Leibniz Bialgebras, Classical Yang–Baxter Equations and Dynamical Systems

Abstract

We extend the notion of bialgebra for Lie algebras to Leibniz algebras and also, the correspondence between the Leibniz bialgebras (for different right or left cases) and its dual is investigated. Moreover, the coboundary Leibniz bialgebras, the classical $r$-matrices and Yang-Baxter equations related to the Leibniz algebras are defined, and some examples are given. Finally, a method for construction of a dynamical system on a Leibniz manifold via Leibniz bialgebra is presented.