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We compute the monoidal and braided auto-equivalences of the modular tensorcategories $\mathcal{C}(\mathfrak{sl}_{r+1},k)$,$\mathcal{C}(\mathfrak{so}_{2r+1},k)$, $\mathcal{C}(\mathfrak{sp}_{2r},k)$, and$\mathcal{C}(\mathfrak{g}_{2},k)$. Along with the expected simple currentauto-equivalences, we show the existence of the charge conjugationauto-equivalence of $\mathcal{C}(\mathfrak{sl}_{r+1},k)$, and exceptionalauto-equivalences of $\mathcal{C}(\mathfrak{so}_{2r+1},2)$,$\mathcal{C}(\mathfrak{sp}_{2r},r)$, $\mathcal{C}(\mathfrak{g}_{2},4)$. We endthe paper with a section discussing potential applications of thesecomputations.

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