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