On the Algebra of Constants of Free Metabelian Lie Algebras

Abstract

Let $K$ be a field of characteristic zero, $X_n=\{x_1,\dots,x_n\}$ be a set of variables, $K[X_n]$ be the polynomial algebra and $F_n$ be the free metabelian Lie algebra of rank $n$ generated by $X_n$ over the base field $K$. Well known result of Weitzenb\"ock states that $K[X_n]^\delta=\big \{u\in K[X_n] \big\vert\ \delta(u)=0\big \}$ is finitely generated as an algebra, where $\delta$ is a locally nilpotent linear derivation of $K[X_n]$. Extending this ideal to the non commutative algebras, recently the algebra $F_n^\delta$ of constants in the free metabelian Lie algebras have been investigated. According to the findings, $F_n^\delta$ is not finitely generated as a Lie algebra; whereas, $F_n^\delta \cap F_n^\prime$ is finitely generated $K[X_n]^\delta$-module and a list of generators for $n\le 4$ was given. In this work, in filling the gap in the list of small $n'$s we work in $F_5$ and give a list of generators of $F_5^\delta$ where $\delta(x_5)=x_4$, $\delta(x_4)=0$, $\delta(x_3)=x_2$, $\delta(x_2)=x_1$ and $\delta(x_1)=0$.