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

TL;DR

This paper explores generalized divided difference operators that create representations of Nichols-Woronowicz algebras linked to Weyl groups, focusing on type A root systems and their connections to elliptic Dunkl operators.

Contribution

It introduces a class of generalized divided difference operators and analyzes their role in representing Nichols-Woronowicz algebras and deformations of quadratic algebras for Weyl groups.

Findings

Representation of Nichols-Woronowicz algebras via generalized divided difference operators.

Conditions for deformations of Fomin-Kirillov quadratic algebra to admit such representations.

Identification of Dunkl elements with truncated elliptic Dunkl operators.

Abstract

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.