  Additive deformations of bialgebras in the sense of Wirth are deformations of
the multiplication map of the bialgebra fulfilling a compatibility condition
with the coalgebra structure and a continuity condition. Two problems
concerning additive deformations are considered. With a deformation theory a
cohomology theory should be developed. Here a variant of the Hochschild
cohomology is used. The main result in the first partad of this paper is the
characterization of the trivial deformations, i.e. deformations generated by a
coboundary. Starting with a Hopf algebra, one would expect the deformed
multiplications to have some analogue of the antipode, which we call deformed
antipodes. We prove, that deformed antipodes always exist, explore their
properties, give a formula to calculate them given the deformation and the
antipode of the original Hopf algebra and show in the cocommutative case, that
each deformation splits into a trivial part and into a part with constant
antipodes.
