  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^{5/2},
where D is the global dimension of the category (the sum of squares of
dimensions of simple objects). Both results generalize Vafa's theorem, saying
that in a modular category twists are roots of unity, and square of the
braiding has finite order. We also discuss the notion of the quasi-exponent of
a finite rigid tensor category, which is motivated by results 1 and 2 and the
paper math/0109196 of S.Gelaki and the author.
