Contributions to the Classification of Simple Modular Lie Algebras

Abstract

Abstract.We develop results directed towards the problem of classifying the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic p > 1. 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.

Tori in the Melikian Algebra

“…If α ∈ * p β then gr −3 G α = G −3 and G α ∩ G 0 operates in G −3 as sl G −3 by Theorem 4.1 (2). It follows gr 0 I ⊃ sl G −3 under identification G 0 ∼ = gl G −3 .…”

“…By Lemma 5.2(5) all roots α such that T α ⊂ G −1 are either proper or improper simultaneously. If T β ⊂ G −1 then β is improper Witt by Lemma 5.2 (2). Suppose that there exists a proper root α.…”

“…In this case T β has a zero weight on gr i G because gr Suppose T is of type (−3 −2). If T α ⊂ G −2 then α is improper by Lemma 5.2 (2). Consider now the set of roots = α T α ⊂ G −2 .…”