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

Abstract

We define a notion of super-transitivity for ètale 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 ètale 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 ètale algebras in $\mathcal{C}({\mathfrak{sl}}_N,k)$, and includes all but 16 of the known non-pointed ètale algebra objects in these categories. These remaining 16 known examples have super-transitivities between 2 and 4.

34 pages with 8 page appendix