Modular tensor categories from semisimple algebras

Modular tensor categories are finitely semisimple Abelian categories with some additional structure that allows us to obtain combinatorial invariants of oriented framed tangles. Reshetikhin and Turaev invented them in order to construct their 3-manifold invariant.

Algebraically, these categories can be constructed as follows: There exist very special finite-dimensional and non-semisimple Hopf algebras. Restrict the category H-Mod of such a Hopf algebra H to its full subcategory of tilting modules. The modular tensor category then appears as a quotient of the category of tilting modules.

I find this recipe too complicated. Question: Is every modular tensor category equivalent to A-Mod for some finite-dimensional semismple algebra A? I show how to fix Tannaka-Krein reconstruction in order to give an affirmative answer to this question.

As the term 'modular' suggests, there are plenty of similarities with representation theory in positive characteristic p. Using quantum groups, however, one obains equivalent categories by merely working in characteristic zero and one can even treat the case in which `p' is not prime.