Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.03636 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915526429638656 |
|---|---|
| author | Centrone, Lucio Findik, Sehmus Souza, Manuela da Silva |
| author_facet | Centrone, Lucio Findik, Sehmus Souza, Manuela da Silva |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03636 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Nowicki Conjecture for the free Lie algebra of rank 2 Centrone, Lucio Findik, Sehmus Souza, Manuela da Silva Rings and Algebras 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. |
| title | On the Nowicki Conjecture for the free Lie algebra of rank 2 |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2501.03636 |