Finite Symmetric Integral Tensor Categories with the Chevalley Property with an Appendix by Kevin Coulembier and Pavel Etingof. (28th May 2019) - Etingof, Pavel; Gelaki, Shlomo
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 $ε \in k\mathcal{G}$ 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 $Rep(\mathcal{G}, ε )$ . In particular, when $\mathcal{C}$ is unipotent, the functor lands in $Vec$, so $\mathcal{C}$ is symmetric tensor equivalent to $Rep(U)$ 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.