  Let g be a Lie bialgebra and let V be a finite-dimensional g-module. We study
deformations of the symmetric algebra of V which are equivariant with respect
to an action of the quantized enveloping algebra of g. In particular we
investigate such quantizations obtained from the quantization of certain Lie
bialgebra structures on the semidirect product of g and V. We classify these
structure in the important special case, when g is complex, simple, with
quasitriangular Lie bialgebra structure and V is a simple g-module. We then
introduce more a general notion, co-Poisson module algebras and their
quantizations, to further address the problem and show that many known examples
of quantized symmetric algebras can be described in this language.
