This paper is the first of a pair that aims to classify a large number of the type
I
I
II
quantum subgroups of the categories
C
(
s
l
r
+
1
,
k
)
\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
I
quantum subgroups of
C
(
s
l
r
+
1
,
k
)
\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
C
(
s
l
2
,
16
)
Rep
(
Z
2
)
0
\mathcal {C}(\mathfrak {sl}_{2}, 16)^0_{\operatorname {Rep}(\mathbb {Z}_{2})}
,
C
(
s
l
3
,
9
)
Rep
(
Z
3
)
0
\mathcal {C}(\mathfrak {sl}_{3}, 9)^0_{\operatorname {Rep}(\mathbb {Z}_{3})}
,
C
(
s
l
4
,
8
)
Rep
(
Z
4
)
0
\mathcal {C}(\mathfrak {sl}_{4}, 8)^0_{\operatorname {Rep}(\mathbb {Z}_{4})}
, and
C
(
s
l
5
,
5
)
Rep
(
Z
5
)
0
\mathcal {C}(\mathfrak {sl}_{5}, 5)^0_{\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
C
(
s
l
r
+
1
,
k
)
\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
-
D
D
case, which is used to construct one of the exceptionals. 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
I
I
II
quantum subgroups of the categories
C
(
s
l
r
+
1
,
k
)
\mathcal {C}(\mathfrak {sl}_{r+1}, k)
. This will essentially finish the type
I
I
II
classification for
s
l
n
\mathfrak {sl}_n
modulo type
I
I
classification. When paired with Gannon’s type
I
I
classification for
r
≤
6
r\leq 6
, our results will complete the type
I
I
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
C
(
s
l
r
+
1
,
k
)
\mathcal {C}(\mathfrak {sl}_{r+1}, k)
.