The Nowicki Conjecture for relatively free algebras

Abstract

A linear locally nilpotent derivation of the polynomial algebra $K[X_m]$ in $m$ variables over a field $K$ of characteristic 0 is called a Weitzenb\"ock derivation. It is well known from the classical theorem of Weitzenb\"ock that the algebra of constants $K[X_{m}]^{\delta}$ of a Weitzenb\"ock derivation $\delta$ is finitely generated. Assume that $\delta$ acts on the polynomial algebra $K[X_{2d}]$ in $2d$ variables as follows: $\delta(x_{2i})=x_{2i-1}$, $\delta(x_{2i-1})=0$, $i=1,\ldots,d$. The Nowicki conjecture states that the algebra $K[X_{2d}]^{\delta}$ is generated by $x_1,x_3.\ldots,x_{2d-1}$, and $x_{2i-1}x_{2j}-x_{2i}x_{2j-1}$, $1\leq i<j\leq d$. The conjecture was proved by several authors based on different techniques. We apply the same idea to two relatively free algebras of rank $2d$. We give the infinite set of generators of the algebra of constants in the the free metabelian associative algebras $F_{2d}(\mathfrak A)$, and finite set of generators in the free algebra $F_{2d}(\mathcal G)$ in the variety determined by the identities of the infinite dimensional Grassmann algebra.