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Existence of ad-nilpotent elements and simple Lie algebras with subalgebras of codimension one. (English) Zbl 0723.17016
Let \({\mathcal L}\) be a finite dimensional Lie algebra over a field of prime characteristic. Let \({\mathcal D}\) be a nonzero nilpotent derivation of \({\mathcal L}\) and \(\delta\in Ker {\mathcal D}\). It is proved that ad \(\delta\) is nilpotent provided ad \(\delta| Ker {\mathcal D}\) is nilpotent. This result allows to generalize slightly some results of A. A. Premet established earlier for restricted Lie algebras to arbitrary finite dimensional Lie algebras over perfect fields.
The author shows that the following statements are equivalent: (1) Any Lie algebra over a perfect field \({\mathcal F}\) contains an ad-nilpotent element. (2) There are no simple Lie algebras over \({\mathcal F}\) all of whose subalgebras are abelian. This implies that any Lie algebra over a perfect field of type \((C_ 1)\) contains an ad-nilpotent element.
The proofs are based on Premet’s theorem stating that any two toral Cartan subalgebras of a finite dimensional restricted Lie algebra over an algebraically closed field of characteristic \(p>0\) have the same (minimal possible) dimension [see A. A. Premet, Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk 1986, No.1, 9-14 (1986; Zbl 0597.17007)]. As a consequence the author gives an easy proof of the fact that Zassenhaus algebras and \({\mathfrak sl}(2)\) are the only simple Lie algebras with a subalgebra of codimension 1 provided the ground field is perfect and of characteristic \(\neq 2\).

MSC:
17B50 Modular Lie (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
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