3-Graded Lie Algebras with Jordan Finiteness Conditions#

Abstract

A notion of socle is introduced for 3-graded Lie algebras (over a ring of scalars Φ containing ) whose associated Jordan pairs are non-degenerate. The 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. Non-degenerate 3-graded Lie algebras having a large socle are essentially determined by TKK-algebras of simple Jordan pairs with minimal inner ideals and their derivation algebras, which are also 3-graded. Classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space provide a source of examples of infinite dimensional strongly prime 3-graded Lie algebras with non-zero socle. Other examples can be found within the class of finitary simple Lie algebras