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

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\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} .