From relative Rota-Baxter operators and relative averaging operators on Lie algebras to relative Rota-Baxter operators on Leibniz algebras: a uniform approach

TL;DR

This paper develops a unified framework connecting relative Rota-Baxter and averaging operators on Lie and Leibniz algebras, establishing categorical relations, exact sequences, and cohomological links.

Contribution

It constructs subcategories of relative Rota-Baxter operators on Leibniz algebras and relates them to Lie algebra operators via exact sequences and cohomology.

Findings

Established subcategories using symmetric and antisymmetric representations.

Derived short exact sequences linking Leibniz and Lie algebra operators.

Presented long exact sequences relating their cohomology groups.

Abstract

In this paper, first we construct two subcategories (using symmetric representations and antisymmetric representations) of the category of relative Rota-Baxter operators on Leibniz algebras, and establish the relations with the categories of relative Rota-Baxter operators and relative averaging operators on Lie algebras. Then we show that there is a short exact sequence describing the relation between the controlling algebra of relative Rota-Baxter operators on a Leibniz algebra with respect to a symmetric (resp. antisymmetric) representation and the controlling algebra of the induced relative Rota-Baxter operators (resp. averaging operators) on the canonical Lie algebra associated to the Leibniz algebra. Finally, we show that there is a long exact sequence describing the relation between the cohomology groups of a relative Rota-Baxter operator on a Leibniz algebra with respect to a symmetric (resp. antisymmetric) representation and the cohomology groups of the induced relative Rota-Baxter operator (resp. averaging operator) on the canonical Lie algebra.