Vol. 264, No. 1, 2013

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ISSN: 0030-8730
On the center of fusion categories

Alain Bruguières and Alexis Virelizier

Vol. 264 (2013), No. 1, 1–30
Abstract

Müger proved in 2003 that the center of a spherical fusion category 𝒞 of nonzero dimension over an algebraically closed field is a modular fusion category whose dimension is the square of that of 𝒞. We generalize this theorem to a pivotal fusion category 𝒞 over an arbitrary commutative ring  𝕜, without any condition on the dimension of the category. (In this generalized setting, modularity is understood as 2-modularity in the sense of Lyubashenko.) Our proof is based on an explicit description of the Hopf algebra structure of the coend of the center of 𝒞. Moreover we show that the dimension of 𝒞 is invertible in 𝕜 if and only if any object of the center of 𝒞 is a retract of a “free” half-braiding. As a consequence, if 𝕜 is a field, then the center of 𝒞 is semisimple (as an abelian category) if and only if the dimension of 𝒞 is nonzero. If in addition 𝕜 is algebraically closed, then this condition implies that the center is a fusion category, so that we recover Müger’s result.

Keywords
categorical center, fusion categories, Hopf monads, modularity
Mathematical Subject Classification 2010
Primary: 18D10, 16T05, 18C20
Milestones
Received: 21 March 2012
Revised: 28 August 2012
Accepted: 4 September 2012
Published: 5 July 2013
Authors
Alain Bruguières
Département de mathématiques
Université Montpellier II
Case Courrier 051
Place Eugène Bataillon
34095 Montpellier Cedex 5
France
Alexis Virelizier
Département de mathématiques
Université Lille 1
59655 Villeneuve d’Ascq
France