Fixed-point-free automorphisms of solvable Lie algebras

TL;DR

This paper explores conditions under which finite-dimensional Lie algebras admit fixed-point-free automorphisms, establishing criteria for various classes and linking automorphisms to algebraic properties.

Contribution

It provides new necessary and sufficient conditions for fixed-point-free automorphisms in complex almost abelian and filiform Lie algebras, extending prior results.

Findings

Lie algebra with fixed-point-free automorphism is solvable

Such algebras are strongly unimodular

Existence criteria for complex almost abelian and filiform Lie algebras

Abstract

In this paper, we investigate the existence of fixed-point-free automorphisms for finite-dimensional Lie algebras. By a result of Jacobson, a Lie algebra admitting a fixed-point-free automorphism is solvable. We prove that such a Lie algebra must be even strongly unimodular. We find a necessary and sufficient criterion such that a complex almost abelian Lie algebra admits a fixed-point-free automorphism. For complex filiform Lie algebras we show that the existence of a fixed-point-free automorphism is equivalent to not being characteristically nilpotent.