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Braided Doubles and rational Cherednik algebras

SPEAKER: Arkady Berenstein

TITLE: Braided Doubles and rational Cherednik algebras.

ABSTRACT: In my talk (based on a recent joint paper with Yuri Bazlov) I will introduce a general class of algebras
with triangular decomposition which we call “braided doubles”. Braided doubles provide a unifying framework
for all classical and quantum universal enveloping algebras and recently discovered rational Cherednik algebras.

Quite surprisingly, one can completely classify free braided doubles.
The classification is achieved in terms of Yetter-Drinfeld (YD-)modules over Hopf algebras and their genralizations.
In particular, to each R-matrix one associates a canonical YD-module so that the corresponding braided double U(R)
is a deformation of the Weyl algebra, where the role of polynomial algebras is played by Nichols-Woronowicz algebras.

The main result is that any rational Cherednik algebra canonically embeds into the double U(R) attached to the
R-matrix emerging from each complex reflection group. This embedding gives a new definition of
rational Cherednik algebras and the instantaneous proof of their triangular decomposition.