Type $II$ quantum subgroups of $\mathfrak{sl}_N$. $I$: Symmetries of local modules

TL;DR

This paper classifies the symmetries of local modules in categories related to quantum groups, identifying all but four exceptional cases, and develops skein theoretic tools applicable to orbifold categories.

Contribution

It provides a comprehensive classification of braided auto-equivalences for known type I quantum subgroups, introduces skein theoretic descriptions of orbifold quantum subgroups, and uncovers new connections involving quadratic categories.

Findings

Most symmetries are non-exceptional, with four specific orbifold cases as exceptions.

Developed skein theoretic descriptions for orbifold quantum subgroups.

Established a connection between quadratic categories and exceptional auto-equivalences.

Abstract

This paper is the first of a pair that aims to classify a large number of the type $II$ quantum subgroups of the categories $\mathcal{C}(\mathfrak{sl}_{r+1},k)$. In this work we classify the braided auto-equivalences of the categories of local modules for all known type $I$ quantum subgroups of $\mathcal{C}(\mathfrak{sl}_{r+1},k)$. We find that the symmetries are all non-exceptional except for four cases (up to level-rank duality). These exceptional cases are the orbifolds $\mathcal{C}( \mathfrak{sl}_{2},16)_{\operatorname{Rep}(\mathbb{Z}_2)}$, $\mathcal{C}( \mathfrak{sl}_{3},9)_{\operatorname{Rep}(\mathbb{Z}_3)}$, $\mathcal{C}( \mathfrak{sl}_{4},8)_{\operatorname{Rep}(\mathbb{Z}_4)}$, and $\mathcal{C}( \mathfrak{sl}_{5},5)_{\operatorname{Rep}(\mathbb{Z}_5)}$. We develop several technical tools in this work. We give a skein theoretic description of the orbifold quantum subgroups of $\mathcal{C}(\mathfrak{sl}_{r+1},k)$. Our methods here are general, and the techniques developed will generalise to give skein theory for any orbifold of a braided tensor category. We also give a formulation of orthogonal level-rank duality in the type $D$-$D$ case, which is used to construct one of the exceptionals. Finally we uncover an unexpected connection between quadratic categories and exceptional braided auto-equivalences of the orbifolds. We use this connection to construct two of the four exceptionals. In the sequel to this paper we will use the classified braided auto-equivalences to construct the corresponding type $II$ quantum subgroups of the categories $\mathcal{C}(\mathfrak{sl}_{r+1},k)$. When paired with Gannon's type $I$ classification for $r\leq 6$, this will complete the type $II$ classification for these same ranks. This paper includes an appendix by Terry Gannon, which provides useful results on the dimensions of objects in the categories $\mathcal{C}(\mathfrak{sl}_{r+1},k)$.