  We show that for dually paired bialgebras, every comodule algebra over one of
the paired bialgebras gives a comodule algebra over their Drinfeld double via a
crossed product construction. These constructions generalize to working with
bialgebra objects in a braided monoidal category of modules over a
quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over
the braided Drinfeld double (the double bosonization) are provided.
Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It
takes a pair of 2-cocycles over dually paired Hopf algebras and produces a
2-cocycle over their Drinfeld double. This construction also has an analogue
for braided Drinfeld doubles.
