Key words: Zhedanov's algebra AW(3); double affine Hecke algebra in rank one; Askey-Wilson polynomials; non-symmetric Askey-Wilson polynomials. |
| Document type | Article |
| Note | In Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics |
| Language | English |
| Published at | https://doi.org/10.3842/SIGMA.2007.063 |
| Published at | http://arxiv.org/abs/math/0612730 |
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