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DIRAC STRUCTURES OF OMNI-LIE ALGEBROIDS

ZHUO CHEN

Department of Mathematics, Tsinghua University, Beijing 100084, P. R. China

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ZHANG JU LIU

Department of Mathematics and LMAM, Peking University, Beijing 100871, P. R. China

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, and

YUNHE SHENG

Department of Mathematics, Jilin University, Changchun 130012, Jilin, P. R. China

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https://doi.org/10.1142/S0129167X11007215Cited by:7 (Source: Crossref)

Abstract

Omni-Lie algebroids are generalizations of Alan Weinstein's omni-Lie algebras. A Dirac structure in an omni-Lie algebroid 𝔇E ⊕ 𝔍E is necessarily a Lie algebroid together with a representation on E. We study the geometry underlying these Dirac structures in the light of reduction theory. In particular, we prove that there is a one-to-one correspondence between reducible Dirac structures and projective Lie algebroids in ; we establish the relation between the normalizer N_L of a reducible Dirac structure L and the derivation algebra Der(b (L)) of the projective Lie algebroid b(L); we study the cohomology group H^•(L, _ρL) and the relation between N_L and H¹(L, ρ_L); we describe Lie bialgebroids using the adjoint representation; we study the deformation of a Dirac structure L, which is related with H²(L, ρ_L).

Keywords:

AMSC: 17B66, 58H05

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