Coradical splittings and a generalization of Kostant's theorem

AbstractSeveral questions are addressed concerning the way the coradical of a Hopf algebra sits inside the Hopf algebra. It is proved that for a Hopf algebra H with cocommutative coradical H0, H0 is a sub Hopf algebra if and only if H0 is coseparable as a coalgebra. Using this result, a generalization of Kostant's theorem on the structure of cocommutative Hopf algebras is proved: a cocommutative Hopf algebra H over any field k is Hopf algebra isomorphic to the smash product H1# H0 if and only if H0 is a sub Hopf algebra. We also prove that for an exact sequence (∗) k→K→H→L→k of Hopf algebras over the field k, H is cosemisimple if and only if K and L are cosemisimple. Under suitable conditions on the sequence (∗), H0 is a sub Hopf algebra if and only if K0 and L0 are sub Hopf algebras