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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Simple Lie algebras of characteristic $ p$ with dependent roots


Authors: Georgia Benkart and J. Marshall Osborn
Journal: Trans. Amer. Math. Soc. 318 (1990), 783-807
MSC: Primary 17B20; Secondary 17B50
MathSciNet review: 955488
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Abstract: We investigate finite dimensional simple Lie algebras over an algebraically closed field $ {\mathbf{F}}$ of characteristic $ p \geqslant 7$ having a Cartan subalgebra $ H$ whose roots are dependent over $ {\mathbf{F}}$. We show that $ H$ must be one-dimensional or for some root $ \alpha $ relative to $ H$ there is a $ 1$-section $ {L^{(\alpha )}}$ such that the core of $ {L^{(\alpha )}}$ is a simple Lie algebra of Cartan type $ H{(2:\underline m :\Phi )^{(2)}}$ or $ W(1:\underline n )$ for some $ n > 1$. The results we obtain have applications to studying the local behavior of simple Lie algebras and to classifying simple Lie algebras which have a Cartan subalgebra of dimension less than $ p - 2$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0955488-2
PII: S 0002-9947(1990)0955488-2
Article copyright: © Copyright 1990 American Mathematical Society