Automorphisms of free metabelian Lie algebras, I

TL;DR

This paper proves that certain automorphisms of free metabelian Lie algebras are tame under specific conditions, challenging previous assumptions, and introduces new classifications for automorphisms based on their degree and variable movement.

Contribution

It establishes tameness of Chein and exponential automorphisms of free metabelian Lie algebras for degrees above certain thresholds, contradicting earlier results, and characterizes automorphisms moving only two variables as almost tame.

Findings

Chein automorphisms of degree ≥ 4 are tame

Exponential automorphisms of degree ≥ 5 are tame

Automorphisms moving only two variables are almost tame

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.