  We obtain Koszul-type dualities for categories of graded modules over a
graded associative algebra which can be realized as the semidirect product of a
bialgebra coinciding with its degree zero part and a graded module algebra for
the latter. In particular, this applies to graded representations of the
universal enveloping algebra of the Takiff Lie algebra (or the truncated
current algebra) and its (super)analogues, and also to semidirect products of
quantum groups with braided symmetric and exterior module algebras in case the
latter are flat deformations of classical ones.
