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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Classification of the maximal subalgebras of exceptional Lie algebras over fields of good characteristic
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by Alexander Premet and David I. Stewart HTML | PDF
J. Amer. Math. Soc. 32 (2019), 965-1008 Request permission


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
  • MR Author ID: 190461
  • Email:
  • David I. Stewart
  • Affiliation: University of Newcastle, Newcastle upon Tyne, NE1 7RU, United Kingdom
  • MR Author ID: 884527
  • Email:
  • Received by editor(s): December 4, 2017
  • Received by editor(s) in revised form: March 18, 2019
  • Published electronically: July 19, 2019
  • © Copyright 2019 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 32 (2019), 965-1008
  • MSC (2010): Primary 17B45
  • DOI:
  • MathSciNet review: 4013738