GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES

By V. Drinfeld, Shlomo Gelaki, D. Nikshych, V. Ostrik. 2007

88 citations recorded by the source corpus.

We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted dou- bles 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 in the sense of (ENO). 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 given in (Dr)).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/450dc0262b861bf2109ba4f4c34768102ff24cf7

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Citations in the corpus

Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups (2007, background): “D DMITRI NIKSHYCH Abstract. We classify Lagrangiansubcategoriesofthe representationcategoryofa 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 d”
Fusion subcategories of representation categories of twisted quantum doubles of finite groups (2008, background): “A fusion subcategory D of a premodular category C with positive categorical dimensions is said to be isotropic if the twist of C restricts to the identity on D [ DGNO ].”
Centers of graded fusion categories (2009, background): “Grouptheoretical properties of such categories were studied in [Drinfeld et al. 2007].”
A Finiteness Property for Braided Fusion Categories (2009, background): “Any (unitary) modular category provides both C-valued multiplicative link invariants (e.g. the Jones polynomial) and a model for a (theoretical) 2-dimensional physical system (e.g. fractional quantum Hall liquids).”