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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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Інститут математики НАН України
2010
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/146531 |
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irk-123456789-1465312019-02-10T01:25:07Z Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations Ito, T. Terwilliger, P. 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. 2010 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/146531 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Ito, T. Terwilliger, P. |
| spellingShingle |
Ito, T. Terwilliger, P. Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Ito, T. Terwilliger, P. |
| author_sort |
Ito, T. |
| title |
Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations |
| 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 |
| publisher |
Інститут математики НАН України |
| publishDate |
2010 |
| url |
http://dspace.nbuv.gov.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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT itot doubleaffineheckealgebrasofrank1andthez3symmetricaskeywilsonrelations AT terwilligerp doubleaffineheckealgebrasofrank1andthez3symmetricaskeywilsonrelations |
| first_indexed |
2023-05-20T17:24:41Z |
| last_indexed |
2023-05-20T17:24:41Z |
| _version_ |
1796153229839433728 |