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Journal of Lie Theory 24 (2014), No. 3, 849--864
Copyright Heldermann Verlag 2014

On Properties of Principal Elements of Frobenius Lie Algebras

Andre Diatta
CNRS -- Institut Fresnel, Université Aix-Marseille, Campus Universitaire de Saint Jérôme, Av. Escadrille Normandie-Niemen, 13397 Marseille, France

Bakary Manga
Département de Mathématiques et Informatique, Université Cheikh Anta Diop, Dakar, Senegal

We investigate the properties of principal elements of Frobenius Lie algebras. We prove that any Lie algebra with a left symmetric algebra structure can be embedded as a subalgebra of some sl(m,K) where K are the real R or the complex numbers 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.

Keywords: Frobenius Lie algebra, affine Lie algebra, left symmetric algebra, invariant affine connection, classical Yang Baxter equation, affine motion, symplectic Lie algebra, Kähler algebra, seaweed Lie algebra.

MSC: 17B05, 17B08, 22E60

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