Tannaka–Kreıˇn reconstruction and a characterization of modular tensor categories

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 finitedimensional, 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-Kreǐn 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. In the more general case of a finitely semisimple additive ribbon category, not necessarily modular, the reconstructed Weak Hopf Algebra is finite-dimensional, split cosemisimple, coribbon and has trivially intersecting base algebras.

Finitely semisimple spherical categories and modular categories are self-dual

“…We show that the universal coend, H = coend(C, ω), of our spherical category C with respect to the long forgetful functor ω : C → Vect k is cospherical and that the category C is equivalent as a spherical category to the category of finite-dimensional right H -comodules. This generalizes [25] to our class of spherical categories. In order to construct coend(C, ω), we need C to be small.…”

“…The following theorem was shown in [25] for modular categories. Its proof is exactly the same for our spherical categories.…”

“…The special case of modular categories was addressed in the article [25]. The functor ω is the long version of Hayashi's canonical forgetful functor [12].…”

“…We also use Sweedler's notation and write (x) = x ⊗ x for the comultiplication of x ∈ H as an abbreviation of the expression (x) = k a k ⊗ b k with some a k , b k ∈ H . Similarly, we write We extend Sweedler's notation to the right H -comodules and write [23,25] …”

“…First, we study the properties of the long forgetful functor ω : C → Vect k associated with our spherical category C. It turns out that Section 3 of [25] applies to the spherical case without any significant change, and so we keep this section brief. Definition 3.1.…”