Nilpotent fusion categories

Details Nilpotent fusion categories Nilpotent fusion categories

2006-10-24

arxiv.org/abs/math/0610726v2

Advances in Mathematics, 217 (2008), no. 3, 1053--1071.

Mathematics

Quantum Algebra

18 pages, AMS-Latex. Theorem 6.11 and 2 references added

Scientific paper

In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series of C. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Muger.

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