Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Type $II$ quantum subgroups of $\mathfrak{sl}_N$. $I$: Symmetries of
  local modules   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
$\mathcal{C}(\mathfrak{sl}_{r+1},k)$.

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The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an