Classification of 1-super-transitive quantum subgroups in type A

We define a notion of super-transitivity for \`etale algebra objects $A \in \mathcal{C}(\mathfrak{sl}_N, k)$. This definition is a direct analogue of the notion of super-transitivity for subfactors, and measures at what depth the first ``new stuff'' appears in the category of $A$-modules internal to $\mathcal{C}(\mathfrak{sl}_N, k)$. Our main theorem gives a classification of all 1-super-transitive \`etale algebra objects in $\mathcal{C}(\mathfrak{sl}_N, k)$ running over all $N,k \in \mathbb{N}$. Our classification captures all known infinite families of non-pointed \`etale algebras in $\mathcal{C}(\mathfrak{sl}_N, k)$, and includes all but 16 of the known non-pointed \`etale algebra objects in these categories. These remaining 16 known examples have super-transitivities between 2 and 4.