Type II quantum subgroups of quantum ${\mathfrak{sl}}_N$. II: Classification

Abstract

In this paper, we study the indecomposable module categories over $\mathcal{C}({\mathfrak{sl}}_N,k)$, the category of integrable level-$k$ representations of affine Kac–Moody ${\mathfrak{sl}}_N$. Our first main result classifies these module categories in the case of generic $k$; i.e., $k$ is sufficiently large relative to $N$. As $\mathcal{C}({\mathfrak{sl}}_N,k)$ is a braided tensor category, there is a relative tensor product structure on its category of module categories. In the generic setting, we obtain a formula for the relative tensor product rules between the indecomposable module categories. Our second main result classifies the indecomposable module categories over $\mathcal{C}({\mathfrak{sl}}_N,k)$ for $N\le 7$, with no restrictions on $k$. In this non-generic setting, exceptional module categories are obtained. This work relies heavily on previous results by the two authors. In previous literature, module category classification results were known only for ${\mathfrak{sl}}_2$ and ${\mathfrak{sl}}_3$.