A monoidal approach to splitting morphisms of bialgebras

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. Let us consider a Hopf algebra $A$ such that its Jacobson radical $J$ is a nilpotent Hopf ideal and $H:=A/J$ is a semisimple algebra. We prove that the canonical projection of $A$ on $H$ has a section which is an $H$-colinear algebra map. Furthermore, if $H$ is cosemisimple too, then we can choose this section to be an $(H,H)$-bicolinear algebra morphism. This fact allows us to describe $A$ as a `generalized bosonization' of a certain algebra $R$ in the category of Yetter-Drinfeld modules over $H$. As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. Let $A$ be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of $H$ into $A$ which is an $H$-linear coalgebra morphism. Furthermore, if $H$ is semisimple too, then we can choose this retraction to be an $(H,H)$-bilinear coalgebra morphism. Then, also in this case, we can describe $A$ as a `generalized bosonization' of a certain coalgebra $R$ in the category of Yetter-Drinfeld modules over $H$.