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