In this paper,we give Maurer-Cartan characterizations as well as a cohomology theory for compatible Lie algebras.Explicitly,we first introduce the notion of a bidifferential graded Lie algebra and thus give Maurer-Cartan characterizations of compatible Lie algebras.Then we introduce a cohomology theory of compatible Lie algebras and use it to classify infinitesimal deformations and abelian extensions of compatible Lie algebras.In particular,we introduce the reduced cohomology of a compatible Lie algebra and establish the relation between the reduced cohomology of a compatible Lie algebra and the cohomology of the corresponding compatible linear Poisson structures introduced by Dubrovin and Zhang(2001)in their study of bi-Hamiltonian structures.Finally,we use the Maurer-Cartan approach to classify nonabelian extensions of compatible Lie algebras.
