Toward a Socle for Lie Algebras.

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

Revista:Marruecos

Abstract: A notion of socle is introduced for 3-graded Lie algebras (over a ring of scalars $\Phi$ containing ${1\over 6}$) whose associated Jordan pairs are nondegenerate. This notion of socle does not depend on certain 3-gradings and this allows us to define a Jordan socle for non-necessarily 3-graded Lie algebras. The Jordan socle turns out to be a 3-graded ideal and is the sum of minimal 3-graded inner ideals each of which is a central extension of the TKK-algebra of a division Jordan pair. Nondegenerate Lie algebras having a large Jordan socle are essentially determined by TKK-algebras of simple Jordan pairs with minimal inner ideals and their derivation algebras, which are also 3-graded.