# Clifford theory for graded fusion categories

### Abstract

We develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category $\mathcal{C}$ graded by a group $G$ as induced from module categories over fusion subcategories associated with the subgroups of $G$. We define invariant $\mathcal{C}_e$-module categories and extensions of $\mathcal{C}_e$ -module categories. The construction of module categories over $\mathcal{C}$ is reduced to determining invariant module categories for subgroups of $G$ and the indecomposable extensions of these module categories. We associate a $G$-crossed product fusion category to each $G$-invariant $\mathcal{C}_e$ -module category and give a criterion for a graded fusion category to be a group-theoretical fusion category. We give necessary and sufficient conditions for an indecomposable module category to be extendable.

Publication
Israel J. Math. 192 (2012), no. 2, 841–867.
##### César Galindo
###### Associate Professor of Mathematics

My research interests include representation theory, category theory and their applications to Mathematical-Physics.