# Free differential Lie Rota-Baxter algebras and Gr\"obner-Shirshov bases

## TL;DR

This paper develops a Gr"obner-Shirshov bases framework for differential Lie $ ext{Omega}$-algebras and constructs a linear basis for free differential Lie Rota-Baxter algebras, advancing algebraic structure theory.

## Contribution

It introduces the Gr"obner-Shirshov bases theory for differential Lie $ ext{Omega}$-algebras and provides a basis for free differential Lie Rota-Baxter algebras, which was previously unknown.

## Key findings

- Established Gr"obner-Shirshov bases for differential Lie $ ext{Omega}$-algebras
- Constructed a linear basis for free differential Lie Rota-Baxter algebra
- Advanced the understanding of algebraic structures involving Rota-Baxter operators

## Abstract

We establish the Gr\"obner-Shirshov bases theory for differential Lie $\Omega$-algebras. As an application, we give a linear basis of a free differential Lie Rota-Baxter algebra on a set.