Fiber functors and reconstruction of Hopf algebras

Abstract

The main objective of the present paper is to present a version of the Tannaka-Krein type reconstruction Theorems: If F:B→C is an exact faithful monoidal functor of tensor categories, one would like to realize B as category of representations of a braided Hopf algebra H(F) in C. We prove that this is the case iff B has the additional structure of a monoidal C-module category compatible with F, which equivalently means that F admits a monoidal section. For Hopf algebras, this reduces to a version of the Radford projection theorem. The Hopf algebra is constructed through the relative coend for module categories. We expect this basic result to have a wide range of applications, in particular in the absence of fibre functors, and we give some applications. One particular motivation was the logarithmic Kazhdan-Lusztig conjecture.