Construction of free Lie Rota-Baxter superalgebra via Gr\"{o}bner-Shirshov bases theory

TL;DR

This paper develops a method to construct free Lie Rota-Baxter superalgebras using Gr"{o}bner-Shirshov bases, providing a systematic way to understand their structure and basis.

Contribution

It introduces a new approach to construct free Lie Rota-Baxter superalgebras via Gr"{o}bner-Shirshov bases, extending the theory to superalgebras with $ ext{Z}_2$-grading.

Findings

Established Gr"{o}bner-Shirshov bases for operated Lie superalgebras

Constructed a linear basis for free Lie Rota-Baxter superalgebras

Provided a composition-diamond lemma for operated Lie superalgebras

Abstract

In this paper, we construct free Lie Rota-Baxter superalgebra by using Gr\"{o}bner-Shirshov bases theory. We firstly construct free operated Lie superalgebras by the operated super-Lyndon-Shirshov monomials. Secondly, we establish Gr\"{o}bner-Shirshov bases theory for operated Lie superalgebras. Thirdly, we find a Gr\"{o}bner-Shirshov basis of a free Lie Rota-Baxter superalgebra on a $\mathbb{Z}_2$-graded set. Consequently, we can obtain a linear basis of a free Lie Rota-Baxter superalgebra by the composition-diamond lemma for operated Lie superalgebras.