Auto-equivalences of the modular tensor categories of type $A$, $B$, $C$ and $G$

We compute the monoidal and braided auto-equivalences of the modular tensor categories $\mathcal{C}({\mathfrak{s}\mathfrak{l}}_{r+1},k)$, $\mathcal{C}({\mathfrak{s}\mathfrak{o}}_{2r+1},k)$, $\mathcal{C}({\mathfrak{s}\mathfrak{p}}_{2r},k)$, and $\mathcal{C}({\mathfrak{g}}_2,k)$. Along with the expected simple current auto-equivalences, we show the existence of the charge conjugation auto-equivalence of $\mathcal{C}({\mathfrak{s}\mathfrak{l}}_{r+1},k)$, and exceptional auto-equivalences of $\mathcal{C}({\mathfrak{s}\mathfrak{o}}_{2r+1},2)$, $\mathcal{C}({\mathfrak{s}\mathfrak{p}}_{2r},r)$, $\mathcal{C}({\mathfrak{g}}_2,4)$. We end the paper with a section discussing potential applications of these computations.