Extending structures of Rota-Baxter Lie algebras

TL;DR

This paper introduces a new framework for extending Rota-Baxter Lie algebras by defining an extending datum and constructing a unified product, thereby unifying previous extension and factorization problems.

Contribution

It develops a comprehensive approach to extend Rota-Baxter Lie algebras using a unified product, generalizing prior extension and factorization problems.

Findings

Defined the notion of an extending datum for Rota-Baxter Lie algebras

Constructed a unified product to facilitate extensions

Solved the extending structures problem, unifying previous studies

Abstract

In this paper, we first introduce the notion of an extending datum of a Rota-Baxter Lie algebra through a vector space. We then construct a unified product for the Rota-Baxter Lie algebra with a vector space as a main ingredient in our approach. Finally, we solve the extending structures problem of Rota-Baxter Lie algebras, which generalizes and unifies two problems in the study of Rota-Baxter Lie algebras: the extension problem studied by Mishra-Das-Hazra and the factorization problem investigated by Lang-Sheng.