Maurer-Cartan characterizations and cohomologies of compatible Lie algebras

Abstract

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 in their study of bi-Hamiltonian structures. Finally, we use the Maurer-Cartan approach to classify nonabelian extensions of compatible Lie algebras.

arXiv comment24 pages

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Cited by in corpus (3)

• Cohomology and deformations of compatible Hom-Lie algebras
• A classification of nilpotent compatible Lie algebras
• Compatible $L_{\infty }$-algebras