Leibniz bialgebras, relative Rota–Baxter operators, and the classical Leibniz Yang–Baxter equation

Abstract

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.