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

TL;DR

This paper classifies indecomposable module categories over the affine $ ext{sl}_N$ category at various levels, providing a comprehensive understanding for generic and specific cases up to N=7, including exceptional categories.

Contribution

It extends the classification of module categories from $ ext{sl}_2$ and $ ext{sl}_3$ to higher N, offering formulas for tensor products and identifying exceptional cases.

Findings

Classifies module categories for generic levels $k$ and $N extless=7$.

Provides formulas for relative tensor product rules.

Identifies exceptional module categories in non-generic settings.

Abstract

In this paper we study the indecomposable module categories over $\mathcal{C}(\mathfrak{sl}_N, k)$, the category of integrable level-$k$ respresentations 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 $N\leq 7$, 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 $\mathfrak{sl}_3$.