On a class of finitary Lie algebras characterized through derivations

Abstract

Let L L be an infinite-dimensional simple Lie algebra over a field of characteristic 0 0 . Then there exist a derivation d d on L L and n ≥ 2 n\ge 2 such that d n d^n is a nonzero finite rank map if and only if L L is finitary and contains a nonzero finite-dimensional abelian inner ideal. This is a partial statement of our main theorem. As auxiliary results needed for the proof we establish some properties of derivations in general nonassociative algebras.