ON THE ALGEBRA OF CONSTANTS OF FREE METABELIAN LIE ALGEBRAS

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