Orbits in free algebras of rank two

Abstract

Let Fbe an algebra which is free in a class of algebras. In this paper we consider the following problem:If an endomorphism ψ of Fpreserves the orbit (Aut F)(ω) of a nontrivial element ω ∈ F, does this imply that ψ is an automorphism? We give positive solutions of this problem when Fis generated by two elements and is the free Lie algebra, free metabelian Lie algebra or free metabelian associative algebra, and ω is a certain kind of element.