Invariants and semi-direct products for finite group actions on tensor categories

Abstract

Suppose a group G acts on a tensor category \mathscr{C} over a field k. Then we have the tensor category \mathscr{C}^G of G-invariant objects in \mathscr{C}, and the semi-direct product tensor category \mathscr{C}[G]. We show that if G is finite and k[G] is semi-simple, there exists a one-to-one correspondence between categories with action of \mathscr{C}^G and categories with action of \mathscr{C}[G].