ŽWe study forms of coalgebras and Hopf algebras i.e., coalgebras and Hopf . algebras which are isomorphic after a suitable extension of the base field . We classify all forms of grouplike coalgebras according to the structure of their simple subcoalgebras. For Hopf algebras, given a W *-Galois field extension K : L for W a finite-dimensional semisimple Hopf algebra and a K-Hopf algebra H, we showWe apply this result to enveloping algebras, duals of finite-dimensional Hopf algebras, and adjoint actions of finite-dimensional semisimple cocommutative Hopf algebras.
We investigate the coradical filtration of pointed coalgebras. First, we generalize a theorem of Taft and Wilson using techniques developed by Radford in [Rad78] and [Rad82]. We then look at the coradical filtration of duals of inseparable field extensions L * upon extension of the base field K, where K ⊆ L is a field extension. We reduce the problem to the case that the field extension is purely inseparable.
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