Cohomology of weighted Rota-Baxter Lie algebras and Rota-Baxter paired operators

TL;DR

This paper develops a cohomology theory for weighted Rota-Baxter Lie algebras and paired operators, enabling the study of their extensions and deformations with new mathematical tools.

Contribution

It introduces the cohomology framework for weighted Rota-Baxter Lie algebras and paired operators, advancing understanding of their structure and deformation theory.

Findings

Cohomology of weighted Rota-Baxter Lie algebras is established.

Applications to abelian extensions and formal deformations are demonstrated.

A cohomology theory for weighted Rota-Baxter paired operators is developed.

Abstract

In this paper, we define representations and cohomology of weighted Rota-Baxter Lie algebras. As applications of cohomology, we study abelian extensions and formal $1$-parameter deformations weighted Rota-Baxter Lie algebras. Finally, we consider weighted Rota-Baxter paired operators that induces a weighted Rota-Baxter Lie algebra and a representation of it. We define suitable cohomology for such paired operators that govern deformation.