Lucio Centrone ; Sehmus Findik ; Manuela da Silva Souza - On the Nowicki Conjecture for the free Lie algebra of rank 2

cm:15036 - Communications in Mathematics, May 29, 2025, Volume 33 (2025), Issue 3 (Special issue: European Non-Associative Algebra Seminar) - https://doi.org/10.46298/cm.15036
On the Nowicki Conjecture for the free Lie algebra of rank 2Article

Authors: Lucio Centrone ; Sehmus Findik ; Manuela da Silva Souza

    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 Weitzenböck 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 Weitzenböck 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.


    Volume: Volume 33 (2025), Issue 3 (Special issue: European Non-Associative Algebra Seminar)
    Published on: May 29, 2025
    Accepted on: May 14, 2025
    Submitted on: March 14, 2025
    Keywords: Mathematics - Rings and Algebras

    Consultation statistics

    This page has been seen 149 times.
    This article's PDF has been downloaded 101 times.