Coradical splittings and a generalization of Kostant's theorem (Q752816)

Let H be a cocommutative Hopf algebra over a field k, \(H_ 0\) its coradical. The author first shows that \(H_ 0\) is a Hopf subalgebra if and only if \(H_ 0\) is coseparable. Using this, he shows that H splits as the smash product \(H^ 1\#H_ 0\) iff \(H_ 0\) is a Hopf subalgebra. Dropping the cocommutativity assumption, he shows that for an exact sequence \(k\to K\to H\to L\to k\) of Hopf algebras, H is cosemi-simple iff K and L are cosemi-simple. He also discusses the condition that \(H_ 0\) be a Hopf subalgebra in terms of conditions on \(K_ 0\) and \(L_ 0\).