Group-theoretical properties of nilpotent modular categories

Understand

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.