The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
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)...
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2007-04-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Online Access: | http://www.emis.de/journals/SIGMA/2007/063/ |
| _version_ | 1818550605284638720 |
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| author | Tom H. Koornwinder |
| author_facet | Tom H. Koornwinder |
| author_sort | Tom H. Koornwinder |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-12T08:48:42Z |
| format | Article |
| id | doaj.art-41d70a14197b4b809cb4f986832109d9 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-12T08:48:42Z |
| publishDate | 2007-04-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-41d70a14197b4b809cb4f986832109d92022-12-22T00:30:20ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592007-04-013063The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One CaseTom H. KoornwinderZhedanov'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.http://www.emis.de/journals/SIGMA/2007/063/Zhedanov's algebra AW(3)double affine Hecke algebra in rank oneAskey-Wilson polynomialsnon-symmetric Askey-Wilson polynomials |
| spellingShingle | Tom H. Koornwinder The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case Symmetry, Integrability and Geometry: Methods and Applications Zhedanov's algebra AW(3) double affine Hecke algebra in rank one Askey-Wilson polynomials non-symmetric Askey-Wilson polynomials |
| title | The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case |
| title_full | The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case |
| title_fullStr | The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case |
| title_full_unstemmed | The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case |
| title_short | The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case |
| title_sort | relationship between zhedanov s algebra aw 3 and the double affine hecke algebra in the rank one case |
| topic | Zhedanov's algebra AW(3) double affine Hecke algebra in rank one Askey-Wilson polynomials non-symmetric Askey-Wilson polynomials |
| url | http://www.emis.de/journals/SIGMA/2007/063/ |
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