### Pfeiffer on Modular Tensor Categories

#### Posted by John Baez

Modular tensor categories play an important role as a bridge from physics to topology. They arise from rational conformal field theories and quantum groups, and they give rise to 3d topological quantum field theories!

What sort of gadget has representations that form a modular tensor category? Some sort of quantum group, you might guess… but what’s the *exact* answer? And: can you “reconstruct” this gadget from its category of representations?

The answers are all here:

Here’s the abstract:

We show that every modular category is equivalent as an additive ribbon category to the category of finite-dimensional comodules of a Weak Hopf Algebra. This Weak Hopf Algebra is finite-dimensional, split cosemisimple, weakly cofactorizable, coribbon and has trivially intersecting base algebras. In order to arrive at this characterization of modular categories, we develop a generalization of Tannaka-Krein reconstruction to the long version of the canonical forgetful functor which is lax and oplax monoidal, but not in general strong monoidal, thereby avoiding all the difficulties related to non-integral Frobenius–Perron dimensions.

I’m not sure what “the long version” means. The rest of course is perfectly clear.