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Abstract: Additive deformations of bialgebras in the sense of Wirth are deformations ofthe multiplication map of the bialgebra fulfilling a compatibility conditionwith the coalgebra structure and a continuity condition. Two problemsconcerning additive deformations are considered. With a deformation theory acohomology theory should be developed. Here a variant of the Hochschildcohomology is used. The main result in the first part of this paper is thecharacterization of the trivial deformations, i.e. deformations generated by acoboundary. When one starts with a Hopf algebra, one would expect the deformedmultiplications to have some analogue to the antipode, which we call deformedantipodes. We prove, that deformed antipodes always exist, explore theirproperties, give a formula to calculate them given the deformation and theantipode of the original Hopf algebra and show in the cocommutative case, thateach deformation splits into a trivial part and into a part with constantantipodes.



Author: Malte Gerhold

Source: https://arxiv.org/







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