Omni-Lie algebroids

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Abstract

A generalized Courant algebroid structure is defined on the direct sum bundle DEJE, where DE and JE are, respectively, the gauge Lie algebroid and the jet bundle of a vector bundle E. 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 JEDE and Lie algebroid (local Lie algebra) structures on E when rank(E)2 (E is a line bundle).

MSC

17B66

Keywords

Gauge Lie algebroid
Jet bundle
Omni-Lie algebroid
Dirac structure
Local Lie algebra

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Research partially supported by NSFC (10871007, 10911120391/A0109) and CPSF (20060400017).