Double Affine Hecke Algebras of Rank 1 and the Z_3-Symmetric Askey-Wilson Relations

• Main Authors: Paul Terwilliger, Tatsuro Ito
• Format: Article
• Language: English
• Published: National Academy of Science of Ukraine 2010-08-01
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Subjects: Askey-Wilson polynomials; Askey-Wilson relations; braid group
• Online Access: http://dx.doi.org/10.3842/SIGMA.2010.065

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_3-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 (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 Z3-symmetric Askey-Wilson relations. Applying the homomorphism H^ → H we find that the elements x, y, z in H satisfy similar relations.