Wojciech J. Gajda

Abstract: Hopf algebras first arose in topology. Roughly speaking they are algebras provided with an additional structure (comultiplication). Geometrically they can be described via algebraic groups, i.e. group objects in a suitable category of schemes. A prototypical example is given by any group ring R[G] over a commutative base ring R. If H is a co-commutative Hopf algebra over R, there is a concept of H-Galois extensions A/R; for H = R[G] this specializes back to classical G-extensions (assume R is a field if you like). Generalizing the classical Galois correspondence, one finds a map sending (suitable) subobjects of H to (certain) subalgebras of A. This map is in general injective but not surjective; nevertheless it is called ``(Hopf) Galois correspondence''. We discuss a new way of pinpointing those subalgebras of A which are hit by the correspondence. So far we implicitly supposed that our extensions are ``finite'' in an appropriate sense. We discuss a generalization to the infinite-dimensional case, when R = K is a field. Large parts of this talk are based on recent work of Bui, Vercruysse, and Wiese.