Abstract
We propose a conjectural extension in the positive characteristic case of well known Deligne’s theorem on the existence of super fiber functors. We prove our conjecture in the special case of semisimple categories with finitely many isomorphism classes of simple objects.
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Notes
We refer the reader to [18, Section 8] for the classification of semisimple group schemes in the category \(\text {sVec}\).
See [17].
It was shown by Shimizu [26, Theorem 6.5] that a pivotal fusion category \(\mathcal {C}\) is non-degenerate if and only if the ring \(K(\mathcal {C})\otimes \mathbf {k}\) is semisimple. Thus the assumption on the dimensions of irreducible representations in Proposition 2.9 can be dropped and the expectation in Remark 2.10 is correct in the pivotal case.
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Acknowledgements
It is my great pleasure to thank Pierre Deligne, Pavel Etingof, Michael Finkelberg, Shlomo Gelaki, Alexander Kleshchev, Dmitri Nikshych, Julia Pevtsova, Alexander Polishchuk, and Vadim Vologodsky for very useful conversations. I am also indebted to an anonymous reviewer for very useful comments and spotting some mistakes. This material is based on work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California in Spring 2020. I was partially supported by the HSE University Basic Research Program, Russian Academic Excellence Project ‘5-100’ and by the NSF grant DMS-1702251.
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To the memory of my parents, Rabina Tatiana Fedorovna and Rabin Vladimir Akimovich.
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