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. © 2006 Elsevier Inc. All rights reserved.
CITATION STYLE
Towers, D. A., & Varea, V. R. (2007). Elementary Lie algebras and Lie A-algebras. Journal of Algebra, 312(2), 891–901. https://doi.org/10.1016/j.jalgebra.2006.11.034
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