(26). The socle of a nondegenerate Lie algebras.

Autores: C. Draper Fountanals, A. Fernández López, E. García y M. Gómez Lozano.

Revista:Journal of Algebra 319 (2008) 2372–2394    (JCR: 89 de 214. Factor de impacto: 0.668)

Abstract: In this paper we give a definition of socle for nondegenerate Lie algebras which is only based on minimal inner ideals. The socle turns out to be an ideal of the whole algebra, and it is sum of simple components. All the minimal inner ideals contained in a simple component are conjugated under elementary automorphisms, which allows us to associate a division Jordan algebra to any of the simple component containing abelian minimal inner ideals. All Classical Lie algebras coincide with their socles, while relevant examples of infinite dimensional simple Lie algebras with socle can be found within the class of finitary Lie algebras. The notion of socle is compatible with the associative and Jordan definitions of socle, and satisfies the descending chain condition on principal inner ideals. Furthermore, we give a structure theory for nondegenerate Lie algebras containing abelian minimal inner ideals, and show that a simple Lie algebra over an algebraically closed field of characteristic zero is finitary if and only if it is nondegenerate and contains nonzero reduced elements. i.e., contains one-dimensional inner ideals.