Natural lifts of Dorfman brackets

TL;DR

This paper demonstrates that Dorfman brackets on a vector bundle are equivalent to certain lifts to linear sections of the tangent and cotangent bundle, revealing the universality of the Courant-Dorfman bracket and enabling characterization of twistings and symmetries.

Contribution

It establishes a correspondence between Dorfman brackets and lifts to linear sections, highlighting the universality of the Courant-Dorfman bracket and facilitating analysis of their twistings and symmetries.

Findings

Dorfman brackets are equivalent to specific lifts to linear sections.

The Courant-Dorfman bracket is universal for these structures.

Characterization of twistings and symmetries of Dorfman brackets is achieved.

Abstract

In this note we prove that, for a vector bundle $E$ over a manifold $M$, a Dorfman bracket on $TM\oplus E^*$ anchored by $\operatorname{pr}_{TM}$ and with $E$ a vector bundle over $M$, is equivalent to a lift from $\Gamma(TM\oplus E^*)$ to linear sections of $TE\oplus T^*E\to E$, that intertwines the given Dorfman bracket with the Courant-Dorfman bracket on sections of $TE\oplus T^*E$. This shows a universality of the Courant-Dorfman bracket, and allows us to caracterise twistings and symmetries of transitive Dorfman brackets via the corresponding lifts.