A bialgebra theory for Compatible Hom-Lie algebras

In this paper, we introduce the notions of matched pairs and Manin triple for compatible Hom-Lie algebras. Then, we give a bialgebra theory of compatible Hom-Lie algebras with emphasis on its compatibility with Manin triple of compatible Hom-Lie algebras associated to a nondegenerate symmetric bilinear form. Moreover, we study coboundary compatible Hom-Lie bialgebras. Finally, we investigate some properties of a representation of a Hom-Nijenhuis Hom-Lie algebra and introduce the notion of a Hom-Nijenhuis Hom-Lie coalgebra. Furthermore, a Hom-Nijenhuis Hom-Lie bialgebra can be established by a Hom-Nijenhuis Hom-Lie algebra and a Hom-Nijenhuis Hom-Lie coalgebra satisfying some compatible conditions.