Centers of graded fusion categories

Understand

Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading.

• The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center Z_D(C).
• We use this result to obtain a criterion for C to be group-theoretical and apply it to Tambara-Yamagami fusion categories.
• We also find several new series of modular categories by analyzing the centers of Tambara-Yamagami categories.