Double Affine Hecke Algebras of Rank 1 and the Z_3Symmetric AskeyWilson Relations
We consider the double affine Hecke algebra H=H(k_0,k_1,k_0^v,k_1^v;q) associated with the root system (C_1^v,C_1). We display three elements x, y, z in H that satisfy essentially the Z_3symmetric AskeyWilson relations. We obtain the relations as follows. We work with an algebra H^ that is more ge...
Main Authors:  , 

Format:  Article 
Language:  English 
Published: 
National Academy of Science of Ukraine
20100801

Series:  Symmetry, Integrability and Geometry: Methods and Applications 
Subjects:  
Online Access:  http://dx.doi.org/10.3842/SIGMA.2010.065 
id 
doaj848fbee4681b442a803349257ddabe23 

record_format 
Article 
spelling 
doaj848fbee4681b442a803349257ddabe2320201125T00:03:43ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications18150659201008016065Double Affine Hecke Algebras of Rank 1 and the Z_3Symmetric AskeyWilson RelationsPaul TerwilligerTatsuro ItoWe consider the double affine Hecke algebra H=H(k_0,k_1,k_0^v,k_1^v;q) associated with the root system (C_1^v,C_1). We display three elements x, y, z in H that satisfy essentially the Z_3symmetric AskeyWilson relations. We obtain the relations as follows. We work with an algebra H^ that is more general than H, called the universal double affine Hecke algebra of type (C_1^v,C_1). An advantage of H^ over H is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism H^ → H. We define some elements x, y, z in H^ that get mapped to their counterparts in H by this homomorphism. We give an action of Artin's braid group B3 on H^ 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 H^ satisfy three equations that resemble the Z3symmetric AskeyWilson relations. Applying the homomorphism H^ → H we find that the elements x, y, z in H satisfy similar relations.http://dx.doi.org/10.3842/SIGMA.2010.065AskeyWilson polynomialsAskeyWilson relationsbraid group 
collection 
DOAJ 
language 
English 
format 
Article 
sources 
DOAJ 
author 
Paul Terwilliger Tatsuro Ito 
spellingShingle 
Paul Terwilliger Tatsuro Ito Double Affine Hecke Algebras of Rank 1 and the Z_3Symmetric AskeyWilson Relations Symmetry, Integrability and Geometry: Methods and Applications AskeyWilson polynomials AskeyWilson relations braid group 
author_facet 
Paul Terwilliger Tatsuro Ito 
author_sort 
Paul Terwilliger 
title 
Double Affine Hecke Algebras of Rank 1 and the Z_3Symmetric AskeyWilson Relations 
title_short 
Double Affine Hecke Algebras of Rank 1 and the Z_3Symmetric AskeyWilson Relations 
title_full 
Double Affine Hecke Algebras of Rank 1 and the Z_3Symmetric AskeyWilson Relations 
title_fullStr 
Double Affine Hecke Algebras of Rank 1 and the Z_3Symmetric AskeyWilson Relations 
title_full_unstemmed 
Double Affine Hecke Algebras of Rank 1 and the Z_3Symmetric AskeyWilson Relations 
title_sort 
double affine hecke algebras of rank 1 and the z_3symmetric askeywilson relations 
publisher 
National Academy of Science of Ukraine 
series 
Symmetry, Integrability and Geometry: Methods and Applications 
issn 
18150659 
publishDate 
20100801 
description 
We consider the double affine Hecke algebra H=H(k_0,k_1,k_0^v,k_1^v;q) associated with the root system (C_1^v,C_1). We display three elements x, y, z in H that satisfy essentially the Z_3symmetric AskeyWilson relations. We obtain the relations as follows. We work with an algebra H^ that is more general than H, called the universal double affine Hecke algebra of type (C_1^v,C_1). An advantage of H^ over H is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism H^ → H. We define some elements x, y, z in H^ that get mapped to their counterparts in H by this homomorphism. We give an action of Artin's braid group B3 on H^ 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 H^ satisfy three equations that resemble the Z3symmetric AskeyWilson relations. Applying the homomorphism H^ → H we find that the elements x, y, z in H satisfy similar relations. 
topic 
AskeyWilson polynomials AskeyWilson relations braid group 
url 
http://dx.doi.org/10.3842/SIGMA.2010.065 
work_keys_str_mv 
AT paulterwilliger doubleaffineheckealgebrasofrank1andthez3symmetricaskeywilsonrelations AT tatsuroito doubleaffineheckealgebrasofrank1andthez3symmetricaskeywilsonrelations 
_version_ 
1725432443707588608 