Coradical splittings and a generalization of Kostant's theorem

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Several 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 H 0 , H 0 is a sub Hopf algebra if and only if H 0 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 H 1 # H 0 if and only if H 0 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 (∗), H 0 is a sub Hopf algebra if and only if K 0 and L 0 are sub Hopf algebras.