Braided doubles and rational Cherednik algebras

We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasi-Yetter–Drinfeld (QYD) modules over Hopf algebras which turn out to be a generalisation of the ordinary Yetter–Drinfeld modules. To each braiding (a solution to the braid equation) we associate a QYD-module and the corresponding braided Heisenberg double—this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by Nichols–Woronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds in the braided Heisenberg double attached to the corresponding complex reflection group.