On higher analogues of Courant algebroids
On higher analogues of Courant algebroids
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DOI：
10.1007/s11425-010-4142-0
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发表时间：
2010-03
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通讯作者：
Yanhui Bi;Y. Sheng
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作者：
Yanhui Bi;Y. Sheng
本文研究了m维流形上直和T_m ~* M上的Courant代数体结构的高级类似的代数性质。作为应用，我们重新审视Nambu-Poisson结构和多辛结构。证明了（n+1）-向量场π的图在高阶Dorfman括号下是闭的当且仅当π是Nambu-Poisson结构.因此，在T *M上存在一个诱导的莱布尼茨代数体结构。（n+1）-形式ω的图在高阶Dorfman括号下是闭的当且仅当ω是n阶的预多辛结构，即，dω= 0。此外，在容许的代数A nT*M上还存在一个李代数胚结构.特别地，对于2-plectic结构，它导出了在（Baez，Hoffnung and Rogers，2010）中给出的Lie 2-代数结构。
In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundleTM⊕ ∧nT*Mfor anm-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an (n+1)-vector fieldπis closed under the higher-order Dorfman bracket iffπis a Nambu-Poisson structure. Consequently, there is an induced Leibniz algebroid structure on ∧nT*M. The graph of an (n+1)-formωis closed under the higher-order Dorfman bracket iffωis a premultisymplectic structure of ordern, i.e.,dω= 0. Furthermore, there is a Lie algebroid structure on the admissible bundleA⊂ ∧nT*M. In particular, for a 2-plectic structure, it induces the Lie 2-algebra structure given in (Baez, Hoffnung and Rogers, 2010).