(Rota-Baxter)3-(Hom)-Lie Bialgebra And 3-lie Classical Hom-Yang-Baxter Equation

n-Lie algebras are a kind of algebraic systems appearing in many fields in math-ematical physics.Specially,the structure of 3-Lie algebras is applied to the study of the supersymmetry and gange symmetry transformations of the world-volume theory.In this paper,we firstly introduce 3-Hom-Lie algebra,3-Hom-Lie coalgebra and 3-Hom-Lie bialgebra whose compatibility conditions between the multiplication and comultiplica-tion are given by derivation conditions.We give some techniques such as the dual or homomorphism of an algebra,to construct 3-Hom-Lie bialgebras.some examples of low dimension of 3-Hom-Lie bialgebras are given.Secondly,for any Lie algebra,following the Drinfeld's theorem,there is a solution of the classical Yang-Bater equation.Inspired by Drinfeld's idea,we hope that we can give the solutions of 3-Lie Classical Hom-Yang-Bater correspondence with this structure,we introduce 3-Hom-Lie bialgebras whose compatibil-ity conditions between the multiplication and comultiplication are given by local cocycle conditions.We study a twisted 3-ary version of the Yang-Baxter Equation,called the 3-Lie classical Hom-Yang-Baxter Equation(3-Lie CHYBE for short),which is a general form of 3-Lie classical Yang-Baxter Equation and prove that the bialgebras induced by the solutions of 3-Lie CHYBE induce the coboundary local cocycle 3-Hom-Lie bialgebras.Finally,we introduce the notions of Rota-Baxter coassociative 3-coalgebra and Rota-Baxter 3-Lie coalgebra which can be viewed as the dual notion of Rota-Baxter 3-algebra and Rota-Baxter 3-Lie algebra.We give some constructions of Rota-Baxter coassocia-tive 3-coalgebra from group-like elements,Polynomial algebras and 3-dimensional linear spaces and also derive the structures of 3-pre-Lie coalgebras via Rota-Baxter 3-Lie coal-gebras of weight 0.We give the notion of Rota-Baxter 3-Lie bialgebra and an example of Rota-Baxter 3-Lie bialgebra in 3-dimensional linear space.