By definition, the endomorphism spaces of tensor powers of objects of a braided tensor category carries a representation of the braid group. For Lie types A and C, this can be used to classify all braided tensor categories whose fusion ring is the one of the representation category of the related Lie algebra. We also discuss the situation for other classical Lie types and some exceptional types. There are several different ways how to construct TQFTs and modular functors. One of the motivations for these categorical questions was to decide when these constructions yield the same results.