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 ${\mathrm{H}}^2(L,{\rho }_L)$.

23 pages, no figure, to appear in International Journal of Mathematics

Cited by in corpus (4)

• AV-Courant algebroids and generalized CR structures
• Omni-Lie 2-algebras and their Dirac structures
• On deformations of Lie algebroids
• Semidirect products of representations up to homotopy