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


17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
17B30 Solvable, nilpotent (super)algebras
17B50 Modular Lie (super)algebras


Zbl 0134.036
Full Text: EuDML


[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.