On higher analogues of Courant algebroids

In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle $TM\oplus\wedge^nT^*M$ 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 $\wedge^nT^*M$. 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. $\dMω=0$. Furthermore, there is a Lie algebroid structure on the admissible bundle $A\subset\wedge^{n}T^*M$. In particular, for a 2-plectic structure, it induces the Lie 2-algebra structure given in \cite{baez:classicalstring}.

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