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

## TL;DR

This paper introduces Leibniz bialgebras, explores their relation to Rota-Baxter operators, and constructs solutions to the classical Leibniz Yang-Baxter equation, advancing the algebraic framework for Leibniz structures.

## Contribution

It defines Leibniz bialgebras, establishes their equivalence with matched pairs and Manin triples, and links Rota-Baxter operators to solutions of the Leibniz Yang-Baxter equation.

## Key findings

- Leibniz bialgebras are equivalent to matched pairs and Manin triples.
- Constructed a graded Lie algebra characterizing Rota-Baxter operators.
- Provided solutions to 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.