scholarly journals Koszul duality and the PBW theorem in symmetric tensor categories in positive characteristic

2018 ◽  
Vol 327 ◽  
pp. 128-160 ◽  
Author(s):  
Pavel Etingof
Keyword(s):  
Positive Characteristic ◽  
Symmetric Tensor ◽  
Koszul Duality ◽  
Tensor Categories ◽  
Pbw Theorem

scholarly journals On the Frobenius functor for symmetric tensor categories in positive characteristic

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Pavel Etingof ◽  
Victor Ostrik
Keyword(s):  
Positive Characteristic ◽  
Symmetric Tensor ◽  
Tensor Categories ◽  
Large Power ◽  
Exact Category ◽  
Fusion Categories ◽  
Moderate Growth ◽  
Monoidal Functor ◽  
New Feature ◽  
The One

AbstractWe develop a theory of Frobenius functors for symmetric tensor categories (STC) {\mathcal{C}} over a field {\boldsymbol{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 {\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k}}(\mathbb{Z}/p)}); a similar construction of the underlying additive functor appeared independently in [K. Coulembier, Tannakian categories in positive characteristic, preprint 2019]. This generalizes the usual Frobenius twist functor in modular representation theory and also the one defined in [V. Ostrik, On symmetric fusion categories in positive characteristic, Selecta Math. (N.S.) 26 2020, 3, Paper No. 36], 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. One of our main results is that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to {{\rm Ver}_{p}}. This is the strongest currently available characteristic p version of Deligne’s theorem (stating that a STC of moderate growth in characteristic zero is the representation category of a supergroup). 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 (namely, one having a fiber functor into the category of representations of the triangular Hopf algebra {\boldsymbol{k}[d]/d^{2}} with d primitive and R-matrix {R=1\otimes 1+d\otimes d}), and show that a STC with Chevalley property is (almost) Frobenius exact. Finally, as a by-product, we resolve Question 2.15 of [P. Etingof and S. Gelaki, Exact sequences of tensor categories with respect to a module category, Adv. Math. 308 2017, 1187–1208].

scholarly journals Symmetric tensor categories in characteristic 2

2019 ◽  
Vol 351 ◽  
pp. 967-999 ◽  
Author(s):  
Dave Benson ◽  
Pavel Etingof
Keyword(s):  
Symmetric Tensor ◽  
Tensor Categories ◽  
Characteristic 2

scholarly journals $p$-adic dimensions in symmetric tensor categories in characteristic $p$

2018 ◽  
Vol 9 (1) ◽  
pp. 119-140 ◽  
Author(s):  
Pavel Etingof ◽  
Nate Harman ◽  
Viktor Ostrik
Keyword(s):  
Symmetric Tensor ◽  
Characteristic P ◽  
Tensor Categories

scholarly journals Finite Symmetric Integral Tensor Categories with the Chevalley Property with an Appendix by Kevin Coulembier and Pavel Etingof

2019 ◽  
Author(s):  
Pavel Etingof ◽  
Shlomo Gelaki
Keyword(s):  
Hopf Algebras ◽  
Tensor Category ◽  
Group Scheme ◽  
Symmetric Tensor ◽  
Closed Field ◽  
Characteristic P ◽  
Tensor Categories ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Chevalley Property

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 $\epsilon \in k\mathcal{G}$ 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 $\textrm{Rep}(\mathcal{G},\epsilon )$. 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.

scholarly journals Finite symmetric tensor categories with the Chevalley property in characteristic 2

2020 ◽  
pp. 2140010
Author(s):  
Pavel Etingof ◽  
Shlomo Gelaki
Keyword(s):  
Tensor Category ◽  
Group Scheme ◽  
Symmetric Tensor ◽  
Function Algebra ◽  
Equivalence Classes ◽  
Closed Field ◽  
Arbitrary Field ◽  
Tensor Categories ◽  
Characteristic 2 ◽  
Chevalley Property

We prove an analog of Deligne’s theorem for finite symmetric tensor categories [Formula: see text] with the Chevalley property over an algebraically closed field [Formula: see text] of characteristic [Formula: see text]. Namely, we prove that every such category [Formula: see text] admits a symmetric fiber functor to the symmetric tensor category [Formula: see text] of representations of the triangular Hopf algebra [Formula: see text]. Equivalently, we prove that there exists a unique finite group scheme [Formula: see text] in [Formula: see text] such that [Formula: see text] is symmetric tensor equivalent to [Formula: see text]. Finally, we compute the group [Formula: see text] of equivalence classes of twists for the group algebra [Formula: see text] of a finite abelian [Formula: see text]-group [Formula: see text] over an arbitrary field [Formula: see text] of characteristic [Formula: see text], and the Sweedler cohomology groups [Formula: see text], [Formula: see text], of the function algebra [Formula: see text] of [Formula: see text].

scholarly journals Modular Koszul duality

2013 ◽  
Vol 150 (2) ◽  
pp. 273-332 ◽  
Author(s):  
Simon Riche ◽  
Wolfgang Soergel ◽  
Geordie Williamson
Keyword(s):  
Positive Characteristic ◽  
Reductive Group ◽  
Flag Variety ◽  
Koszul Duality ◽  
Technical Result ◽  
Dg Algebra

AbstractWe prove an analogue of Koszul duality for category$ \mathcal{O} $of a reductive group$G$in positive characteristic$\ell $larger than$1$plus the number of roots of$G$. However, there are no Koszul rings, and we do not prove an analogue of the Kazhdan–Lusztig conjectures in this context. The main technical result is the formality of the dg-algebra of extensions of parity sheaves on the flag variety if the characteristic of the coefficients is at least the number of roots of$G$plus$2$.

scholarly journals Monoidal abelian envelopes

2021 ◽  
Vol 157 (7) ◽  
pp. 1584-1609
Author(s):  
Kevin Coulembier
Keyword(s):  
Existence Theorem ◽  
Positive Characteristic ◽  
Monoidal Categories ◽  
Tilting Modules ◽  
Tensor Categories

We prove a constructive existence theorem for abelian envelopes of non-abelian monoidal categories. This establishes a new tool for the construction of tensor categories. As an example we obtain new proofs for the existence of several universal tensor categories as conjectured by Deligne. Another example constructs interesting tensor categories in positive characteristic via tilting modules for ${\rm SL}_2$ .

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