Module categories, weak Hopf algebras and modular invariants

Abstract

We develop a theory of module categories over monoidal categories (this is a straightforward categorization of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and we classify module categories over the fusion category of sl̂(2) at a positive integer level where we meet once again the ADE classification pattern.