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Linearization of the higher analogue of Courant algebroids

H. Lang,Y. Sheng

Published 2020 in The Journal of Geometric Mechanics

ABSTRACT

In this paper, we show that the spaces of sections of the $n$-th differential operator bundle $\dev^n E$ and the $n$-th skew-symmetric jet bundle $\jet_n E$ 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 $\dev E\oplus\jet_n E$ introduced by Bi-Vitagliago-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 $\dev E\oplus \wedge^n\jet E$ 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.

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