Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 6 (2010), 065, 9 pages      arXiv:1001.2764      http://dx.doi.org/10.3842/SIGMA.2010.065

Double Affine Hecke Algebras of Rank 1 and the Z3-Symmetric Askey-Wilson Relations

Tatsuro Ito a and Paul Terwilliger b
a) Division of Mathematical and Physical Sciences, Graduate School of Natural Science nd Technology, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan
b) Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706-1388, USA

Received January 23, 2010, in final form August 10, 2010; Published online August 17, 2010

Abstract
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 H^ that is more general than H, called the universal double affine Hecke algebra of type (C1v,C1). 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 xyzx 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 Z3-symmetric Askey-Wilson relations. Applying the homomorphism H^H we find that the elements x, y, z in H satisfy similar relations.

Key words: Askey-Wilson polynomials; Askey-Wilson relations; braid group.

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