Tannaka–Krěın reconstruction and a characterization of modular tensor categories
نویسنده
چکیده
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–Krěın 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 finitedimensional, split cosemisimple, coribbon and has trivially intersecting base algebras. Mathematics Subject Classification (2000): 16W30, 18D10 keywords: Modular category, tensor category, Weak Hopf Algebra, Tannaka–Krěın reconstruction
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