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On the classification of solvable and complemented Lie algebras. (Chinese. English summary) Zbl 0782.17008

A Lie algebra \(M\) is said to be complemented if the lattice of subalgebras of \(M\) is complemented. E. L. Stitzinger has shown that if \(L\) is a complemented solvable Lie algebra, then \(L=A_ 1\oplus \dots\oplus A_ \ell \dot+C\) where the \(A_ i\) are 1-dimensional ideals and \(C\) a commutative subalgebra of \(L\). The author first describes \(L\) as a Lie algebra defined by generators and relations. Then she associates to every \(L\) a matrix \(I(L)\) shows that \(L\) and \(L'\) are isomorphic if and only if \(I(L)\) and \(I(L')\) are “equivalent” in a certain sense. The author finally classifies the complemented solvable Lie algebras by working out the equivalence relation of matrices in detail. The Lie algebra \(L\) is simply the holomorph of the commutative Lie algebra \(C\) and a direct sum of its 1-dimensional modules. The structure of \(L\) is clearly completely determined by the \(\ell\) weights of \(C\) on the \(A_ i\). In the reviewer’s opinion, the approach of the article under review is unnecessarily technical and complicated.

MSC:

17B30 Solvable, nilpotent (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
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