On symplectic double groupoids and the duality of Poisson groupoids

Details On symplectic double groupoids and the duality of Poisson groupoids On symplectic double groupoids and the duality of Poisson groupoids

1998-08-03

arxiv.org/abs/math/9808005v1

Mathematics

Differential Geometry

21 pages. Sections 1 and 2 are revisions of sections 4 and 5 of dg-ga/9712013. The revisions consist of additional background

Scientific paper

We prove that the cotangent of a double Lie groupoid S has itself a double groupoid structure with sides the duals of associated Lie algebroids, and double base the dual of the Lie algebroid of the core of S. Using this, we prove a result outlined by Weinstein in 1988, that the side groupoids of a general symplectic double groupoid are Poisson groupoids in duality. Further, we prove that any double Lie groupoid gives rise to a pair of Poisson groupoids (and thus of Lie bialgebroids) in duality. To handle the structures involved effectively we extend to this context the dualities and canonical isomorphisms for tangent and cotangent structures of the author and Ping Xu.

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