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

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.