Manin triples for lie bialgebroids

Zhang Ju Liu, Alan Weinstein, Ping Xu

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329 Scopus citations


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 → 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 → 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 ⨁ 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.

Original languageEnglish (US)
Pages (from-to)547-574
Number of pages28
JournalJournal of Differential Geometry
Issue number3
StatePublished - 1997

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology


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