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On the existence of ad-nilpotent elements. (English) Zbl 0359.17008
This note studies conditions which are sufficient to guarantee that a derivation on a finite dimensional Lie algebra over an algebraically closed field is nilpotent. It is shown that if a derivation is nilpotent on the span of the set of eigenvectors of \(\mathrm{ad}_x\) for some \(x\) in the Lie algebra, then it is nilpotent. This result is used to prove the following theorem: Let \(L\) be a finite dimensional Lie algebra over an algebraically closed field. Let \(X\subseteq L\) be a non-empty subset such that for each \(x\in L\) all eigenvectors of \(\mathrm{ad}_x\) lie in \(X\). Then \(\mathrm{ad}_y\) is nilpotent for some \(y\in X\). A consequence of the theorem when \(X=L\setminus\{0\}\) is an elementary proof of the fact that every finite dimensional Lie algebra over an algebraically closed field possesses an ad-nilpotent element.
Reviewer: George M. Benkart

MSC:
17B05 Structure theory for Lie algebras and superalgebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
17B50 Modular Lie (super)algebras
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[1] G. M. Benkart, On inner ideals and ad-nilpotent elements, Trans. Amer. Math. Soc. (to appear). · Zbl 0373.17003
[2] G. M. Benkart, I. M. Isaacs, and J. M. Osborn, Lie algebras with self-centralizing ad-nilpotent elements, J. Algebra 57 (1979), no. 2, 279 – 309. · Zbl 0402.17013 · doi:10.1016/0021-8693(79)90225-4 · doi.org
[3] Helmut Strade, Nonclassical simple Lie algebras and strong degeneration, Arch. Math. (Basel) 24 (1973), 482 – 485. · Zbl 0275.17004 · doi:10.1007/BF01228244 · doi.org
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