On extensions of the Jacobson-Morozov theorem to even characteristic
David I. Stewart, Adam R. Thomas
TL;DR
This work tackles the problem of extending the Jacobson–Morozov framework to characteristic $2$, asking when nilpotent elements in $ rak g= ext{Lie}(G)$ can be embedded into subalgebras isomorphic to the 3‑dimensional simple algebra $ rak s$, or into copies of $ ext{Lie}( ext{SL}_2)$ or $ ext{Lie}( ext{PGL}_2)$. It develops explicit criteria based on standard forms of nilpotent orbits for both classical and exceptional groups, using root-vector realizations, Borel–de Siebenthal inputs, and Bala–Carter data, and computes automiser dimensions to understand orbitwise rigidity and $G$-isogeny effects. The main results classify all nilpotent orbits that admit $ rak s$, $ rak{pgl}_2$, or $ rak{sl}_2$-overalgebras, and show that the automiser dimension is typically $1$ (with sporadic exceptions in $p=2$) while exceptional types require extensive Magma computations. For exceptional groups, the authors provide an explicit computational framework that yields representatives and verifies the embedding properties, consolidating a complete picture of nilpotent extensions in even characteristic and enabling further representation-theoretic analysis in this delicate setting.
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.