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arXiv · 1512.05614

A modular analogue of Morozov's theorem on maximal subalgebras of simple Lie algebras

Alexander Premet
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Abstract

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p>0$ and suppose that $p$ is a very good prime for $G$. We prove that any maximal Lie subalgebra $M$ of $\mathfrak{g} = {\rm Lie}(G)$ with ${\rm rad}(M) \ne 0$ has the form $M = {\rm Lie}(P)$ for some maximal parabolic subgroup $P$ of $G$. We show that the assumption on $p$ is necessary by providing a counterexample for groups type ${\rm E}_8$ over fields of characteristic $5$. Our arguments rely on the main results and methods of the classification theory of finite dimensional simple Lie algebras over fields prime characteristic.

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Alexander Premet. 2017-03-08. A modular analogue of Morozov's theorem on maximal subalgebras of simple Lie algebras. https://arxiv.org/abs/1512.05614

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