ON THE ALGEBRA OF CONSTANTS OF FREE METABELIAN LIE ALGEBRAS

Andre DUSHIMIRIMANA

doi:https://doi.org/10.33773/jum.1143787

Article Details

ON THE ALGEBRA OF CONSTANTS OF FREE METABELIAN LIE ALGEBRAS

Authors:

Andre DUSHIMIRIMANA

Publisher:

Gökhan ÇUVALCIOĞLU

Main Topics:

Mathematics

Journal:

Journal of Universal Mathematics 2022 Vol.5 Issue 2, pp.185-192

Document Type:

Article

Publication Date:

7/2022

Document Language:

English

Document ID (Org No.):

2000509345

Frequency:

Bi-Annual

eISSN:

2618-5660

pISSN:

2618-5660

DOI:

https://doi.org/10.33773/jum.1143787

Abstract:

 

Let KKK be a field of characteristic zero, Xn={x1,…,xn}Xn={x1,…,xn}X_n=\{x_1,\dots,x_n\} be a set of variables, K[Xn]K[Xn]K[X_n] be the polynomial algebra and FnFnF_n be the free metabelian Lie algebra of rank nnn generated by XnXnX_n over the base field KKK. Well known result of Weitzenb\"ock states that K[Xn]δ={u∈K[Xn]∣∣ δ(u)=0}K[Xn]δ={u∈K[Xn]| δ(u)=0}K[X_n]^\delta=\big \{u\in K[X_n]  \big\vert\ \delta(u)=0\big \} is finitely generated as an algebra, where δδ\delta is a locally nilpotent linear derivation of K[Xn]K[Xn]K[X_n]. Extending this ideal to the non commutative algebras, recently the algebra FδnFnδF_n^\delta of constants in the free metabelian Lie algebras have been investigated. According to the findings,  FδnFnδF_n^\delta is not finitely generated as a Lie algebra; whereas, Fδn∩F′nFnδ∩Fn′F_n^\delta \cap F_n^\prime is finitely generated K[Xn]δK[Xn]δK[X_n]^\delta-module and a list of generators for  n≤4n≤4n\le 4 was given. In this work, in filling the gap in the list of small n′n′n's we work in F5F5F_5 and give a list of generators of Fδ5F5δF_5^\delta where δ(x5)=x4δ(x5)=x4\delta(x_5)=x_4, δ(x4)=0δ(x4)=0\delta(x_4)=0, δ(x3)=x2δ(x3)=x2\delta(x_3)=x_2, δ(x2)=x1δ(x2)=x1\delta(x_2)=x_1 and δ(x1)=0δ(x1)=0\delta(x_1)=0.

Article Details

ON THE ALGEBRA OF CONSTANTS OF FREE METABELIAN LIE ALGEBRAS

Authors: Andre DUSHIMIRIMANA

Publisher: Gökhan ÇUVALCIOĞLU

Main Topics: Mathematics

Journal: Journal of Universal Mathematics 2022 Vol.5 Issue 2, pp.185-192

Document Type: Article

Publication Date: 7/2022

Document Language: English

Document ID (Org No.): 2000509345

Frequency: Bi-Annual

eISSN: 2618-5660

pISSN: 2618-5660

DOI: https://doi.org/10.33773/jum.1143787

Issues

Authors:

Andre DUSHIMIRIMANA

Publisher:

Gökhan ÇUVALCIOĞLU

Main Topics:

Mathematics

Journal:

Journal of Universal Mathematics 2022 Vol.5 Issue 2, pp.185-192

Document Type:

Article

Publication Date:

7/2022

Document Language:

English

Document ID (Org No.):

2000509345

Frequency:

Bi-Annual

eISSN:

2618-5660

pISSN:

2618-5660

DOI:

https://doi.org/10.33773/jum.1143787