Tannaka-Krein reconstruction and a characterization of modular tensor categories

Details Tannaka-Krein reconstruction and a characterization of modular tensor categories Tannaka-Krein reconstruction and a characterization of modular tensor categories

2007-11-09

arxiv.org/abs/0711.1402v3

Journal of Algebra 321 No. 12 (2009) 3714-3763

Mathematics

Quantum Algebra

52 pages; LaTeX2e; xypic and pstricks macros; v2: typos corrected

Scientific paper

10.1016/j.jalgebra.2009.02.026

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.

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