On the center of fusion categories

Alain Bruguières, Alexis Virelizier
2013 Pacific Journal of Mathematics  
Müger proved in 2003 that the center of a spherical fusion category Ꮿ of nonzero dimension over an algebraically closed field is a modular fusion category whose dimension is the square of that of Ꮿ. We generalize this theorem to a pivotal fusion category Ꮿ 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
more » ... licit description of the Hopf algebra structure of the coend of the center of Ꮿ. Moreover we show that the dimension of Ꮿ is invertible in k if and only if any object of the center of Ꮿ is a retract of a "free" half-braiding. As a consequence, if k is a field, then the center of Ꮿ is semisimple (as an abelian category) if and only if the dimension of Ꮿ 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.
doi:10.2140/pjm.2013.264.1 fatcat:pmyqfpd3bjeunhwwhxyl7sgeau