Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations
We consider the double affine Hecke algebra H=H(k₀,k₁,k₀v,k₁v;q) associated with the root system (C₁v,C₁). We display three elements x, y, z in H that satisfy essentially the Z₃-symmetric Askey-Wilson relations. We obtain the relations as follows. We work with an algebra Ĥ that is more general than...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2010 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2010
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146531 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations / T. Ito, P. Terwilliger // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 17 назв. — англ. |
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Ito, T. Terwilliger, P. 2019-02-09T20:27:33Z 2019-02-09T20:27:33Z 2010 Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations / T. Ito, P. Terwilliger // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 17 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D80; 33D45 DOI:10.3842/SIGMA.2010.065 https://nasplib.isofts.kiev.ua/handle/123456789/146531 We consider the double affine Hecke algebra H=H(k₀,k₁,k₀v,k₁v;q) associated with the root system (C₁v,C₁). We display three elements x, y, z in H that satisfy essentially the Z₃-symmetric Askey-Wilson relations. We obtain the relations as follows. We work with an algebra Ĥ that is more general than H, called the universal double affine Hecke algebra of type (C₁v,C₁). An advantage of Ĥ over H is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism Ĥ → H. We define some elements x, y, z in Ĥ that get mapped to their counterparts in H by this homomorphism. We give an action of Artin's braid group B₃ on Ĥ that acts nicely on the elements x, y, z; one generator sends x → y → z → x and another generator interchanges x, y. Using the B₃ action we show that the elements x, y, z in Ĥ satisfy three equations that resemble the Z₃-symmetric Askey-Wilson relations. Applying the homomorphism Ĥ → H we find that the elements x, y, z in H satisfy similar relations. We thank Alexei Zhedanov for mentioning to us around 2005 that AW(3) has the presentation (1)–(3); this knowledge motivated us to search for a result like Theorem 2.4. We also thank Zhedanov for several illuminating conversations on DAHA during his visit to Kanazawa in December 2007. We thank the two referees for clarifying how the present paper is related to the previous literature. The second author thanks Tom Koornwinder, Alexei Oblomkov, and Xiaoguang Ma for useful recent conversations on the general subject DAHA. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations |
| spellingShingle |
Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations Ito, T. Terwilliger, P. |
| title_short |
Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations |
| title_full |
Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations |
| title_fullStr |
Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations |
| title_full_unstemmed |
Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations |
| title_sort |
double affine hecke algebras of rank 1 and the z₃-symmetric askey-wilson relations |
| author |
Ito, T. Terwilliger, P. |
| author_facet |
Ito, T. Terwilliger, P. |
| publishDate |
2010 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We consider the double affine Hecke algebra H=H(k₀,k₁,k₀v,k₁v;q) associated with the root system (C₁v,C₁). We display three elements x, y, z in H that satisfy essentially the Z₃-symmetric Askey-Wilson relations. We obtain the relations as follows. We work with an algebra Ĥ that is more general than H, called the universal double affine Hecke algebra of type (C₁v,C₁). An advantage of Ĥ over H is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism Ĥ → H. We define some elements x, y, z in Ĥ that get mapped to their counterparts in H by this homomorphism. We give an action of Artin's braid group B₃ on Ĥ that acts nicely on the elements x, y, z; one generator sends x → y → z → x and another generator interchanges x, y. Using the B₃ action we show that the elements x, y, z in Ĥ satisfy three equations that resemble the Z₃-symmetric Askey-Wilson relations. Applying the homomorphism Ĥ → H we find that the elements x, y, z in H satisfy similar relations.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146531 |
| citation_txt |
Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations / T. Ito, P. Terwilliger // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 17 назв. — англ. |
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AT itot doubleaffineheckealgebrasofrank1andthez3symmetricaskeywilsonrelations AT terwilligerp doubleaffineheckealgebrasofrank1andthez3symmetricaskeywilsonrelations |
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2025-12-07T20:38:50Z |
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2025-12-07T20:38:50Z |
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