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We determine the $$L_\infty $$L∞ -algebrathat controls deformations of a relative Rota–Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying $$\mathsf {Lie}\mathsf {Rep}$$LieRep pair by the dg Lie algebra controlling deformations of the relative Rota–Baxter operator. Consequently, we define thecohomologyof relative Rota–Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota–Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of ahomotopyrelative Rota–Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota–Baxter Lie algebras is intimately related topre-Lie $$_\infty $$∞ -algebras.

In this paper, first we introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras, and Leibniz bialgebras are equivalent. Then we introduce the notion of a (relative) Rota–Baxter operator on a Leibniz algebra and construct the graded Lie algebra that characterizes relative Rota–Baxter operators as Maurer–Cartan elements. By these structures and the twisting theory of twilled Leibniz algebras, we further define the classical Leibniz Yang–Baxter equation, classical Leibniz r -matrices, and triangular Leibniz bialgebras. Finally, we construct solutions of the classical Leibniz Yang–Baxter equation using relative Rota– Baxter operators and Leibniz-dendriform algebras.