Locally solvable finitary lie algebras

Abstract

A Lie subalgebra L of gl(V) is finitary if L consists of elements of finite rank. We show here that an irreducible finitary algebra with a non-zero abelian ideal is finite dimensional. Additional results require that for some positive integer d, the finitary Lie algebra L be d-bounded; i.e.L has a generating set consisting of elements of rank at most d. In particular, we show that if L is an irreducible, finitaryd-bounded Lie subalgebra of gl(V) and the locally solvable radical ls(L) is non-zero, then L is finite dimensional. If, in additionL is locally solvable, then L is solvable of derived length at most 64d ².