Manin Triples for Lie Bialgebroids
| dc.creator | Liu, Zhang-Ju | |
| dc.creator | Weinstein, Alan | |
| dc.creator | Xu, Ping | |
| dc.date | 1995-08-28 | |
| dc.date | 1997-02-28 | |
| dc.date.accessioned | 2026-07-07T08:59:08Z | |
| dc.date.available | 2026-07-07T08:59:08Z | |
| dc.description | In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does not satisfy the Jacobi identity except on certain subspaces. In this paper we systematize the properties of this bracket in the definition of a Courant algebroid. This structure on a vector bundle $E\rightarrow M$, consists of an antisymmetric bracket on the sections of $E$ whose ``Jacobi anomaly'' has an explicit expression in terms of a bundle map $E\rightarrow TM$ and a field of symmetric bilinear forms on $E$. When $M$ is a point, the definition reduces to that of a Lie algebra carrying an invariant nondegenerate symmetric bilinear form. For any Lie bialgebroid $(A,A^{*})$ over $M$ (a notion defined by Mackenzie and Xu), there is a natural Courant algebroid structure on $A\oplus A^{*}$ which is the Drinfel'd double of a Lie bialgebra when $M$ is a point. Conversely, if $A$ and $A^*$ are complementary isotropic subbundles of a Courant algebroid $E$, closed under the bracket (such a bundle, with dimension half that of $E$, is called a Dirac structure), there is a natural Lie bialgebroid structure on $(A,A^{*})$ whose double is isomorphic to $E$. The theory of Manin triples is thereby extended from Lie algebras to Lie algebroids. Our work gives a new approach to bihamiltonian structures and a new way of combining two Poisson structures to obtain a third one. We also take some tentative steps toward generalizing Drinfel'd's theory of Poisson homogeneous spaces from groups to groupoids. | |
| dc.description | 24 pages, LaTeX2e (minor corrections, added section at end), final version of paper to appear in J. Diff. Geom | |
| dc.identifier | https://arxiv.org/abs/dg-ga/9508013 | |
| dc.identifier | http://arxiv.org/abs/dg-ga/9508013 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/147600 | |
| dc.subject | Differential Geometry | |
| dc.subject | Symplectic Geometry | |
| dc.title | Manin Triples for Lie Bialgebroids | |
| dc.type | text |