Omni-Lie algebroids

TL;DR

This paper introduces omni-Lie algebroids, a generalized structure on the sum of gauge Lie algebroids and jet bundles, characterizing Lie algebroids via Dirac structures.

Contribution

It defines omni-Lie algebroids and establishes a correspondence between Lie algebroid structures and Dirac structures within this framework.

Findings

Defines omni-Lie algebroids on D(E)+J(E).

Characterizes Lie algebroids as Dirac structures in the omni-Lie algebroid.

Generalizes the omni-Lie algebra concept to vector bundles.

Abstract

A generalized Courant algebroid structure is defined on the direct sum bundle D(E) +J(E), where D(E) and J(E) are the gauge Lie algebroid and the jet bundle of a vector bundle E respectively. 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 any Lie algebroid structure on E is characterized by a Dirac structure as the graph of a bundle map from J(E) to D(E).