The socle of a nondegenerate Lie algebra
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.
(This paper has appeared in
J. Algebra 319 (2008), no. 6, 2372--2394)
C. Draper Fontanals < cdf@uma.es >
A. Fernández López < emalfer@agt.cie.uma.es >
E. García González < esther.garcia@urjc.es >
Miguel Gómez Lozano < magomez@agt.cie.uma.es >