Module categories, weak Hopf algebras and modular invariants

TL;DR

The paper develops a categorical Morita framework for module categories over semisimple monoidal categories, showing that every indecomposable semisimple module category is equivalent to modules over an internal semisimple algebra in the ambient category. It connects module categories to weak Hopf algebras, via a constructive correspondence that realizes a given monoidal category as Rep of a weak Hopf algebra, and it interprets dualities and centers in this setting. Applying the theory to Rep(G) yields a classification of module categories in terms of subgroup data and 2-cocycles, while the analysis of the fusion category at level l of affine sl2 unveils an ADE-type classification of indecomposable module categories, with explicit constructions and a nonexistence result for tadpole diagrams. The work ties together boundary conformal field theory, subfactor theory, and dynamical twists, highlighting how modular invariants, alpha-induction, and centers emerge from a unified categorical perspective and linking these structures to well-known ADE classifications.

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.