On higher analogues of Courant algebroids

Abstract

In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle $TM\oplus {\wedge }^n{T}^{\ast }M$ 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 $\pi$ is closed under the higher-order Dorfman bracket iff $\pi$ is a Nambu-Poisson structure. Consequently, there is an induced Leibniz algebroid structure on ${\wedge }^n{T}^{\ast }M$. The graph of an $(n+1)$-form $\omega$ is closed under the higher-order Dorfman bracket iff $\omega$ is a premultisymplectic structure of order $n$, i.e. $\dMω=0$. Furthermore, there is a Lie algebroid structure on the admissible bundle $A\subset {\wedge }^n{T}^{\ast }M$. In particular, for a 2-plectic structure, it induces the Lie 2-algebra structure given in \cite{baez:classicalstring}.

13 pages

References in corpus (1)

• Modular Classes of Loday Algebroids

Cited by in corpus (4)

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• Dissections and automorphisms of regular Courant algebroids