Lie type algebras with an automorphism of finite order

Abstract

An algebra L over a field F, in which product is denoted by [,], is called a Lie type algebra if for all elements a,b,c∈L there exist α,β∈F (depending on a,b,c) such that α≠0 and [[a,b],c]=α[a,[b,c]]+β[[a,c],b]. Examples of Lie type algebras include 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.