Lie algebras acting semitransitively

Abstract

A set S of linear operators on a vector space acts semitransitively if, given nonzero vectors x,y, there exists an operator a∈S such that either ax=y or ay=x. We show that for a Lie algebra g acting on a finite-dimensional complex vector space X this is equivalent to the existence of a (necessarily unique) maximal chain 0=Y0<Y1<⋯<Yn=X of g-invariant subspaces such that the orbit gx, x∈X, coincides with the smallest Yi containing x. In particular, a Lie algebra acting irreducibly and semitransitively actually acts transitively. We also show that every Lie algebra acting semitransitively contains a solvable subalgebra with the same property.