On the Nowicki Conjecture for the free Lie algebra of rank 2

TL;DR

This paper proves a conjecture related to the structure of constants in free Lie algebras under specific derivations, introducing pseudodeterminants and characterizing Hall monomials as constants.

Contribution

It extends Nowicki's conjecture to free Lie algebras, introduces pseudodeterminants, and characterizes constants, providing explicit generators for low degrees.

Findings

Constants are characterized as pseudodeterminants.

Complete list of generators for degrees less than 7.

Hall monomials are closely related to pseudodeterminants.

Abstract

Let K[X_n]=K[x_1,\ldots,x_n] be the polynomial algebra in n variables over a field K of characteristic zero. A locally nilpotent linear derivation \delta of K[X_n] is called Weitzenb\"ock due to his well known result from 1932 stating that the algebra \text{\rm ker}(\delta)=K[X_n]^{\delta} of constants of $\delta$ is finitely generated. The explicit form of a generating set of $K[X_n,Y_n]^{\delta}$ was conjectured by Nowicki in 1994 in the case \delta was such that \delta(y_{i})=x_{i}$, $\delta(x_i)=0, i=1,\ldots,n. Nowicki's conjecture turned out to be true and, recently, has been applied to several relatively free associative algebras. In this paper, we consider the free Lie algebra \mathcal{L}(x,y) of rank 2 generated by x and y over K and we assume the Weitzenb\"ock derivation \delta sending y to x, and x to zero. We introduce the idea of pseudodeterminants and we present a characterization of Hall monomials that are constants showing they are not so far from being pseudodeterminants. We also give a complete list of generators of the constants of degree less than 7 which are, of course, pseudodeterminants.