Linearization of the higher analogue of Courant algebroids

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 $E^{\ast }$ 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 $T{E}^{\ast }\oplus {\wedge }^n{T}^{\ast }{E}^{\ast }$. 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 $T{E}^{\ast }\oplus {\wedge }^n{T}^{\ast }{E}^{\ast }$. 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.

23 pages