Contributions to the classification of simple modular Lie algebras

Benkart, Georgia; Osborn, J. Marshall; Strade, Helmut

doi:10.1090/S0002-9947-1994-1129435-0

Contributions to the classification of simple modular Lie algebras
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by Georgia Benkart, J. Marshall Osborn and Helmut Strade PDF

Trans. Amer. Math. Soc. 341 (1994), 227-252 Request permission

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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