Invertible bimodule categories over the representation category of a Hopf algebra

Abstract

For any finite-dimensional Hopf algebra H we construct a group homomorphism BiGal(H)→BrPic(Rep(H)), from the group of equivalence classes of H-biGalois objects to the group of equivalence classes of invertible exact Rep(H)-bimodule categories. We discuss the injectivity of this map. We exemplify in the case H=Tq is a Taft Hopf algebra and for this we classify all exact indecomposable Rep(Tq)-bimodule categories.