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تسجيل مستخدم جديدOn the Frobenius functor for symmetric tensor categories in positive characteristic
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We develop a theory of Frobenius functors for symmetric tensor categories (STC) $mathcal{C}$ over a field $bf k$ of characteristic $p$, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor $F: mathcal{C}to mathcal{C}boxtimes {rm Ver}_p$, where ${rm Ver}_p$ is the Verlinde category (the semisimplification of ${rm Rep}_{bf k}(mathbb{Z}/p)$). This generalizes the usual Frobenius twist functor in modular representation theory and also one defined in arXiv:1503.01492, where it is used to show that if $mathcal{C}$ is finite and semisimple then it admits a fiber functor to ${rm Ver}_p$. The main new feature is that when $mathcal{C}$ is not semisimple, $F$ need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor $mathcal{C}to {rm Ver}_p$. We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of $F$, and use it to show that for categories with finitely many simple objects $F$ does not increase the Frobenius-Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which $F$ is exact, and define the canonical maximal Frobenius exact subcategory $mathcal{C}_{rm ex}$ inside any STC $mathcal{C}$ with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius-Perron dimension is preserved by $F$. We prove that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to ${rm Ver}_p$. We also show that a sufficiently large power of $F$ lands in $mathcal{C}_{rm ex}$. Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category and show that a STC with Chevalley property is (almost) Frobenius exact.
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We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $bf k$. If ${rm char}({bf k})=p>0$, we use this method to construct generalizations ${rm Ver}_{p^n}$, ${rm Ver}_
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A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine
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This is an expanded version of the notes by the second author of the lectures on symmetric tensor categories given by the first author at Ohio State University in March 2019 and later at ICRA-2020 in November 2020. We review some aspects of the curre
nt state of the theory of symmetric tensor categories and discuss their applications, including ones unavailable in the literature.
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