Omni-Lie Superalgebras and Lie 2-superalgebras

We introduce the notion of omni-Lie superalgebra as a super version of the omni-Lie algebra introduced by Weinstein. This algebraic structure gives a nontrivial example of Leibniz superalgebra and Lie 2-superalgebra. We prove that there is a one-to-one correspondence between Dirac structures of the omni-Lie superalgebra and Lie superalgebra structures on subspaces of a super vector space.Comment: 14page

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

Deformations of Lie 2-algebras

In this paper, we consider deformations of Lie 2-algebras via the cohomology theory. We prove that a 1-parameter infinitesimal deformation of a Lie 2-algebra \g corresponds to a 2-cocycle of \g with the coefficients in the adjoint representation. The Nijenhuis operator for Lie 2-algebras is introduced to describe trivial deformations. We also study abelian extensions of Lie 2-algebras from the viewpoint of deformations of semidirect product Lie 2-algebras.Comment: 20 page

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

Coadjoint orbits of Lie groupoids

For a Lie groupoid $\mathcal{G}$ with Lie algebroid $A$, we realize the symplectic leaves of the Lie-Poisson structure on $A^*$ as orbits of the affine coadjoint action of the Lie groupoid $\mathcal{J}\mathcal{G}\ltimes T^*M$ on $A^*$, which coincide with the groupoid orbits of the symplectic groupoid $T^*\mathcal{G}$ over $A^*$. It is also shown that there is a fiber bundle structure on each symplectic leaf. In the case of gauge groupoids, a symplectic leaf is the universal phase space for a classical particle in a Yang-Mills field

The Atiyah class of generalized holomorphic vector bundles

For a generalized holomorphic vector bundle, we introduce the Atiyah class, which is the obstruction of the existence of generalized holomorphic connections on this bundle. Similar to the holomorphic case, such Atiyah classes can be defined by three approaches: the $\rm{\check{C}}$ech cohomology, the extension class of the first jet bundle as well as the Lie pair

Omni-Lie 2-algebras and their Dirac structures

We introduce the notion of omni-Lie 2-algebra, which is a categorification of Weinstein's omni-Lie algebras. We prove that there is a one-to-one correspondence between strict Lie 2-algebra structures on 2-sub-vector spaces of a 2-vector space \V and Dirac structures on the omni-Lie 2-algebra \gl(\V)\oplus \V . In particular, strict Lie 2-algebra structures on \V itself one-to-one correspond to Dirac structures of the form of graphs. Finally, we introduce the notion of twisted omni-Lie 2-algebra to describe (non-strict) Lie 2-algebra structures. Dirac structures of a twisted omni-Lie 2-algebra correspond to certain (non-strict) Lie 2-algebra structures, which include string Lie 2-algebra structures.Comment: 23 page