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

Abstract

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