Lie algebras with Frobenius dihedral groups of automorphisms

TL;DR

This paper investigates Lie algebras with specific Frobenius dihedral automorphism groups, establishing bounds on their derived length under certain fixed-point conditions.

Contribution

It provides a new bound on the derived length of Lie algebras admitting Frobenius dihedral automorphisms with particular fixed-point properties.

Findings

Derived length of such Lie algebras is bounded by a constant.

Fixed-point subalgebra of the kernel is trivial.

Fixed-point subalgebra of the complement is metabelian.

Abstract

Suppose that a Lie algebra $L$ admits a finite Frobenius group of automorphisms $FH$ with cyclic kernel $F$ and complement $H$ of order 2, such that the fixed-point subalgebra of $F$ is trivial and the fixed-point subalgebra of $H$ is metabelian. Then the derived length of $L$ is bounded by a constant.