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

We study simple extensions of pointed finite tensor categories, that is, tensor categories C admitting an abelian decomposition C D ⊕ M where D is a pointed tensor subcategory and 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 sRep(W⊕ W^*) with non-trivial braiding for which sRep(W) is Lagrangian. Moreover, any braided non-semisimple near-group category C arises canonically as an extension of such a category by Rep(G), where G is the Picard group of a symmetric subcategory determined by the unique simple projective object of C.