On the minimal extension and structure of weakly group-theoretical braided fusion categories

We show that any slightly degenerate weakly group-theoretical fusion category admits a minimal non-degenerate extension. Let $d$ be a positive square-free integer, given a weakly group-theoretical non-degenerate fusion category $\mathcal{C}$, assume that $\text{FPdim}(\mathcal{C})=nd$ and $(n,d)=1$. If $(\text{FPdim}(X)^2,d)=1$ for all simple objects $X$ of $\mathcal{C}$, then we show that $\mathcal{C}$ contains a non-degenerate fusion subcategory $\mathcal{C}(\mathbb{Z}_d,q)$. In particular, we obtain that integral fusion categories of FP-dimensions $p^md$ such that $\mathcal{C}'\subseteq \text{sVec}$ are nilpotent and group-theoretical, where $p$ is a prime and $(p,d)=1$.