Automorphisms of Free Metabelian Lie Algebras, II

Abstract

We show that every automorphism of the free metabelian Lie algebra $M_n$ of rank $n\geq 4$ over an arbitrary field $K$ is almost tame; that is, it can be expressed as a product of Chein automorphisms (also called one-row transformations). Moreover, we prove that, for $n\geq 4$, the group $\mathrm{Aut}(M_n)$ of all automorphisms of the free metabelian Lie algebra $M_n$ over a field $K$ of characteristic not equal to $3$ is generated by its linear automorphisms, together with a single quadratic and a single cubic automorphism. The same result holds for fields of any characteristic if $n\geq 5$.