On automorphisms of certain free nilpotent-by-abelian Lie algebras

TL;DR

This paper investigates the automorphism groups of certain free nilpotent-by-abelian Lie algebras, showing that a specific subgroup generated by tame automorphisms and explicit automorphisms is dense in the full automorphism group.

Contribution

It introduces a new dense subgroup of automorphisms generated by tame and explicit automorphisms in free nilpotent-by-abelian Lie algebras.

Findings

The subgroup generated by tame and explicit automorphisms is dense in Aut(R_n).

Explicit automorphisms can approximate any automorphism in the formal power series topology.

The result applies to Lie algebras of rank n ≥ 4.

Abstract

For a positive integer $n$, with $n \geq 4$, let $R_{n}$ be a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank $n$. We show that the subgroup of Aut$(R_{n})$ generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of $R_{n}$ is dense in Aut$(R_{n})$ with respect to the formal power series topology.