Lie type algebras with an automorphism of finite order

TL;DR

This paper investigates Lie type algebras with finite order automorphisms, proving they contain a soluble ideal of finite codimension with bounded derived length, extending understanding of their structure under symmetry constraints.

Contribution

It establishes that Lie type algebras with finite order automorphisms and finite-dimensional fixed points have a soluble ideal with bounded properties, generalizing previous results to a broader class.

Findings

Existence of a soluble ideal with finite codimension

Bounded derived length of the ideal

Results apply to various algebraic structures like Lie and Leibniz algebras

Abstract

An algebra $L$ over a field $\Bbb F$, in which product is denoted by $[\,,\,]$, is said to be \textit{ Lie type algebra} if for all elements $a,b,c\in L$ there exist $\alpha, \beta\in \Bbb F$ such that $\alpha\neq 0$ and $[[a,b],c]=\alpha [a,[b,c]]+\beta[[a,c],b]$. Examples of Lie type algebras are associative algebras, Lie algebras, Leibniz algebras, etc. It is proved that if a Lie type algebra $L$ admits an automorphism of finite order $n$ with finite-dimensional fixed-point subalgebra of dimension $m$, then $L$ has a soluble ideal of finite codimension bounded in terms of $n$ and $m$ and of derived length bounded in terms of $n$.