Group-theoretical properties of nilpotent modular categories
Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik
TL;DR
The paper develops a structural theory for nilpotent and modular fusion categories, linking them to group-theoretical fusion categories and twisted group doubles. It proves a Sylow-type decomposition for braided nilpotent fusion categories and provides a Lagrangian-subcategory criterion that identifies when a modular category is the center of a twisted group double, with central-charge invariance under modularization. As consequences, semisimple quasi-Hopf algebras of prime-power dimension are shown to be group-theoretical, and a categorical reconstruction analogous to Manin pairs is established. The framework relies on modularization techniques, centralizers, and lattice-theoretic arguments for isotropic and Lagrangian subcategories to connect hyperbolic modular categories with twisted doubles and to characterize when a modular category is group-theoretical.
Abstract
We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted doubles of finite p-groups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects of C have integral Frobenius-Perron dimensions then C is group-theoretical. As a consequence, we obtain that semisimple quasi-Hopf algebras of prime power dimension are group-theoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (this is reminiscent to the characterization of doubles of quasi-Lie bialgebras in terms of Manin pairs).