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Double Affine Hecke Algebras of Rank 1 and the $\mathbb Z_3$-Symmetric Askey–Wilson Relations
Symmetry, integrability and geometry: methods and applications, Tome 6 (2010) Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the double affine Hecke algebra $H=H(k_0,k_1,k^\vee_0,k^\vee_1;q)$ associated with the root system $(C^\vee_1,C_1)$. We display three elements $x$, $y$, $z$ in $H$ that satisfy essentially the $\mathbb Z_3$-symmetric Askey–Wilson relations. We obtain the relations as follows. We work with an algebra $\hat H$ that is more general than $H$, called the universal double affine Hecke algebra of type $(C_1^\vee,C_1)$. An advantage of $\hat H$ over $H$ is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism $\hat H\to H$. We define some elements $x$, $y$, $z$ in $\hat H$ that get mapped to their counterparts in $H$ by this homomorphism. We give an action of Artin's braid group $B_3$ on $\hat H$ that acts nicely on the elements $x$, $y$, $z$; one generator sends $x\mapsto y\mapsto z \mapsto x$ and another generator interchanges $x$, $y$. Using the $B_3$ action we show that the elements $x$, $y$, $z$ in $\hat H$ satisfy three equations that resemble the $\mathbb Z_3$-symmetric Askey–Wilson relations. Applying the homomorphism ${\hat H}\to H$ we find that the elements $x$, $y$, $z$ in $H$ satisfy similar relations.
Keywords: Askey–Wilson polynomials; Askey–Wilson relations; braid group. 
@article{SIGMA_2010_6_a64,
     author = {Tatsuro Ito and Paul Terwilliger},
     title = {Double {Affine} {Hecke} {Algebras} of {Rank~1} and the $\mathbb Z_3${-Symmetric} {Askey{\textendash}Wilson} {Relations}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2010},
     volume = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a64/}
}
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Tatsuro Ito; Paul Terwilliger. Double Affine Hecke Algebras of Rank 1 and the $\mathbb Z_3$-Symmetric Askey–Wilson Relations. Symmetry, integrability and geometry: methods and applications, Tome 6 (2010). http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a64/

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