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A new characterization of semisimple Lie algebras. (English) Zbl 0891.17005

The author proves that a finite-dimensional quadratic Lie algebra over an algebraically closed field of characteristic 0 is semisimple if and only if its Casimir element is invertible.
Reviewer: G.Brown (Boulder)

MSC:

17B20 Simple, semisimple, reductive (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
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References:

[1] Saïd Benayadi, Une propriété nécessaire et suffisante pour qu’une algèbre de Lie sympathique quadratique soit semi-simple, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 11, 1155 – 1158 (French, with English and French summaries). · Zbl 0814.17007
[2] Saïd Benayadi, Structures de certaines algèbres de Lie quadratiques, Comm. Algebra 23 (1995), no. 10, 3867 – 3887 (French). · Zbl 0835.17004 · doi:10.1080/00927879508825437
[3] N. Bourbaki, Éléments de mathématique. Fasc. XXVI. Groupes et algèbres de Lie. Chapitre I: Algèbres de Lie, Seconde édition. Actualités Scientifiques et Industrielles, No. 1285, Hermann, Paris, 1971 (French). · Zbl 0213.04103
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