2010/01/31 by Tatsuro Ito, Paul Terwilliger · 1 citation
Mathematics · #math.RA #math.QA #msc:33D80 #msc:33D45
paper · pdf · doi:10.3842/sigma.2010.065
published as SIGMA 6 (2010), 065, 9 pages
arxiv created 2010/08/17 · arxiv updated 2010/08/18
We consider the double affine Hecke algebra H=H(k0,k1,k^\vee0,k^\vee1;q) associated with the root system (C^\vee1,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 (C1^\vee,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 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.