Compatible Lie Bialgebras

A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket.We construct a bialgebra theory of compatible Lie algebras as an analogue of a Lie bialgebra.They can also be regarded as a "compatible version" of Lie bialgebras,that is,a pair of Lie bialgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra.Many properties of compatible Lie bialgebras as the "compatible version" of the corresponding properties of Lie bialgebras are presented.In particular,there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in Lie algebras.Furthermore,a notion of compatible pre-Lie algebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie algebras which leads to a construction of the solutions of the latter.As a byproduct,the compatible Lie bialgebras St into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov.