Manin triples for Lie bialgebroids.

*(English)*Zbl 0885.58030In 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 \(E\to M\) consists of an antisymmetric bracket on the sections of \(E\) whose “Jacobi anomaly” has an explicit expression in terms of a bundle map \(E\to TM\) 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 algebroid \((A,A^*)\) 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.

For any Lie algebroid \((A,A^*)\) 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.

Reviewer: A.Weinstein (Berkeley)

##### MSC:

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

17B66 | Lie algebras of vector fields and related (super) algebras |

22A22 | Topological groupoids (including differentiable and Lie groupoids) |

53C99 | Global differential geometry |

58H05 | Pseudogroups and differentiable groupoids |