A Generalized Biproduct Theorem

We develop the Radford's biproduct theorem which plays an important role in giving a negative answer to a conjecture of I Kaplansky. Let B, H be two Hopf algebras with H acting weakly on B and α, β : B → H H be two linear maps verifying suitable conditions. We consider in this paper a twisted Hopf crossed coproduct B ×βα H and derive a necessary and sufficient condition for B # ×βα H with a Hopf smash product structure to be a bialgebra which generalizes in [14, Theorem 1.1] and the well-known Radford biproduct theorem [10, Theorem 1] .

Some Results on Cohomology of Twisted Smash Coproduct

With the main motivation to present some alternatives to the cosemisimplicity of twisted Smash coproduct,this paper first gives some basic constructions which make sense to consider the derived functors of some left exact functors from the category of comodules to vector spaces.Then,the Hochschild–Serre spectral sequence for twisted Smash coproduct coalgebras is obtained,which extends some results on homological properties given in the category of comodules over Smash coproduct in the literature.

On Corepresentations of Multiplier Hopf Algebras

We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate some properties of functors, integrals and morphisms related to the categories of the crossed modules and covariant modules over multiplier Hopf algebras. By doing so, we can apply our theory to the case of group-cograded multiplier Hopf algebras, in particular, the case of Hopf group-coalgebras.