REPRESENTATION
THEORY AND RELATED TOPICS SEMINAR
Arkady Berenstein
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 generalizations. 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.
For further information visit http://www.math.neu.edu/~martsinkovsky/p/rtrt.html or contact Alex Martsinkovsky alexmart >at< neu >dot< edu