Maurer-Cartan characterizations and cohomologies of compatible Lie algebras

TL;DR

This paper develops Maurer-Cartan characterizations and a cohomology theory for compatible Lie algebras, enabling classification of their deformations, extensions, and relations to bi-Hamiltonian structures.

Contribution

It introduces a Maurer-Cartan framework and a cohomology theory specific to compatible Lie algebras, advancing their structural understanding.

Findings

Maurer-Cartan characterizations of compatible Lie algebras

A cohomology theory for classifying deformations and extensions

Relation between reduced cohomology and bi-Hamiltonian structures

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.