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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?

Thanks!

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I don't understand you second question? What do you mean by «define»? How could the property of semi solvability determine the comultiplication? –  Mariano Suárez-Alvarez May 12 '12 at 18:36
    
I mean maybe there is certain properties of comultiplication that must be satisfied for a semisimple Hopf algebra to be semisolvable? –  grozhd May 12 '12 at 22:34
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