Inner Ideals of Finitary Lie Algebras
Inner ideals of infinite dimensional finitary simple Lie
algebras over a field of characteristic zero are described in geometric
terms. We also study when these inner ideals are principal or minimal,
and characterize those elements which are von Neumann regular. As a
consequence we prove that any finitary central simple Lie algebra over a
field of characteristic zero satisfies the descending chain condition on
principal inner ideals. We also characterize when these algebras are
Artinian,
proving in particular that a finitary simple Lie algebra over an
algebraically closed field of characteristic zero is Artinian if and only
if it is finite dimensional. Because it is useful for our approach, we
provide a characterization of the trace of a finite rank operator on a
vector space over a division algebra which is intrinsic in the sense that
it avoids embeddings into finite matrices.
(This paper has appeared in J. Lie Theory 16 (2006), no. 1, 97-114)
Antonio Fernández López < emalfer@agt.cie.uma.es >
Esther García < egarciag@mat.ucm.es >
Miguel Gómez Lozano < magomez@agt.cie.uma.es >