Selecta Mathematica

, Volume 14, Issue 3, pp 325–372

Noncommutative Dunkl operators and braided Cherednik algebras


DOI: 10.1007/s00029-009-0525-x

Cite this article as:
Bazlov, Y. & Berenstein, A. Sel. math., New ser. (2009) 14: 325. doi:10.1007/s00029-009-0525-x


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 20G42Secondary 16S80, 20F55


Braided doubleDunkl operator

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

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