Suppose a group acts on a tensor category over a field . Then we have the tensor category of -invariant objects in , and the semi-direct product tensor category . We show that if is finite and is semi-simple, there exists a one-to-one correspondence between categories with action of and categories with action of .
Daisuke TAMBARA. "Invariants and semi-direct products for finite group actions on tensor categories." J. Math. Soc. Japan 53 (2) 429 - 456, April, 2001. https://doi.org/10.2969/jmsj/05320429