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

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.