Coalgebra deformations of bialgebras by Harrison cocycles, copairings of Hopf algebras and double crosscoproducts

We study how the comultiplication on a Hopf algebra can be modified in such a way that the new comultiplication together with the original multiplication and a suitable antipode gives a new Hopf algebra. To this end, we have to study Harrison type cocycles, and it turns out that there is a relation with the Yang-Baxter equation. The construction is applied to deform the coalgebra structure on the tensor product of two bialgebras using a copairing. This new bialgebra can be viewed as a double crosscoproduct. It is also shown that a crossed coproduct over an inner comeasuring is isomorphic to a twisted coproduct.