Compatible Lie Bialgebras* *Supported by National Natural Science Foundation of China under Grant Nos. 11271202, 11221091, 11425104 and Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20120031110022

Abstract

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 fit into the framework to construct non-constant solutions of the classical Yang–Baxter equation given by Golubchik and Sokolov.