## Modular tensor categories from semisimple algebras

- Date: 04/30/2007

Hendryk Pfeiffer (Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge)

University of British Columbia

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.

Algebraic Geometry Seminar 2007