I have stumbled upon the definitions of upper and lower semisolvability for Hopf algebras in the paper on classification of semisimple Hopf algebras of certain dimension.
And I have two questions:
1) Is upper semisolvability and lower are equivalent for finite dimensional semisimple Hopf algebras?
2) How well does semisolvability define a semisimple finite dimensional Hopf algebra? Can we deduce how comultiplication acts on different elements of our algebra if the algebra is semisolvable?