The topic of this talk is the representation theory of Hopf-Galois extensions. We consider the following questions.
Let H be a Hopf algebra, and A, B right H-comodule algebras. Assume that A and B are faithfully flat H-Galois extensions.
1. If A and B are Morita equivalent, does it follow that the subalgebras A^coH and B^coH of H-coinvariant elements are also Morita equivalent?
2. Conversely, if A^coH and B^coH are Morita equivalent, when does it follow that A and B are Morita equivalent?
As an application, we investigate H-Morita autoequivalences of the H-Galois extension A, introduce the concept of H-Picard group, and we establish an exact sequence linking the H-Picard group of A and
the Picard group of A^coH.(joint work with Stefaan Caenepeel)