
Journal of Lie Theory 24 (2014), No. 3, 849864 Copyright Heldermann Verlag 2014 On Properties of Principal Elements of Frobenius Lie Algebras Andre Diatta CNRS  Institut Fresnel, Université AixMarseille, Campus Universitaire de Saint Jérôme, Av. Escadrille NormandieNiemen, 13397 Marseille, France andre.diatta@fresnel.fr Bakary Manga Département de Mathématiques et Informatique, Université Cheikh Anta Diop, Dakar, Senegal bakary.manga@ucad.edu.sn 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 YangBaxter 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 quasiFrobenius 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 [ Fulltextpdf (320 KB)] for subscribers only. 