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Elementary Lie Algebras and Lie A-Algebras.

Research output: Contribution to journalJournal article


Journal publication date15/06/2007
JournalJournal of Algebra
Number of pages11
Original languageEnglish


A finite-dimensional Lie algebra L over a field F is called elementary if each of its subalgebras has trivial Frattini ideal; it is an A-algebra if every nilpotent subalgebra is abelian. The present paper is primarily concerned with the classification of elementary Lie algebras. In particular, we provide a complete list in the case when F is algebraically closed and of characteristic different from 2,3, reduce the classification over fields of characteristic 0 to the description of elementary semisimple Lie algebras, and identify the latter in the case when F is the real field. Additionally it is shown that over fields of characteristic 0 every elementary Lie algebra is almost algebraic; in fact, if L has no non-zero semisimple ideals, then it is elementary if and only if it is an almost algebraic A-algebra.

Bibliographic note

The final, definitive version of this article has been published in the Journal, Journal of Algebra 312 (2), 2007, © ELSEVIER.