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- W2947515110 abstract "Abstract We prove that every finite symmetric integral tensor category $mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p>2$ admits a symmetric fiber functor to the category of supervector spaces. This proves Ostrik’s conjecture [25, Conjecture 1.3] in this case. Equivalently, we prove that there exists a unique finite supergroup scheme $mathcal{G}$ over $k$ and a grouplike element $epsilon in kmathcal{G}$ of order $le 2$, whose action by conjugation on $mathcal{G}$ coincides with the parity automorphism of $mathcal{G}$, such that $mathcal{C}$ is symmetric tensor equivalent to $textrm{Rep}(mathcal{G},epsilon )$. In particular, when $mathcal{C}$ is unipotent, the functor lands in $textrm{Vec}$, so $mathcal{C}$ is symmetric tensor equivalent to $textrm{Rep}(U)$ for a unique finite unipotent group scheme $U$ over $k$. We apply our result and the results of [17] to classify certain finite dimensional triangular Hopf algebras with the Chevalley property over $k$ (e.g., local), in group scheme-theoretical terms. Finally, we compute the Sweedler cohomology of restricted enveloping algebras over an algebraically closed field $k$ of characteristic $p>0$, classify associators for their duals, and study finite dimensional (not necessarily triangular) local quasi-Hopf algebras and finite (not necessarily symmetric) unipotent tensor categories over an algebraically closed field $k$ of characteristic $p>0$. The appendix by K. Coulembier and P. Etingof gives another proof of the above classification results using the recent paper [4], and more generally, shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic $ne 2$ is always a Serre subcategory." @default.
- W2947515110 created "2019-06-07" @default.
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- W2947515110 date "2019-05-28" @default.
- W2947515110 modified "2023-09-23" @default.
- W2947515110 title "Finite Symmetric Integral Tensor Categories with the Chevalley Property with an Appendix by Kevin Coulembier and Pavel Etingof" @default.
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- W2947515110 doi "https://doi.org/10.1093/imrn/rnz093" @default.
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