The Nowicki Conjecture for relatively free algebras

TL;DR

This paper extends the Nowicki conjecture, originally about polynomial algebras, to relatively free algebras, providing explicit generators for their algebras of constants under specific derivations.

Contribution

It applies techniques from the Nowicki conjecture to find generators of constants in free metabelian and Grassmann algebra varieties.

Findings

Infinite generators for constants in free metabelian algebras.

Finite generators for constants in free Grassmann algebra varieties.

Extension of the Nowicki conjecture to new algebraic contexts.

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.