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## "Module categories, weak hopf algebras, and modular invariants", Viktor Ostrik, 2001

*Reviewed March 23, 2024*

*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.