We study the induction and restriction functor from a Hopf subalgebra of a semisimple Hopf algebra. The image of the induction functor is described when the Hopf subalgebra is normal. In this situation, at the level of characters this image is isomorphic to the image of the restriction functor. A criterion for subcoalgebras to be invariant under the adjoint action is given generalizing Masuoka’...

Recently, Brzezi\'nski, Koenig and K\"ulshammer have introduced the notion of normal exact Borel subalgebra a quasihereditary algebra. They shown that there exists one-to-one correspondence between directed bocses algebras with homological subalgebra. In this short note, we prove every is automatically normal. As corollary, conclude bocs has group-like element. These results simplify K\"ulshamm...