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On Properties of Principal Elements of Frobenius Lie Algebras

We investigate the properties of principal elements of Frobenius Lie algebras. We prove that any Lie algebra with a left symmetric algebra struc- ture can be embedded as a subalgebra of some sl(m, K) for K = R or C. Hence, the work of Belavin and Drinfeld on solutions of the Classical Yang-Baxter Equation on simple Lie algebras, applied to the particular case of sl(m, K) alone, paves the way to the complete classification of Frobenius and more generally quasi-Frobenius Lie algebras. We prove that, if a Frobenius Lie algebra has the property that every derivation is an inner derivation, then every principal element is semisimple. As an important case, we prove that in the Lie algebra of the group of affine motions of the Euclidean space of finite dimension, every derivation is inner. We also bring examples of Frobenius Lie algebras that are subalgebras of sl(m, K) , but nevertheless have nonsemisimple principal elements as well as some with semisimple principal elements having nonrational eigenvalues, where K = R or C.

Auteur(s) : André Diatta, Bakary Manga
Pages : 849--864
Année de publication : 2014
Revue : Journal of Lie Theory
N° de volume : 24 (3)
Type : Article
Statut Editorial : publié
Mise en ligne par : MANGA Bakary