E-Courant algebroids

In this paper, we introduce the notion of $E$-Courant algebroids, where $E$ is a vector bundle. It is a kind of generalized Courant algebroid and contains Courant algebroids, Courant-Jacobi algebroids and omni-Lie algebroids as its special cases. We explore novel phenomena exhibited by $E$-Courant algebroids and provide many examples. We study the automorphism groups of omni-Lie algebroids and classify the isomorphism classes of exact $E$-Courant algebroids. In addition, we introduce the concepts of $E$-Lie bialgebroids and Manin triples.Comment: 29 pages, no figur

Dirac structures of omni-Lie algebroids

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 $\mathrm{H}^2(L,\rho_{L})$.Comment: 23 pages, no figure, to appear in International Journal of Mathematic