Modular transformations for tensor categories Original Research Article

For an abelian tensor 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.