Tannaka-Krein reconstruction and a characterization of modular tensor categories

TL;DR

This paper characterizes modular tensor categories as categories of comodules over a specific type of Weak Hopf Algebra, extending Tannaka-Krein reconstruction to handle non-strong monoidal functors.

Contribution

It generalizes Tannaka-Krein reconstruction to non-strong monoidal functors, providing a new characterization of modular categories via Weak Hopf Algebras.

Findings

Every modular category is equivalent to comodules of a finite-dimensional Weak Hopf Algebra.

The Weak Hopf Algebra has specific properties: split cosemisimple, weakly cofactorizable, and coribbon.

The generalized reconstruction avoids issues with non-integral Frobenius-Perron dimensions.

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.