Fixed Algebras of Residually Nilpotent Lie Algebras

Abstract

Let ${L_m}$ be the free Lie algebra of rank $m > 1$ over a field $K$, and let $J$ be an ideal of ${L_m}$ such that $J \subset L_m^{''}$ and the algebra ${L_m}/J$ is residually nilpotent. Let $G \ne \langle 1\rangle$ be a finite group of automorphisms of ${L_m}/J$ and the order of $G$ be invertible in $K$. We establish that the algebra of fixed points ${({L_m}/J)^G}$ is not finitely generated. The class of algebras under consideration contains the free Lie algebra over an arbitrary field and the relatively free algebras in nonnilpotent varieties of Lie algebras over infinite fields of characteristic different from $2$ and $3$.