Lie algebras admitting a metacyclic Frobenius group of automorphisms

TL;DR

This paper investigates the structure of Lie algebras with a Frobenius group of automorphisms, showing how fixed point subalgebras influence the existence of large nilpotent subalgebras with bounded properties.

Contribution

It establishes bounds on the nilpotency class and codimension of subalgebras in Lie algebras admitting a Frobenius automorphism group with cyclic kernel.

Findings

Existence of a nilpotent subalgebra of finite codimension bounded by parameters

Nilpotency class of subalgebra bounded by automorphism group size and fixed point nilpotency

Cyclicity of the kernel is essential for the main results

Abstract

Suppose that a Lie algebra $L$ admits a finite Frobenius group of automorphisms $FH$ with cyclic kernel $F$ and complement $H$ such that the characteristic of the ground field does not divide $|H|$. It is proved that if the subalgebra $C_L(F)$ of fixed points of the kernel has finite dimension $m$ and the subalgebra $C_L(H)$ of fixed points of the complement is nilpotent of class $c$, then $L$ has a nilpotent subalgebra of finite codimension bounded in terms of $m$, $c$, $|H|$, and $|F|$ whose nilpotency class is bounded in terms of only $|H|$ and $c$. Examples show that the condition of the kernel $F$ being cyclic is essential.