Noncommutative Dunkl operators and braided Cherednik algebras

• Yuri Bazlov ^[1] ; Arkady Berenstein ^[2]
  1. [1] University of Warwick

     University of Warwick

     Reino Unido

     
  2. [2] Department of Mathematics, University of Oregon, USA
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 14, Nº. 3-4, 2009, págs. 325-372
• Idioma: inglés
• DOI: 10.1007/s00029-009-0525-x
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• Resumen

  • 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.