Natural lifts of Dorfman brackets

Abstract

In this note we prove that, for a vector bundle $E$ over a manifold $M$, a Dorfman bracket on $TM\oplus E^{\ast }$ anchored by ${\mathrm{pr}}_{TM}$ and with $E$ a vector bundle over $M$, is equivalent to a lift from $\Gamma (TM\oplus E^{\ast })$ to linear sections of $TE\oplus T^{\ast }E\to E$, that intertwines the given Dorfman bracket with the Courant-Dorfman bracket on sections of $TE\oplus T^{\ast }E$. 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.

arXiv comment30 pages, v3: version accepted for publication in "Advances in Theoretical and Mathematical Physics'' in October 2018. Only changes to the contents: the address of the second author and some bibliography entries were updated