Dirac structures of omni-Lie algebroids

Abstract

Omni-Lie algebroids are generalizations of Alan Weinstein's omni-Lie algebras. A Dirac structure in an omni-Lie algebroid $\dev E\oplus \jet E$ 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 $\huaT=TM\oplus E$; we establish the relation between the normalizer $N_{L}$ of a reducible Dirac structure $L$ and the derivation algebra $\Der(\pomnib (L))$ of the projective Lie algebroid $\pomnib (L)$; we study the cohomology group $\mathrm{H}^\bullet(L,\rho_{L})$ and the relation between $N_{L}$ and $\mathrm{H}^1(L,\rho_{L})$; we describe Lie bialgebroids using the adjoint representation; we study the deformation of a Dirac structure $L$, which is related with $\mathrm{H}^2(L,\rho_{L})$.