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 \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.