Compatible structures on Lie algebroids and Monge-Ampère operators

Yvette Kosmann-Schwarzbach; Vladimir Rubtsov

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arXiv · 0812.4838

Compatible structures on Lie algebroids and Monge-Ampère operators

Yvette Kosmann-Schwarzbach·Vladimir Rubtsov

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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 such composite 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.

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Yvette Kosmann-Schwarzbach, Vladimir Rubtsov. 2008-12-30. Compatible structures on Lie algebroids and Monge-Ampère operators. https://arxiv.org/abs/0812.4838

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