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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Existence of ad-nilpotent elements and simple Lie algebras with subalgebras of codimension one


Author: V. R. Varea
Journal: Proc. Amer. Math. Soc. 104 (1988), 363-368
MSC: Primary 17B40; Secondary 17B50
MathSciNet review: 962799
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Abstract: For a perfect field $ F$ of arbitrary characteristic, the following statements are proved to be equivalent: (1) Any Lie algebra over $ F$ contains an ad-nilpotent element. (2) There are no simple Lie algebras over $ F$ having only abelian subalgebras. They are used to guarantee the existence of an ad-nilpotent element in every Lie algebra over a perfect field of type $ ({C_1})$ of arbitrary characteristic (in particular, over any finite field). Furthermore, we give a sufficient condition to insure the existence of ad-nilpotent elements in a Lie algebra over any perfect field. As a consequence of this result we obtain an easy proof of the fact that the Zassenhaus algebras and $ {\text{sl}}(2,F)$ are the only simple Lie algebras which have subalgebras of codimension 1, whenever the ground field $ F$ is perfect with $ {\text{char}}(F) \ne 2$. All Lie algebras considered are finite dimensional.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0962799-X
PII: S 0002-9939(1988)0962799-X
Keywords: Ad-nilpotent element, modular Lie algebras, Zassenhaus algebra
Article copyright: © Copyright 1988 American Mathematical Society