Module categories, weak Hopf algebras and modular invariants

TL;DR

This paper develops a categorical framework for module categories over monoidal categories, showing their equivalence to weak Hopf algebra representations and classifying modules over certain fusion categories, revealing ADE patterns.

Contribution

It generalizes module theory to monoidal categories, proves equivalence with weak Hopf algebra representations, and classifies modules over specific fusion categories.

Findings

Semisimple monoidal categories are equivalent to weak Hopf algebra representations.

Classification of module categories over () at positive levels follows ADE patterns.

Established a categorical framework extending classical module theory.

Abstract

We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification 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 classify module categories over the fusion category of $\hat{sl}(2)$ at a positive integer level where we meet once again ADE classification pattern.