Automorphisms of free metabelian Lie algebras

Abstract: We show that every automorphism of a free metabelian Lie algebra $M_n$ of rank $n\geq 4$ over an arbitrary field $K$ is almost tame, that is, it is a product of so-called Chein automorphisms (or one-row transformations). Moreover, we show that the group of all automorphisms $\mathrm{Aut}(M_n)$ of $M_n$ of rank $n\geq 4$ over a field $K$ of characteristic $\neq 3$ is generated by all linear automorphisms, as well as one quadratic and one cubic automorphism. The same result holds for fields of any characteristic if $n\geq 5$. We also show that all Chein automorphisms of lower degree $\geq 4$ and all exponential automorphisms of lower degree $\geq 5$ are tame, which contradicts the results of \cite{BN,OE}.