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

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.

Exact factorizations and extensions of finite tensor categories

“…Equivalently, Coh(G, ω) is the tensor category Rep (O(G), ω) of finite dimensional representations of the quasi-Hopf algebra (O(G), ω). (See [EG2,G3] for examples of nontrivial 3-cocycles on non constant finite group schemes. )…”

Minimal extensions of Tannakian categories in positive characteristic

“…Finally, in Section 6 we give some examples. In particular, we use the results of [EG2] to show that the group M ext (Rep(µ p )) is trivial (see Example 6.1) and the group M ext (Rep(α p )) is infinite (see Example 6.2), and use [FP] to conclude that if O(Γ) * = u(g) for a semisimple restricted p-Lie algebra g, then M ext (Rep(Γ)) is the trivial group and M ext (Rep(Γ × α p )) ∼ = g * (1) (see Examples 6.3 and 6.4).…”

“…Let G be a finite abelian p-group. Then by[EG2,], H 3 (G D , G m ) = 1, thus by Corollary 1.3, M ext (Rep(G D )) = 1 is the trivial group. For example, if G = Z/pZ then G D = µ p is the Frobenius kernel of the multiplicative group G m (see, e.g., [G, Section 2.2]), so M ext (Rep(µ p )) = 1.Example 6.2.…”

“…Now by[EG2, Theorem 5.4], we haveH 3 (Γ, G m ) = H 3 (Γ, Ga ). Thus by Corollary 1.3, we have a group isomorphism (6.1) M ext (Rep(Γ)) = H 3 (Γ, G a ).…”

“…For example, if G = Z/pZ then G D = µ p is the Frobenius kernel of the multiplicative group G m (see, e.g., [G, Section 2.2]), so M ext (Rep(µ p )) = 1.Example 6.2. Let α p,r denote the r-th Frobenius kernel of the additive group G a (see, e.g.,[EG2, Section 2.6]). Let G := n i=1 α p,r i .…”