Forms of Coalgebras and Hopf Algebras

We 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 show that all L-forms of H are invariant rings [L ⊗ H]W under appropriate actions of W on L⊗H. We apply this result to enveloping algebras, duals of finite-dimensional Hopf algebras, and adjoint actions of finite-dimensional semisimple cocommutative Hopf algebras.