On central automorphisms of free metabelian Lie algebras

Abstract

Let $F_m$ be the free metabelian Lie algebra of rank $m$ over a field $K$ of characteristic 0. An automorphism $\varphi$ of $F_m$ is called central if $\varphi$ commutes with every inner automorphism of $F_m$. Such automorphisms form the centralizer $\text{\rm C}(\text{\rm Inn}(F_m))$ of inner automorphism group $\text{\rm Inn}(F_m)$ of $F_m$ in $\text{\rm Aut}(F_m)$. We provide an elementary proof to show that $\text{\rm C}(\text{\rm Inn}(F_m))=\text{\rm Inn}(F_m)$.