On extensions of the Jacobson–Morozov theorem to even characteristic

IF 1.2 2区数学 Q1 MATHEMATICS Journal of the London Mathematical Society-Second Series Pub Date : 2024-10-21 DOI:10.1112/jlms.70007

David I. Stewart, Adam R. Thomas

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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 $g : = Lie (G)$ and also those with overalgebras isomorphic to the algebras $Lie ({SL}_{2})$ and $Lie ({PGL}_{2})$ . This leads us to calculate the dimension of the Lie automiser $n_{g} (k \cdot e) / c_{g} (e)$ for all nilpotent orbits; in even characteristic this quantity is very sensitive to isogeny.

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关于雅各布森-莫罗佐夫定理向偶数特征的扩展

让 G $G$ 是特征为 2 的代数闭域 k $\mathbb {k}$ 上的一个简单代数群。我们考虑雅各布森-莫罗佐夫定理在这种情况下的相似性。更准确地说，我们将那些在 g : = Lie ( G ) $\mathfrak {g}:=\operatorname{Lie}(G)$中具有简单三维 Lie 上代数的无幂元素以及那些具有与 Lie ( SL 2 ) $\operatorname{Lie}(\mathrm{SL}_2)$ 和 Lie ( PGL 2 ) $\operatorname{Lie}(\mathrm{PGL}_2)$ 同构的上代数的无幂元素进行了分类。这样我们就可以计算 Lie 自动机的维度 n g ( k - e ) / c g ( e ) $\mathfrak {n}_\mathfrak {g}(\mathbb {k}\cdot e)/\mathfrak {c}_\mathfrak {g}(e)$ 适用于所有零势轨道；在偶数特征中，这个量对同源性非常敏感。

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来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.

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