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 Weitzenböck derivation. It is well known from the classical theorem of Weitzenböck that the algebra of constants $K[X_m{]}^{\delta }$ of a Weitzenböck derivation $\delta$ is finitely generated. Assume that $\delta$ acts on the polynomial algebra $K[X_{2d}]$ in $2d$ variables as follows: $\delta (x_{2i})=x_{2i-1}$, $\delta (x_{2i-1})=0$, $i=1,\dots ,d$. The Nowicki conjecture states that the algebra $K[X_{2d}{]}^{\delta }$ is generated by $x_1,x_3.\dots ,x_{2d-1}$, and $x_{2i-1}{x}_{2j}-x_{2i}{x}_{2j-1}$, $1\le i<j\le d$. 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.

14 pages