| This dissertation investigates quadratic Poisson structures on symmetric and exterior algebras. I classify all Poisson-modules over a reductive Lie algebra g ; i.e. modules V for which the classical (skew-symmetrized) r-matrix defines a Poisson structure on the symmetric algebra S(V). I show that if a g -module V is Poisson, then the deformation quantization of S(V) admits a module algebra action of the quantized enveloping algebra Uq( g ). Moreover, it is a quantum symmetric algebra Sq( Vq) of a certain Uq( g )-module Vq. I construct these quantum symmetric algebras as subalgebras of the quantized enveloping algebra Uq( g' ), by using a description of the Poisson modules as radicals corresponding to certain parabolic subalgebras of g' . Finally, I formulate a number of conjectures and open problems. |
