Minimal extensions of Tannakian categories in positive characteristic

TL;DR

This paper extends key theorems about Tannakian categories to positive characteristic, classifies finite non-degenerate braided tensor categories with Tannakian subcategories, and explores the structure of minimal extensions related to group cohomology.

Contribution

It generalizes existing theorems to positive characteristic and characterizes minimal extensions of Tannakian categories via third cohomology groups.

Findings

Classification of braided tensor categories containing Tannakian subcategories as twisted doubles

Identification of the group of minimal extensions with third cohomology group

Explicit computations of minimal extensions for specific group schemes

Abstract

We extend \cite[Theorem 4.5]{DGNO} and \cite[Theorem 4.22]{LKW} to positive characteristic (i.e., to the finite, not necessarily fusion, case). Namely, we prove that if $\D$ is a finite non-degenerate braided tensor category over an algebraically closed field $k$ of characteristic $p>0$, containing a Tannakian Lagrangian subcategory $\Rep(G)$, where $G$ is a finite $k$-group scheme, then $\D$ is braided tensor equivalent to $\Rep(D^{\omega}(G))$ for some $\omega\in H^3(G,\mathbb{G}_m)$, where $D^{\omega}(G)$ denotes the twisted double of $G$ \cite{G2}. We then prove that the group $\mathcal{M}_{{\rm ext}}(\Rep(G))$ of minimal extensions of $\Rep(G)$ is isomorphic to the group $H^3(G,\mathbb{G}_m)$. In particular, we use \cite{EG2,FP} to show that $\mathcal{M}_{\rm ext}(\Rep(\mu_p))=1$, $\mathcal{M}_{\rm ext}(\Rep(\alpha_p))$ is infinite, and if $\O(\Gamma)^*=u(\g)$ for a semisimple restricted $p$-Lie algebra $\g$, then $\mathcal{M}_{\rm ext}(\Rep(\Gamma))=1$ and $\mathcal{M}_{\rm ext}(\Rep(\Gamma\times \alpha_p))\cong \g^{*(1)}$.