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

Abstract

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 $p^{m}d$ such that ${\mathcal{C}}^{\prime }\subseteq \text{sVec}$ are nilpotent and group-theoretical, where $p$ is a prime and $(p,d)=1$.

15 pages, final version

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• Minimal nondegenerate extensions