Maurer-Cartan characterizations and cohomologies of compatible Lie algebras  

作　　者：Jiefeng Liu Yunhe Sheng Chengming Bai 

机构地区：[1]School of Mathematics and Statistics,Northeast Normal University,Changchun 130024,China [2]Department of Mathematics,Jilin University,Changchun 130012,China [3]Chern Institute of Mathematics&LPMC,Nankai University,Tianjin 300071,China

出　　处：《Science China Mathematics》2023年第6期1177-1198,共22页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos.11901501,11922110 and 11931009);supported by the Fundamental Research Funds for the Central Universities;Nankai Zhide Foundation;supported by the National Key Research and Development Program of China (Grant No.2021YFA1002000);the Fundamental Research Funds for the Central Universities (Grant No.2412022QD033)。

摘　　要：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.

关 键 词：compatible Lie algebra Maurer-Cartan element COHOMOLOGY deformation EXTENSION 

分 类 号：O152.5[理学—数学]