Weisfeiler, Boris On subalgebras of simple Lie algebras of characteristic \(p>0\). (English) Zbl 0555.17003 Trans. Am. Math. Soc. 286, 471-503 (1984). This paper studies subalgebras of a simple finite-dimensional Lie algebra \(L\) over an algebraically closed field of characteristic \(p\geq 7\), especially subalgebras associated with filtrations. The first main result implies that if a maximal subalgebra \(L_ 0\) of \(L\) has solvable quotients of dimension \(\geq 2\), then every nilpotent ideal of \(L_0\) acts nilpotently on \(L\). It is shown that if \(L\) contains a solvable maximal subalgebra, then \(L\) is of type \(A_1\) or \(W_1\). In the final part of the article it is shown that certain \(Z\)-graded finite- dimensional simple Lie algebras either are classical or the difference between the number of nonzero positive and negative homogeneous components is large. Reviewer: Gordon Brown Cited in 3 ReviewsCited in 15 Documents MSC: 17B50 Modular Lie (super)algebras 17B70 Graded Lie (super)algebras Keywords:graded Lie algebras; prime characteristic; filtrations; maximal subalgebra; simple Lie algebras PDF BibTeX XML Cite \textit{B. Weisfeiler}, Trans. Am. Math. Soc. 286, 471--503 (1984; Zbl 0555.17003) Full Text: DOI