
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).
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Ophthalmology and Eye Disorders
