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Strong homotopy Lie algebras, homotopy Poisson manifolds and Courant algebroids

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Abstract

We study Maurer–Cartan elements on homotopy Poisson manifolds of degree n. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson \(\mathfrak g\)-manifolds, and twisted Courant algebroids. Using the fact that the dual of an n-term \(L_\infty \)-algebra is a homotopy Poisson manifold of degree \(n-1\), we obtain a Courant algebroid from a 2-term \(L_\infty \)-algebra \(\mathfrak g\) via the degree 2 symplectic NQ-manifold \(T^*[2]\mathfrak g^*[1]\). By integrating the Lie quasi-bialgebroid associated to the Courant algebroid, we obtain a Lie-quasi-Poisson groupoid from a 2-term \(L_\infty \)-algebra, which is proposed to be the geometric structure on the dual of a Lie 2-algebra. These results lead to a construction of a new 2-term \(L_\infty \)-algebra from a given one, which could produce many interesting examples.

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Notes

  1. There are different ways to describe the double of a Lie bialgebroid, e.g. Mackenzie gave the description of Drinfeld doubles for Lie bialgebroids using double Lie algebroids in [24]; Roytenberg and Voronov gave the description of Drinfeld doubles for Lie bialgebroids using graded manifolds in [28, 39] respectively.

  2. By definition, there is an \(L_\infty \)-algebra structure on the complex of section spaces together with some compatibility conditions. See [7, 35] for more details.

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Acknowledgements

We give our warmest thanks to Jim Stasheff and the referee for very helpful comments that improve the paper. We also give our special thanks to Noriaki Ikeda, Zhangju Liu, Pavol Ševera and Chenchang Zhu for very useful comments and discussions. We thank Marco Zambon for pointing out the reference [26] to us.

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Authors and Affiliations

  1. Department of Mathematics, Peking University, Beijing, 100871, China

    Honglei Lang

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

    Yunhe Sheng

  3. Kavli Institute for Theoretical Physics China, CAS, Beijing, 100190, China

    Yunhe Sheng

  4. Section of Mathematics, University of Geneva, 2-4 Rue de Lièvre, c.p. 64, 1211, Geneva 4, Switzerland

    Xiaomeng Xu

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  1. Honglei Lang
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  2. Yunhe Sheng
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Correspondence to Yunhe Sheng.

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Research supported by NSFC (11101179,11471139) and NSF of Jilin Province (20140520054JH). Xiaomeng Xu was partially supported by the SNSF Grants P2GEP2-165118 and NCCR SwissMAP.

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Lang, H., Sheng, Y. & Xu, X. Strong homotopy Lie algebras, homotopy Poisson manifolds and Courant algebroids. Lett Math Phys 107, 861–885 (2017). https://doi.org/10.1007/s11005-016-0925-8

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