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Maurer-Cartan characterizations and cohomologies of compatible Lie algebras

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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 (2001) in their study of bi-Hamiltonian structures. Finally, we use the Maurer-Cartan approach to classify nonabelian extensions of compatible Lie algebras.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11901501, 11922110 and 11931009). Chengming Bai was supported by the Fundamental Research Funds for the Central Universities and Nankai Zhide Foundation. Jiefeng Liu was supported by the National Key Research and Development Program of China (Grant No. 2021YFA1002000) and the Fundamental Research Funds for the Central Universities (Grant No. 2412022QD033).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Jiefeng Liu

  2. Department of Mathematics, Jilin University, Changchun, 130012, China

    Yunhe Sheng

  3. Chern Institute of Mathematics & LPMC, Nankai University, Tianjin, 300071, China

    Chengming Bai

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  1. Jiefeng Liu
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  2. Yunhe Sheng
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  3. Chengming Bai
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Correspondence to Yunhe Sheng.

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Liu, J., Sheng, Y. & Bai, C. Maurer-Cartan characterizations and cohomologies of compatible Lie algebras. Sci. China Math. 66, 1177–1198 (2023). https://doi.org/10.1007/s11425-021-2014-5

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