Omni-Lie algebroids

A generalized Courant algebroid structure is defined on the direct sum bundle DE⊕JEDE⊕JE, where DEDE and JEJE are, respectively, the gauge Lie algebroid and the jet bundle of a vector bundle EE. Such a structure is called an omni-Lie algebroid   since it is reduced to the omni-Lie algebra introduced by A. Weinstein if the base manifold is a point. We prove that there is a one-to-one correspondence between Dirac structures coming from bundle maps JE→DEJE→DE and Lie algebroid (local Lie algebra) structures on EE when rank(E)≥2 (EE is a line bundle).