Automorphisms of free metabelian Lie algebras, I

Abstract

We show that all Chein automorphisms (or one-row transformations) of lower degree $\geq 4$ of a free metabelian Lie algebra $M_n$ of rank $n\geq 4$ over an arbitrary field $K$ of characteristic $\neq 3$ are tame. We then show that all exponential automorphisms of $M_n$ of lower degree $\geq 5$ are also tame under the same conditions. The same results hold for fields of any characteristic when $n\geq 5$. These results contradict some long-standing results in the area. We also prove that a large class of automorphisms of $M_n$ of rank $n\geq 4$ that move only two variables are almost tame, that is, they can be expressed as a product of Chein automorphisms.