Tannaka-Krein reconstruction and a characterization of modular tensor categories
Hendryk Pfeiffer
TL;DR
The paper proves that any modular category over a field k is equivalent to the category of finite-dimensional comodules of a finite-dimensional split cosemisimple coribbon weak Hopf algebra, obtained as the coend of the long forgetful functor. By extending Tannaka--Kreĭn reconstruction to lax/oplax monoidal long forgetful functors, it handles non-integer Frobenius-Perron dimensions and constructs H with coribbon structure that recovers the modular category via comodules. The equivalence is established at the level of monoidal, ribbon, and modular structures, and the paper provides explicit basis-based formulas and an illustrative U_q(sl2) example. This framework unifies modular categories with WHA comodules, clarifying the role of base algebras, weak cofactorizability, and Morita considerations in reconstruction. It broadens the reach of Tannaka-type reconstruction to non-finite FP-dimension settings and offers concrete computational tools for realizations of modular data via WHAs.
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.