Relative Rota-Baxter Leibniz algebras, their characterization and cohomology

TL;DR

This paper introduces the concept of relative Rota-Baxter Leibniz algebras, constructs an $L_$-algebra to characterize them, and develops their cohomology theory with applications to deformations and extensions.

Contribution

It defines rRB Leibniz algebras, constructs an $L_$-algebra for their characterization, and develops their cohomology theory with applications.

Findings

Construction of an $L_$-algebra characterizing rRB Leibniz algebras

Definition of cohomology for rRB Leibniz algebras with coefficients in a representation

Applications to deformations and abelian extensions of rRB Leibniz algebras

Abstract

Recently, relative Rota-Baxter (Lie/associative) algebras are extensively studied in the literature from cohomological points of view. In this paper, we consider relative Rota-Baxter Leibniz algebras (rRB Leibniz algebras) as the object of our study. We construct an $L_\infty$-algebra that characterizes rRB Leibniz algebras as its Maurer-Cartan elements. Then we define representations of an rRB Leibniz algebra and introduce cohomology with coefficients in a representation. As applications of cohomology, we study deformations and abelian extensions of rRB Leibniz algebras.