In this paper, we show that the spaces of sections of the $ n $-th differential operator bundle $ \mathfrak{D}^n E $ and the $ n $-th skew-symmetric jet bundle $ \mathfrak{J}_n E $ of a vector bundle $ E $ are isomorphic to the spaces of linear $ n $-vector fields and linear $ n $-forms on $ E^* $ respectively. Consequently, the $ n $-omni-Lie algebroid $ \mathfrak{D} E\oplus \mathfrak{J}_n E $ introduced by Bi-Vitagliano-Zhang can be explained as certain linearization, which we call pseudo-linearization of the higher analogue of Courant algebroids $ TE^*\oplus \wedge^nT^*E^* $. On the other hand, we show that the omni $ n $-Lie algebroid $ \mathfrak{D} E\oplus \wedge^n \mathfrak{J} E $ can also be explained as certain linearization, which we call Weinstein-linearization of the higher analogue of Courant algebroids $ TE^*\oplus \wedge^nT^*E^* $. We also show that $ n $-Lie algebroids, local $ n $-Lie algebras and Nambu-Jacobi structures can be characterized as integrable subbundles of omni $ n $-Lie algebroids.

中文翻译：

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库兰特代数的高级类似物的线性化

在本文中，我们证明了第n个差分算子束$ \ mathfrak {D} ^ n E $和第n个第n个歪斜对称射束$ \ mathfrak {J} _n的区间空间向量束$ E $的E $分别与$ E ^ * $上的线性$ n $-向量字段和线性$ n $-形式的空间同构。因此，Bi-Vitagliano-Zhang引入的$ n $ -omni-Lie代数$ \ mathfrak {D} E \ oplus \ mathfrak {J} _n E $可以解释为某些线性化，我们称其为伪线性化。库兰特代数$ TE ^ * \ oplus \ wedge ^ nT ^ * E ^ * $的高级类似物。另一方面，我们证明了omni $ n $ -Lie代数$ \ mathfrak {D} E \ oplus \ wedge ^ n \ mathfrak {J} E $也可以解释为某些线性化，我们称之为Weinstein-linearization库兰特代数$ TE ^ * \ oplus \ wedge ^ nT ^ * E ^ * $的高级类似物的形式。