On Vafa’s Theorem for Tensor Categories

In this note we prove two main results. 1. In a rigid braided finite tensor category over C (not necessarily semisimple), some power of the Casimir element and some even power of the braiding is unipotent. 2. In a (semisimple) modular category, the twists are roots of unity dividing the algebraic integer D, where D is the global dimension of the category (the sum of squares of dimensions of simple objects). Both results generalize Vafa’s theorem ([V, AM, BaKi]), saying that in a modular category twists are roots of unity, and square of the braiding has finite order. In the case when the category admits a fiber functor, i.e. is a category of representations of a finite dimensional Hopf algebra, these results can be found in [EG1]. In fact, the method of proof of 1 and 2 is parallel to the proof of Theorems 4.3 and 4.8 in [EG1], modulo two new ingredients: categorical determinants and Frobenius-Perron dimensions. We note that statement 1 in the semisimple case was proved in [Da1], also using determinants. At the end of the note, we discuss the notion of the quasi-exponent of a finite rigid tensor category, which is motivated by results 1 and 2 and the paper [EG3]. Acknowledgements. I thank A.Davydov for very useful discussions which inspired this note, and S.Gelaki for collaboration in [EG1], where the methods used here were introduced. I am grateful to MPIM (Bonn) for hospitality. This research was partially supported by the NSF grant DMS-9988796, and was done in part for the Clay Mathematics Institute.