Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups

TL;DR

This paper classifies Lagrangian subcategories of twisted quantum doubles of finite groups, providing a complete description of braided tensor equivalences and linking them to module categories and Morita equivalence.

Contribution

It offers a complete classification of Lagrangian subcategories and braided tensor equivalences for twisted quantum doubles, connecting them to module categories and Morita theory.

Findings

Classified Lagrangian subcategories of twisted quantum doubles.

Established a bijection between Lagrangian subcategories and module categories.

Showed that group-theoretical fusion categories are weakly Morita equivalent iff their centers are braided tensor equivalent.

Abstract

We classify Lagrangian subcategories of the representation category of a twisted quantum double of a finite group. In view of results of 0704.0195v2 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 the representation category of a twisted quantum double of a finite group G and module categories over the category 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. 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.