Double Lie algebroids and second-order geometry, II

Details Double Lie algebroids and second-order geometry, II Double Lie algebroids and second-order geometry, II

1997-12-22

arxiv.org/abs/dg-ga/9712013v1

Mathematics

Differential Geometry

36 pages, LaTeX

Scientific paper

We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We show that the Lie algebroid structure of an LA--groupoid may be prolonged to the Lie algebroid of its Lie groupoid structure; in the case of a double groupoid this prolonged structure for either LA--groupoid is canonically isomorphic to the Lie algebroid structure associated with the other; this extends many canonical isomorphisms associated with iterated tangent and cotangent structures. We also show that the cotangent of a double Lie groupoid is a symplectic double groupoid, and that the side groupoids of any symplectic double groupoid are Poisson groupoids in duality. Thus any double Lie groupoid gives rise to a dual pair of Poisson groupoids.

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