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A conjecture on Lie algebras admitting a regular automorphism of finite order. (English) Zbl 0582.17007
If a Lie algebra of characteristic \(p\geq 0\) admits a regular automorphism (i.e. having no non-zero fixed elements) of order \(n\geq 2\), then it is solvable and the length of its derived series is not greater than \(2^{n-1}\). For this result see V. A. Kreknin [Dokl. Akad. Nauk SSSR 150, 467-469 (1963; Zbl 0134.036)].
It has been conjectured by O. Kowalski that a better bound for the length is n-1, and this is verified for \(2\leq n\leq 7\) in the present paper. In fact it turns out that for \(n=5\) and \(n=7\), the bound can be improved to n-2.
Reviewer: E.W.Wallace
MSC:
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
17B30 Solvable, nilpotent (super)algebras
17B50 Modular Lie (super)algebras
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References:
[1] JACOBSON N.: Lie algebras. Interscience Publishers, New York-London, 1962. · Zbl 0121.27504
[2] KREKNIN V. A.: O razrešimosti algebr Li s reguľarnym avtomorfizmom konečnogo poriadka. DAN SSSR, 150, 3 1963, 467-469.
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