Noncommutative Dunkl operators and braided Cherednik algebras

TL;DR

This paper introduces braided Dunkl operators acting on q-polynomial algebras, constructs braided Cherednik algebras, classifies them, and provides explicit formulas for new operators with anti-commuting properties.

Contribution

It generalizes Cherednik algebras to a braided setting, classifies all such algebras, and explicitly constructs new operators with anti-commutation relations.

Findings

Introduced braided Dunkl operators on q-polynomial algebras.

Classified all braided Cherednik algebras, including new types.

Explicit formulas for braided Dunkl operators in terms of derivatives.

Abstract

We introduce braided Dunkl operators that are acting on a q-polynomial algebra and q-commute. Generalizing the approach of Etingof and Ginzburg, we explain the q-commutation phenomenon by constructing braided Cherednik algebras for which the above operators form a representation. We classify all braided Cherednik algebras using the theory of braided doubles developed in our previous paper. Besides ordinary rational Cherednik algebras, our classification gives new algebras attached to an infinite family of subgroups of even elements in complex reflection groups, so that the corresponding braided Dunkl operators pairwise anti-commute. We explicitly compute these new operators in terms of braided partial derivatives and divided differences.