Computing the group of minimal non-degenerate extensions of a super-Tannakian category (Q2094613)

Assuming the basics of the theory of braided fusion categories [\textit{V. Drinfeld} et al., Sel. Math., New Ser. 16, No. 1, 1--119 (2010; Zbl 1201.18005); \textit{P. Etingof} et al., Tensor categories. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1365.18001)], this paper aims to describe the group of minimal extensions of a super-Tannakian fusion category \(\mathcal{E}\) and compute it in several concrete examples. The synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] gives preliminaries on symmetric fusion categories and their Picard groups (\S 2.1), the \(2\)-categorical group of minimal extensions of a symmetric category (\S 2.2) and central and braided graded extensions (\S 2.3). \item[\S 3] establishes a version of the Künneth formula \(Mext(\mathcal{R}ep(G)\boxtimes\mathcal{E})\), where \(G\) is a finite group and \(\mathcal{E}\) is a symmetric fusion category. It is established (Theorem 3.8), after [\textit{T. Lan} et al., Commun. Math. Phys. 351, No. 2, 709--739 (2017; Zbl 1361.81184)] that there is a group isomorphism \[ Mext(\mathcal{R}ep(G)\boxtimes\mathcal{E})\cong Mext(\mathcal{E})\times2-Fun(G,\boldsymbol{Pic}(\mathcal{E})) \] where \ is the group of monoidal \(2\)-functors from \(G\) to the \(2\)-categorical Picard group of \(\mathcal{E}\). \item[\S 4] analyzes the structure of the group \(Mext(\mathcal{E} )\) for a pointed symmetric category \(\mathcal{E}\), considering a filtration \[ Mext_{triv}(\mathcal{E})\subset Mext_{pt}(\mathcal{E} )\subset Mext_{int}(\mathcal{E})\subset Mext(\mathcal{E}) \] with the subgroups \(Mext_{triv}(\mathcal{E})\), \(Mext_{pt} (\mathcal{E})\) and \(Mext_{int}(\mathcal{E})\) of trivial, pointed and integral minimal extensions of \(\mathcal{E}\), and computing its composition factors (Theorem 4.18). This description of \ generalizes that of the third cohomology group \((A,\boldsymbol{k} ^{\times})\) of a finite abelian group \(A\) [\textit{A. Davydov} and \textit{D. Simmons}, J. Algebra 471, 149--175 (2017; Zbl 1368.18002); \textit{G. Mason} and \textit{S.-H. Ng}, Trans. Am. Math. Soc. 353, No. 9, 3465--3509 (2001; Zbl 0968.57030)]. A new feature is the appearance of cohomological obstructions from the theory of graded extensions [\textit{A. Davydov} and \textit{D. Nikshych}, Sel. Math., New Ser. 27, No. 4, Paper No. 65, 87 p. (2021; Zbl 1477.18016); \textit{P. Etingof} et al., Quantum Topol. 1, No. 3, 209--273 (2010; Zbl 1214.18007)]. \item[\S 5] applies the results to compute the group of minimal extensions of concrete examples of super-Tannakian categories, namely, \(\mathcal{R}ep(\mathbb{Z}_{2^{n}}^{f})\) and \(\mathcal{R}ep(\mathbb{Z} _{2}\times\mathbb{Z}_{2}^{f})\). \end{itemize}