A Jordan approach to finitary Lie algebras
A Lie algebra L over a field F is said to be finitary
if it is isomorphic to a subalgebra of the Lie algebra of finite
rank linear transformations of a vector space over F. A
nonzero element a in L is said to be extremal if (ada)2 L =
F.a.
By using Baranov's classification, it is not difficult to verify
that any simple finitary Lie algebra over an algebraically closed
field of characteristic 0 is spanned by extremal elements. In
this note, we provide a classification-free proof of this result
by using Jordan theory instead of representation theory.
(This paper appeared in Quart. J. Math. 67 (2016), 565-571)
Antonio Fernández López < emalfer@uma.es >