Representations and cohomologies of relative Rota-Baxter Lie algebras and applications

TL;DR

This paper develops a cohomology theory for relative Rota-Baxter Lie algebras, classifies their extensions and 2-algebras, and extends the framework to Rota-Baxter Lie algebras with applications.

Contribution

It introduces the notions of representations and cohomologies for relative Rota-Baxter Lie algebras and Rota-Baxter Lie algebras, and applies these to classify extensions and 2-algebras.

Findings

Classified abelian extensions using second cohomology.

Classified skeletal Rota-Baxter Lie 2-algebras using third cohomology.

Extended the framework to Rota-Baxter Lie algebras with consistent representations.

Abstract

In this paper, first we give the notion of a representation of a relative Rota-Baxter Lie algebra and introduce the cohomologies of a relative Rota-Baxter Lie algebra with coefficients in a representation. Then we classify abelian extensions of relative Rota-Baxter Lie algebras using the second cohomology group, and classify skeletal relative Rota-Baxter Lie 2-algebras using the third cohomology group as applications. At last, using the established general framework of representations and cohomologies of relative Rota-Baxter Lie algebras, we give the notion of representations of Rota-Baxter Lie algebras, which is consistent with representations of Rota-Baxter associative algebras in the literature, and introduce the cohomologies of Rota-Baxter Lie algebras with coefficients in a representation. Applications are also given to classify abelian extensions of Rota-Baxter Lie algebras and skeletal Rota-Baxter Lie 2-algebras.