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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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