ON THE ALGEBRA OF CONSTANTS OF FREE METABELIAN LIE ALGEBRAS

Authors : Andre DUSHIMIRIMANA

Pages : 185-192

Doi:10.33773/jum.1143787

View : 20 | Download : 14

Publication Date : 2022-07-31

Article Type : Research Paper

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 $K[X_n]^\delta=\big \{u\in K[X_n] \big\vert\ \deltainsert ignore into journalissuearticles values(u);=0\big \}$ 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 $\deltainsert ignore into journalissuearticles values(x_5);=x_4$, $\deltainsert ignore into journalissuearticles values(x_4);=0$, $\deltainsert ignore into journalissuearticles values(x_3);=x_2$, $\deltainsert ignore into journalissuearticles values(x_2);=x_1$ and $\deltainsert ignore into journalissuearticles values(x_1);=0$.
Keywords : Lie Algebras, Algebras of Constants, Weitzenb¨ock Derivations