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-Car...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.展开更多
In this paper, we introduce a notion of J-dendriform algebra with two operations as a Jordan algebraic analogue of a dendriform algebra such that the antieommutator of the sum of the two operations is a Jordan algebra...In this paper, we introduce a notion of J-dendriform algebra with two operations as a Jordan algebraic analogue of a dendriform algebra such that the antieommutator of the sum of the two operations is a Jordan algebra. A dendriform algebra is a J-dendriform algebra. Moreover, J-dendriform algebras fit into a commutative diagram which extends the relationships among associative, Lie, and Jordan algebras. Their relations with some structures such as Rota-Baxter operators, classical Yang-Baxter equation, and bilinear forms are given.展开更多
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+2 种基金Nankai Zhide Foundationsupported by the National Key Research and Development Program of China (Grant No.2021YFA1002000)the Fundamental Research Funds for the Central Universities (Grant No.2412022QD033)。
文摘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.
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 10621101, 10921061）, the National Key Basic Research Development Project （2006CB805905）, and the Specialized Research Fund for the Doctoral Program （200800550015）.
文摘In this paper, we introduce a notion of J-dendriform algebra with two operations as a Jordan algebraic analogue of a dendriform algebra such that the antieommutator of the sum of the two operations is a Jordan algebra. A dendriform algebra is a J-dendriform algebra. Moreover, J-dendriform algebras fit into a commutative diagram which extends the relationships among associative, Lie, and Jordan algebras. Their relations with some structures such as Rota-Baxter operators, classical Yang-Baxter equation, and bilinear forms are given.
