On the Nowicki Conjecture for the free Lie algebra of rank 2
Lucio Centrone, Sehmus Findik, Manuela da Silva Souza
TL;DR
This work studies the algebra of constants for a Weitzenböck-type derivation on the free Lie algebra of rank $2$, focusing on $L(x,y)$ with $\delta(y)=x$ and $\delta(x)=0$. It introduces pseudodeterminants as determinant-like invariants built from Hall monomials and analyzes which Hall monomials remain fixed under $\delta$, connecting these constants to determinant-like expressions. The authors provide both a direct, degree-based proof and an invariant-theory argument showing that all constants of degree $\le 7$ lie in the subalgebra generated by $x$, $[y,x]$, and a specific higher commutator, and they conjecture that the entire constant algebra is generated by $x$ and constant pseudodeterminants, i.e., $L^\delta = U^{(1)}$. This work links invariant theory with noncommutative algebra in the quest for an explicit description of $L^\delta$, guiding future efforts toward a complete characterization.
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 δof K[X_n] is called Weitzenböck due to his well known result from 1932 stating that the algebra \text{\rm ker}(δ)=K[X_n]^δ of constants of $δ$ is finitely generated. The explicit form of a generating set of $K[X_n,Y_n]^δ$ was conjectured by Nowicki in 1994 in the case δwas such that δ(y_{i})=x_{i}$, $δ(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 Weitzenböck derivation δsending 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.