Additive Deformations of Hopf Algebras

TL;DR

This paper studies additive deformations of Hopf algebras, characterizing trivial deformations via a Hochschild cohomology variant and proving the existence of deformed antipodes with specific properties.

Contribution

It develops a cohomology framework for additive deformations and proves the existence and properties of deformed antipodes in Hopf algebras.

Findings

Trivial deformations are characterized as coboundaries.

Deformed antipodes always exist for additive deformations.

In cocommutative cases, deformations split into trivial and constant antipode parts.

Abstract

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.