Selecta Mathematica

, Volume 14, Issue 3–4, pp 325–372

Noncommutative Dunkl operators and braided Cherednik algebras



We introduce braided Dunkl operators \(\underline{\nabla}_1,\ldots,\underline{\nabla}_n\) that act on a q-symmetric algebra \(S_{\bf q}({\mathbb{C}}^n)\) and q-commute. Generalising the approach of Etingof and Ginzburg, we explain the q-commutation phenomenon by constructing braided Cherednik algebras \(\underline{{\mathcal{H}}}\) 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 \(\underline{{\mathcal{H}}}(W_+)\) attached to an infinite family of subgroups of even elements in complex reflection groups, so that the corresponding braided Dunkl operators \(\underline{\nabla}_i\) pairwise anticommute. We explicitly compute these new operators in terms of braided partial derivatives and W+-divided differences.

Mathematics Subject Classification (2000).

Primary 20G42 Secondary 16S80, 20F55 


Braided double Dunkl operator 


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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Department of MathematicsUniversity of OregonEugeneUSA

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