Lie type algebras with an automorphism of finite order

Abstract

An algebra $L$ over a field $\mathbb{F}$, 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 $\alpha ,\beta \in \mathbb{F}$ such that $\alpha \ne 0$ 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 $n$.