Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups
Abstract
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.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- May 2008
- DOI:
- 10.1007/s00220-008-0441-5
- arXiv:
- arXiv:0705.0665
- Bibcode:
- 2008CMaPh.279..845N
- Keywords:
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- Module Category;
- Irreducible Character;
- Simple Object;
- Tensor Category;
- Fusion Category;
- Mathematics - Quantum Algebra
- E-Print:
- 26 pages