Citation: Ostrik, Victor. "Module categories, weak Hopf algebras and modular invariants." Transformation groups 8 (2003): 177-206.
Web: https://arxiv.org/abs/math/0111139
Tags: Mathematical, Defects/boundaries
This paper establishes the basic theory of module categories over monoidal categories. As the author repeatedly stresses, nothing in this paper is surprising to the experts. The point is that lots of people care about these ideas, so it is good to have a paper that does the basics properly. This includes giving a lot of new proofs/reconstructions, and re-casting the ADE classification of sl(2) quantum group modules (which was originally understood by Ocneanu in the setting of operator algebras) in the language of fusion categories.
The main theorem gives a structure theorem for modules over (a more general context which includes) fusion categories. It states that every module category can be decategorified. That is, every semisimple indecomposable module category is module-equivalent to the category of modules over a semisimple indecomposable algebras. This allows us to translate questions about module categories into questions about algebras. For instance, this theorem can be used to show that module categories over Rep(G) are all of the form Rep(H,omega), where H is a subgroup and omega is a U(1)-valued cocycle on H.
As per usual, an interpretation of the results in terms of Hopf algebras is given.