Gr\"obner-Shirshov bases for Lie $\Omega$-algebras and free Rota-Baxter Lie algebras

TL;DR

This paper develops a basis theory for Lie $ ext{Omega}$-algebras, generalizing Lyndon-Shirshov words, and applies it to construct free Rota-Baxter and Nijenhuis Lie algebras, providing explicit bases and Gröbner-Shirshov bases.

Contribution

It introduces Lyndon-Shirshov $ ext{Omega}$-words, establishes their role as bases for free Lie $ ext{Omega}$-algebras, and constructs Gröbner-Shirshov bases for various free Rota-Baxter and Nijenhuis Lie algebras.

Findings

Lyndon-Shirshov $ ext{Omega}$-words form a linear basis for free Lie $ ext{Omega}$-algebras.

Explicit Gröbner-Shirshov bases for free $ ext{lambda}$-Rota-Baxter Lie algebras.

Linear bases for free modified Rota-Baxter and Nijenhuis Lie algebras.

Abstract

In this paper, we generalize the Lyndon-Shirshov words to Lyndon-Shirshov $\Omega$-words on a set $X$ and prove that the set of all non-associative Lyndon-Shirshov $\Omega$-words forms a linear basis of the free Lie $\Omega$-algebra on the set $X$. From this, we establish Gr\"{o}bner-Shirshov bases theory for Lie $\Omega$-algebras. As applications, we give Gr\"{o}bner-Shirshov bases for free $\lambda$-Rota-Baxter Lie algebras, free modified $\lambda$-Rota-Baxter Lie algebras and free Nijenhuis Lie algebras and then linear bases of such three free algebras are obtained.