Group-theoretical properties of nilpotent modular categories

TL;DR

This paper explores the structure of nilpotent modular categories, characterizing their relation to finite p-groups, and establishing conditions under which they are group-theoretical, with implications for quasi-Hopf algebras.

Contribution

It provides a characterization of certain modular categories as twisted doubles of p-groups and introduces an analogue of Sylow decomposition for nilpotent braided fusion categories.

Findings

Nilpotent modular categories of prime power dimension are representation categories of twisted p-group doubles.

A Sylow-like decomposition exists for nilpotent braided fusion categories.

Semisimple quasi-Hopf algebras of prime power dimension are 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).