MODULAR TRANSFORMATIONS FOR TENSOR CATEGORIES

For an abelian tenser category we investigate a Hopf algebra F in it, the ''algebra of functions'' or ''automorphisms of the identity functor''. We show the existence of the object of integrals for any Hopf algebra in a rigid abelian category. If some assumptions of finiteness and non-degeneracy are satisfied, the Hopf algebra F has an integral and there are morphisms S, T:F --> F, called modular transformations. They yield a representation of a modular group. The properties of S are similar to those of the Fourier transform.