Noncommutative Dunkl operators and braided Cherednik algebras

Details Noncommutative Dunkl operators and braided Cherednik algebras Noncommutative Dunkl operators and braided Cherednik algebras

2008-06-04

arxiv.org/abs/0806.0867v2

Selecta Math. 14 (2009), no. 3-4, 325--372

Mathematics

Quantum Algebra

48 pages, no figures. References to papers by T. Khongsap, W. Wang are added. An anti-commutatator version is included (Coroll

Scientific paper

We introduce braided Dunkl operators that are acting on a q-polynomial algebra and q-commute. Generalizing the approach of Etingof and Ginzburg, we explain the q-commutation phenomenon by constructing braided Cherednik algebras 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 attached to an infinite family of subgroups of even elements in complex reflection groups, so that the corresponding braided Dunkl operators pairwise anti-commute. We explicitly compute these new operators in terms of braided partial derivatives and divided differences.

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