Abstract
We classify Lagrangian subcategories of the representation category of a twisted quantum double D ω(G), where G is a finite group and ω is a 3-cocycle on it. In view of results of [DGNO] this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of Rep(D ω(G)) and module categories over the category \({\rm Vec}_{G} ^{\omega}\) of twisted G-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfeld’s characterization of homogeneous spaces of a Poisson-Lie group in terms of Lagrangian subalgebras of the double of its Lie bialgebra [D]. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories.
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Communicated by Y. Kawahigashi
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Naidu, D., Nikshych, D. Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups. Commun. Math. Phys. 279, 845–872 (2008). https://doi.org/10.1007/s00220-008-0441-5
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DOI: https://doi.org/10.1007/s00220-008-0441-5