Resumen de Noncommutative Dunkl operators and braided Cherednik algebras

Yuri Bazlov, Arkady Berenstein

• We introduce braided Dunkl operators ∇––1,…,∇––n that act on a q-symmetric algebra Sq(Cn) and q-commute. Generalising the approach of Etingof and Ginzburg, we explain the q-commutation phenomenon by constructing braided Cherednik algebras H–– 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 H––(W+) attached to an infinite family of subgroups of even elements in complex reflection groups, so that the corresponding braided Dunkl operators ∇––i pairwise anticommute. We explicitly compute these new operators in terms of braided partial derivatives and W+-divided differences.