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Contributions to the classification of simple modular Lie algebras


Authors: Georgia Benkart, J. Marshall Osborn and Helmut Strade
Journal: Trans. Amer. Math. Soc. 341 (1994), 227-252
MSC: Primary 17B50
DOI: https://doi.org/10.1090/S0002-9947-1994-1129435-0
MathSciNet review: 1129435
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Abstract: We develop results directed towards the problem of classifying the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic $ p > 7$. A $ 1$-section of such a Lie algebra relative to a torus $ T$ of maximal absolute toral rank possesses a unique subalgebra maximal with respect to having a composition series with factors which are abelian or classical simple. In this paper we show that the sum $ Q$ of those compositionally classical subalgebras is a subalgebra. This extends to the general case a crucial step in the classification by Block and Wilson of the restricted simple Lie algebras. We derive properties of the filtration which can be constructed using $ Q$ and obtain structural information about the $ 1$-sections and $ 2$-sections of $ Q$ relative to $ T$. We further classify all those algebras in which $ Q$ is solvable.


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DOI: https://doi.org/10.1090/S0002-9947-1994-1129435-0
Article copyright: © Copyright 1994 American Mathematical Society

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