Congruence Subgroups and Generalized Frobenius-schur Indicators

We define generalized Frobenius-Schur indicators for objects in a linear pivotal category C. An equivariant indicator of an object is defined as a functional on the Grothendieck algebra of the quantum double Z(C) of C using the values of the generalized Frobenius-Schur indicators. In a spherical fusion category C with Frobenius-Schur exponent N , we prove that the set of all equivariant indicators admits a natural action of the modular group, and the kernel of the canonical modular representation of Z(C) is a congruence subgroup of level N . Moreover, if C is modular, then the kernel of the projective modular representation of C is also a congruence subgroup of level N , and every modular representation of C has a finite image.