The Nowicki Conjecture for free metabelian Lie algebras

Let $K[X_d]=K[x_1,\ldots,x_d]$ be the polynomial algebra in $d$ variables over a field $K$ of characteristic 0. The classical theorem of Weitzenb\"ock from 1932 states that for linear locally nilpotent derivations $\delta$ (known as Weitzenb\"ock derivations) the algebra of constants $K[X_{d}]^{\delta}$ is finitely generated. When the Weitzenb\"ock derivation $\delta$ acts on the polynomial algebra $K[X_d,Y_d]$ in $2d$ variables by $\delta(y_i)=x_i$, $\delta(x_i)=0$, $i=1,\ldots,d$, Nowicki conjectured that $K[X_d,Y_d]^{\delta}$ is generated by $X_d$ and $x_iy_j-y_ix_j$ for all $1\leq i<j\leq d$. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenb\"ock derivations of the free $d$-generated metabelian Lie algebra $F_d$, with few trivial exceptions, the algebra $F_d^{\delta}$ is not finitely generated. However, the vector subspace $(F_d')^{\delta}$ of the commutator ideal $F_d'$ of $F_d$ is finitely generated as a $K[X_d]^{\delta}$-module. In this paper we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the $K[X_d,Y_d]^{\delta}$-module $(F_{2d}')^{\delta}$.