Cohomology, deformations and extensions of Rota-Baxter Leibniz algebras

TL;DR

This paper develops a cohomology theory for Rota-Baxter Leibniz algebras, explores their deformations and extensions, and establishes connections between cohomology classes and algebraic structures.

Contribution

It introduces a new cohomology framework for Rota-Baxter Leibniz algebras and links it to their deformation and extension theories.

Findings

Cohomology theory for Rota-Baxter Leibniz algebras is established.

Deformation theory is connected to the developed cohomology.

Extensions are classified via cohomology groups.

Abstract

A Rota-Baxter Leibniz algebra is a Leibniz algebra $(\mathfrak{g},[~,~]_{\mathfrak{g}})$ equipped with a Rota-Baxter operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. We define representation and dual representation of Rota-Baxter Leibniz algebras. Next, we define a cohomology theory of Rota-Baxter Leibniz algebras. We also study the infinitesimal and formal deformation theory of Rota-Baxter Leibniz algebras and show that our cohomology is deformation cohomology. Moreover, We define an abelian extension of Rota-Baxter Leibniz algebras and show that equivalence classes of such extensions are related to the cohomology groups.