On extensions of the Jacobson-Morozov theorem to even characteristic

TL;DR

This paper explores extensions of the Jacobson-Morozov theorem for simple algebraic groups over fields of characteristic 2, classifying certain nilpotent elements and analyzing their associated Lie algebra structures and automorphism dimensions.

Contribution

It provides a classification of nilpotent elements with specific Lie algebra overgroups in characteristic 2 and computes related automorphism dimensions, extending classical theorems to even characteristic.

Findings

Classified nilpotent elements with 3-dimensional Lie overgroups in characteristic 2.

Calculated the dimension of Lie automorphism groups for all nilpotent orbits.

Showed that these automorphism dimensions are sensitive to isogeny in even characteristic.

Abstract

Let G be a simple algebraic group over an algebraically closed field k of characteristic 2. We consider analogues of the Jacobson-Morozov theorem in this setting. More precisely, we classify those nilpotent elements with a simple 3-dimensional Lie overalgebra in $\mathfrak{g} := \text{Lie}(G)$ and also those with overalgebras isomorphic to the algebras $\text{Lie}(\text{SL}_2)$ and $\text{Lie}(\text{PGL}_2)$. This leads us to calculate the dimension of Lie automiser $\mathfrak{n}_\mathfrak{g}(k\cdot e)/\mathfrak{c}_\mathfrak{g}(e)$ for all nilpotent orbits; in even characteristic this quantity is very sensitive to isogeny.