Braided differential structure on Weyl groups, quadratic algebras and elliptic functions

We discuss a class of generalized divided difference operators which give rise to a representation of Nichols-Woronowicz algebras associated to Weyl groups. For the root system of type A, we also study the condition for the deformations of the Fomin-Kirillov quadratic algebra, which is a quadratic lift of the Nichols-Woronowicz algebra, to admit a representation given by generalized divided difference operators. The relations satisfied by the mutually commuting elements called Dunkl elements in the deformed Fomin-Kirillov algebra are determined. The Dunkl elements correspond to the truncated elliptic Dunkl operators via the representation given by the generalized divided difference operators. Introduction The rational Dunkl operators, which were introduced in [5] for any finite Coxeter group, constitute a remarkable family of operators of differentialdifference type. The Dunkl operators are defined to be the ones acting on the functions on the reflection representation V of the corresponding Weyl group W. For the root system of type An−1, the Dunkl operators D1, . . . , Dn are defined by the formula Di := ∂ ∂xi + ∑ j 6=i 1− sij xi − xj , ∗Supported by Grant-in-Aid for Scientific Research.