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On higher analogues of Courant algebroids

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Abstract

In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM ⊕ ∧n T*M for an m-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an (n+1)-vector field π is closed under the higher-order Dorfman bracket iff π is a Nambu-Poisson structure. Consequently, there is an induced Leibniz algebroid structure on ∧n T*M. The graph of an (n+1)-form ω is closed under the higher-order Dorfman bracket iff ω is a premultisymplectic structure of order n, i.e., dω = 0. Furthermore, there is a Lie algebroid structure on the admissible bundle A ⊂ ∧n T*M. In particular, for a 2-plectic structure, it induces the Lie 2-algebra structure given in (Baez, Hoffnung and Rogers, 2010).

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Authors and Affiliations

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

    YanHui Bi

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

    YunHe Sheng

  3. School of Mathematics, Dalian University of Technology, Dalian, 116024, China

    YunHe Sheng

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  1. YanHui Bi
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  2. YunHe Sheng
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Correspondence to YunHe Sheng.

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Bi, Y., Sheng, Y. On higher analogues of Courant algebroids. Sci. China Math. 54, 437–447 (2011). https://doi.org/10.1007/s11425-010-4142-0

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  • DOI: https://doi.org/10.1007/s11425-010-4142-0

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