On certain decompositions of solvable Lie algebras (original) (raw)
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The literature contains two different classifications of solvable Lie algebras of dimensions up to and including 4. This paper is devoted to comparing the two classifications and translating each into the other. In particular, we exhibit an isomorphism between each solvable Lie algebra of one classification and the corresponding algebra of the second. The first classification is provided by de Graaf, and the second classification is from a recent book by Snobl and Winternitz.
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Journal of the London Mathematical Society, 1973
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.