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Classification of the maximal subalgebras of exceptional Lie algebras over fields of good characteristic


Authors: Alexander Premet and David I. Stewart
Journal: J. Amer. Math. Soc. 32 (2019), 965-1008
MSC (2010): Primary 17B45
DOI: https://doi.org/10.1090/jams/926
Published electronically: July 19, 2019
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Abstract: Let $ G$ be an exceptional simple algebraic group over an algebraically closed field $ k$ and suppose that $ p={\operatorname {char}}(k)$ is a good prime for $ G$. In this paper we classify the maximal Lie subalgebras $ \mathfrak{m}$ of the Lie algebra $ \mathfrak{g}=\operatorname {Lie}(G)$. Specifically, we show that either $ \mathfrak{m}=\operatorname {Lie}(M)$ for some maximal connected subgroup $ M$ of $ G$, or $ \mathfrak{m}$ is a maximal Witt subalgebra of $ \mathfrak{g}$, or $ \mathfrak{m}$ is a maximal exotic semidirect product. The conjugacy classes of maximal connected subgroups of $ G$ are known thanks to the work of Seitz, Testerman, and Liebeck-Seitz. All maximal Witt subalgebras of $ \mathfrak{g}$ are $ G$-conjugate and they occur when $ G$ is not of type $ {\mathrm {E}}_6$ and $ p-1$ coincides with the Coxeter number of $ G$. We show that there are two conjugacy classes of maximal exotic semidirect products in $ \mathfrak{g}$, one in characteristic $ 5$ and one in characteristic $ 7$, and both occur when $ G$ is a group of type $ {\mathrm {E}}_7$.


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Additional Information

Alexander Premet
Affiliation: School of Mathematics, The University of Manchester, Oxford Road, M13 9PL, United Kingdom
Email: alexander.premet@manchester.ac.uk

David I. Stewart
Affiliation: University of Newcastle, Newcastle upon Tyne, NE1 7RU, United Kingdom
Email: david.stewart@ncl.ac.uk

DOI: https://doi.org/10.1090/jams/926
Received by editor(s): December 4, 2017
Received by editor(s) in revised form: March 18, 2019
Published electronically: July 19, 2019
Article copyright: © Copyright 2019 American Mathematical Society