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Acta Applicandae Mathematicae

, Volume 109, Issue 1, pp 101–135 | Cite as

Compatible Structures on Lie Algebroids and Monge-Ampère Operators

  • Yvette Kosmann-Schwarzbach
  • Vladimir Rubtsov
Article

Abstract

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the relationships between PN-, P Ω- and Ω N-structures. We then show that the non-degenerate Monge-Ampère structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dimensional manifolds, such Monge-Ampère operators give rise to generalized complex structures or generalized product structures on the cotangent bundle of the manifold.

Keywords

Graded Poisson brackets Lie algebroids Poisson structures Symplectic structures Nijenhuis tensors Complementary 2-forms Bi-Hamiltonian structures PN-structures PΩ-structures ΩN-structures Hitchin pairs Dorfman bracket Courant algebroids Generalized complex structures Monge-Ampère operators 

Mathematics Subject Classification (2000)

53D17 17B70 58J60 37K20 37K25 70G45 

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Centre de Mathématiques Laurent SchwartzÉcole PolytechniquePalaiseauFrance
  2. 2.Département de MathématiquesUniversité d’AngersAngersFrance

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