Forms of coalgebras and Hopf algebras

We study forms of coalgebras and Hopf algebras (i.e., coalgebras and Hopf a lgebras which are isomorphic after a suitable extension of the base field). We classify all forms of grouplike coalgebras according to the structure o f their simple subcoalgebras. For Hopf algebras, given a W*-Galois field ex tension K subset of or equal to L for W a finite-dimensional semisimple Hop f algebra anti a K-Hopf algebra H, we show that all L-forms of H are invari ant rings [L circle times H](W) under appropriate actions of W on L circle times H. We apply this result to enveloping algebras, duals of finite-dimen sional Hopf algebras, and adjoint actions of finite-dimensional semisimple cocommutative Hopf algebras. (C) 2001 Academic Press.