On braided simple extensions and braided non-semisimple near-group categories

We study simple extensions of pointed finite tensor categories, that is, tensor categories $\mathcal{C}$ admitting an abelian decomposition $\mathcal{C}\cong \mathcal{D}\oplus \mathcal{M}$ where $\mathcal{D}$ is a pointed tensor subcategory and $\mathcal{M}$ has a unique simple projective object. Such categories provide a natural generalization of near-group categories. Our results concern the braided case. We prove that every non-degenerate braided non-semisimple near-group category is a braided simple extension of $\mathrm{s}\mathrm{R}\mathrm{e}\mathrm{p}(W\oplus W^{\ast })$ with non-trivial braiding for which $\mathrm{s}\mathrm{R}\mathrm{e}\mathrm{p}(W)$ is Lagrangian. Moreover, any braided non-semisimple near-group category $\mathcal{C}$ arises canonically as an extension of such a category by $\mathrm{R}\mathrm{e}\mathrm{p}(G)$, where $G$ is the Picard group of a symmetric subcategory determined by the unique simple projective object of $\mathcal{C}$.