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N SOLVABLE 3 -LIEALGEBRASWITHAMAXIMALHYPO-NILPOTENTIDEAL ELA

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ELA

SOLVABLE 3-LIE ALGEBRAS WITH A MAXIMAL HYPO-NILPOTENT IDEAL N

∗

RUI-PU BAI†, CAI-HONG SHEN†, AND YAO-ZHONG ZHANG‡

Abstract. This paper obtains all solvable 3-Lie algebras with them-dimensional filiform 3-Lie algebraN (m≥5) as a maximal hypo-nilpotent ideal, and proves that them-dimensional filiform 3-Lie algebra N can’t be as the nilradical of solvable non-nilpotent 3-Lie algebras. By means of one dimensional extension of Lie algebras to the 3-Lie algebras, we get some classes of solvable Lie algebras directly.

Key words. 3-Lie algebra, hypo-nilpotent ideal, filiform n-Lie algebra.

AMS subject classifications.17B05, 17D99.

∗Received by the editors on June 10, 2009. Accepted for publication on July 31, 2010. Handling Editors: Roger A. Horn and Fuzhen Zhang.

†College of Mathematics and Computer Science, Key Lab in Machine Learning and Computational Intelligence, Hebei University, Baoding (071002), China ([email protected], [email protected]).

Partially supported by NSF(10871192) of China, NSF(A2007000138) of Hebei Province, China.

‡School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Aus- tralia ([email protected]). Supported by the Australian Research Council.

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 21, pp. 43-62, October 2010

http://math.technion.ac.il/iic/ela

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