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Vol. 51 No. 3–4 (2025)
Serdica Mathematical Journal

Articles

Graded polynomial identities for the Lie algebra of \(3 \times 3\) upper triangular matrices endowed with the canonical grading

https://doi.org/10.55630/serdica.2025.51.491-524

Published 23-12-2025

  • Daniela Martinez Correa
    • ,
  • Dimas José Gonçalves
    • ,
  • Evandro Riva

Daniela Martinez Correa
Universidade de São Paulo

Dimas José Gonçalves
Universidade Federal de São Carlos

Evandro Riva
Universidade Tecnológica Federal do Paraná

Abstract

Let \(\mathbb{F}\) be a finite field of characteristic different from 2, and let \(UT_3^{(-)}=UT_3(\mathbb{F})^{(-)}\) be the Lie algebra of \(3\times 3\) upper triangular matrices over \(\mathbb{F}\) endowed with the canonical \(\mathbb{Z}_3\)-grading. In this paper, we describe the \(T_{\mathbb{Z}_3}\)-ideal \(Id=Id(UT_3^{(-)})\) of all \(\mathbb{Z}_3\)-graded polynomial identities of \(UT_3^{(-)}\) and we produce a linear basis for the corresponding relatively free graded algebra. Moreover, we prove that \(Id\) satisfies the Specht property, that is, \(Id\) and every \(T_{\mathbb{Z}_3}\)-ideal containing \(Id\) is finitely generated as a \(T_{\mathbb{Z}_3}\)-ideal.

2020 Mathematics Subject Classification:

16R10, 16R50, 16W50, 17B70, 17B01

Keywords

Lie algebra, upper triangular matrices, canonical grading, graded polynomial identities, Specht problem

Full text

Author Details

Daniela Martinez Correa

Universidade de São Paulo (USP)
Instituto de Matemática, Estatística e Ciência da Computação
R. do Matão-Butantã, 05508-090, São Paulo, SP, Brazil
e-mail: danielam.correa@ime.usp.br

Dimas José Gonçalves

Universidade Federal de São Carlos (UFSCar)
Departamento de Matemática
13565-905, São Carlos, SP, Brazil
e-mail: dimasjg@ufscar.br

Evandro Riva

Universidade Tecnológica Federal do Paraná (UTFPR)
Departamento de Matemática
Av. Sete de Setembro, 3165, 80230-901, Curitiba, PR, Brazil
e-mail: evandroriva@utfpr.edu.br

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