On subalgebras of simple Lie algebras of characteristic \(p>0\). (English) Zbl 0555.17003

Weisfeiler, Boris

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

MSC:

17B50	Modular Lie (super)algebras
17B70	Graded Lie (super)algebras

Keywords:

graded Lie algebras; prime characteristic; filtrations; maximal subalgebra; simple Lie algebras