Hom-Malcev Bialgebras

In this paper,we mainly study Hom-Malcev algebra.We give the definition of representation,matched pair and Manin triple of Hom-Malcev algebra,construct Hom-Malcev bialgebra.At the same time,we also construct coboundary operators in two different cases which are the trivial representation and the ?s-adjoint representation of Hom-Malcev algebra.In the first part,we mainly introduce the basic concepts and conclusions related to this paper,including the definition of Malcev algebra,the definition of Hom-Malcev algebra and the relationship between Malcev algebra and Hom-Malcev algebra.In the second part,we mainly study representation of Hom-Malcev algebra.First,we give the definition of representation of Hom-Malcev algebra and its adjoint representation is used as an example to illustrate this definition.Then,we get conditions that the dual mapping of representation is a representation of Hom-Malcev algebra.In particular,we find conditions that the dual mapping of adjoint representation is a representation of Hom-Malcev algebra.and we give the definition of compatible Hom-Malcev algebra by these conditions.In the third part,we mainly study matched pair and Manin triple of Hom-Malcev algebra and Hom-Malcev bialgebra.First,we give the definition of matched pair of Hom-Malcev algebra,and we get a new Hom-Malcev algebra on the direct sum of any two Hom-Malcev algebras.We also construct a new Hom-Malcev algebra on the direct sum of a Hom-Malcev algebra and its dual space.Then,we give the definition of Hom-Malcev bialgebra and Manin triple of Hom-Malcev algebra,and we give the relation between matched pair,Manin triple of Hom-Malcev algebra and Hom-Malcev bialgebra.In the fourth part,we mainly construct coboundary operators of Hom-Malcev algebra.We construct coboundary operators about representation of Hom-Malcev algebra,and study the relation between them.We construct coboundary operators about trivial representation and ?s-adjoint representation of Hom-Malcev algebra.First,we give the definition of the trivial representation of Hom-Malcev algebra,and we construct coboundary operators in this case.We also calculate the form of coboundary operators when the extension of Hom-Malcev algebra is a Hom-Malcev algebra.Then,we give the definition of the ?s-adjoint representation of Hom-Malcev algebra,we construct coboundary operators in this case.We also study the deformation of Hom-Malcev algebras,give the definition of Hom-Nijenhuis operators of Hom-Malcev algebras,and we show that the deformation generated by a hom-Nijenhuis operator is trivial.