On symmetric fusion categories in positive characteristic

Ostrik, Victor

doi:10.1007/s00029-020-00567-5

Cited by 23 publications

(16 citation statements)

References 20 publications

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We prove that every finite symmetric integral tensor category 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 [O, Conjecture 1.3] in this case. Equivalently, we prove that there exists a unique finite supergroup scheme G over k and a grouplike element ǫ ∈ kG of order ≤ 2, whose action by conjugation on G coincides with the parity automorphism of G, such that C is symmetric tensor equivalent to Rep(G, ǫ).

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

Etingof

Gelaki

2019

International Mathematics Research Notices

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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.

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confidence: 72%

“…(1) The proof of Theorem 1.1 uses [O,Theorem 1.1] and [EOV,Theorem 8.1], and the method we use in the proof is motivated by Drinfeld's paper [D,Propositions 3.5,3.6].…”

Section: Introductionmentioning

confidence: 99%

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

Etingof

Gelaki

2019

International Mathematics Research Notices

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“…That is, the matrix [c( 1 (1) The proof of (3) ⇒ (1) in Proposition 2.2 comes from the proof of [15,Proposition 2.9]. From this proof one is able to see that d C = ± det[a ij ], where a ij = Tr(X i X j ) for i, j ∈ I.…”

Section: Numerical Invariantsmentioning

confidence: 99%

“…It is interesting to know when a fusion category over a field of positive characteristic is nondegenerate. Ostrik stated that a spherical fusion category C over a field k is nondegenerated, if the Grothendieck algebra Gr(C) ⊗ Z k is semisimple (see [15,Proposition 2.9]). It has been proved by Shimizu that a pivotal fusion category C over an algebraically closed field k is nondegenerate if and only if its Grothendieck algebra Gr(C) ⊗ Z k is semisimple (see [17,Theorem 6.5]).…”

Section: Introductionmentioning

confidence: 99%

The Casimir Number and the Determinant of a Fusion Category

Wang¹,

Liu²,

Li³

2020

Glasgow Math. J.

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Let $\mathcal{C}$ be a fusion category over an algebraically closed field $\mathbb{k}$ of arbitrary characteristic. Two numerical invariants of $\mathcal{C}$ , that is, the Casimir number and the determinant of $\mathcal{C}$ are considered in this paper. These two numbers are both positive integers and admit the property that the Grothendieck algebra $(\mathcal{C})\otimes_{\mathbb{Z}}K$ over any field K is semisimple if and only if any of these numbers is not zero in K. This shows that these two numbers have the same prime factors. If moreover $\mathcal{C}$ is pivotal, it gives a numerical criterion that $\mathcal{C}$ is nondegenerate if and only if any of these numbers is not zero in $\mathbb{k}$ . For the case that $\mathcal{C}$ is a spherical fusion category over the field $\mathbb{C}$ of complex numbers, these two numbers and the Frobenius–Schur exponent of $\mathcal{C}$ share the same prime factors. This may be thought of as another version of the Cauchy theorem for spherical fusion categories.

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“…The structure theory of tensor categories over fields of positive characteristic is in full development; see for instance [BE19, BEO20, Cou20, EG19, EO19, Ost20]. An important tool developed in [Cou20, EO19, Ost20] is the ‘Frobenius twist’ in arbitrary tensor categories. In [BE19] a family of tensor categories in characteristic 2 was constructed in which this functor is not exact.…”

Section: Introductionmentioning

confidence: 99%

Monoidal abelian envelopes

Coulembier¹

2021

Compositio Math.

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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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On symmetric fusion categories in positive characteristic

Cited by 23 publications

References 20 publications

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

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

The Casimir Number and the Determinant of a Fusion Category

Monoidal abelian envelopes

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