A Study On Double Affine Yokonuma-Hecke Algebras

Double affine Hecke algebras,also called Cherednik algebras,are a class of infinite dimensional algebras introduced by I.Cherednik in the proof of Macdonald’s inner product conjectures about orthogonal polynomials for arbitrary root systems.After that,its trigonometric and rational forms were also defined from different perspectives.Yokonuma-Hecke algebras were introduced by Yokonuma as a centralizer algebra associated to the permutation representation of a finite Chevalley group G with respect to a maximal unipotent subgroup of it.In order to study the representation theory of Yokonuma-Hecke algebras,Chlouveraki and Poulain d’Andecy defined and studied affine Yokonuma-Hecke algebras Y_r,n（q）.In the first part,we define a new class of infinite dimensional algebras,called the double affine Yokonuma-Hecke algebras Y_r,n（ζ,t）,We give two equivalent presentations.Moreover,we define a cyclotomic version of double affine Yokonuma-Hecke algebras Y_r,n（ζ,t）.In the second part,we define two new classes of infinite dimensional algebras,called rational double affine Yokonuma-Hecke algebras Y_r,n^κ and trigonometric double affine YokonumaHecke algebras Y_r,n^κ,respectively,and We give two equivalent presentations of trigonometric double affine Yokonuma-Hecke algebras.Moreover,we define a cyclotomic version of trigonometric double affine Yokonuma-Hecke algebras Y_r,n^κ;then we define a new class of algebras,called the cyclotomic rational double affine Yokonuma-Hecke algebras by giving two equivalent presentations.