Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups

By D. Venku Naidu, D. Nikshych. 2007

77 citations recorded by the source corpus.

We classify Lagrangian subcategories of the representation category of a twisted quantum double Dω(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(Dω(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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/61d3e00f9356ce3ddd05937966bf5fbf33aec6f6

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Non-group-theoretical semisimple Hopf algebras from group actions on fusion categories (2007, background): “Such categories were extensively studied (see e.g., [4, 6, 7, 10, 17, 18 , 19]), and many classication results were obtained.”
On finite symmetries and their gauging in two dimensions (2017, background): “There is a theorem by Naidu, Nikshych [29, 30] and Uribe [31] determining exactly when this happens, and if so, explicitly the form of G′ and α′.”
On gauging finite subgroups (2017, background): “When D = 2 this result was known in the mathematical literature [29,30] for some time.”
On braided fusion categories I (2009, background): “An answer to this question is given in [37].”
The Quantum Double Model with Boundary: Condensations and Symmetries (2010, methodology): “This question was first answered by Naidu and Nikshych [11,12] based on the classification of Lagrangian subcategories of Z(G).”