Picard groups in Poisson geometry

Details Picard groups in Poisson geometry Picard groups in Poisson geometry

2003-04-03

arxiv.org/abs/math/0304048v2

Moscow Math. J. 4 (2004), no. 1, 39-66.

Mathematics

Symplectic Geometry

26 pages, to appear in special issue of Moscow Math. J. in honor of Pierre Cartier

Scientific paper

We study isomorphism classes of symplectic dual pairs P <- S -> P-, where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres. For fixed P, these Morita self-equivalences of P form a group Pic(P) under a natural ``tensor product'' operation. We discuss this group in several examples and study variants of this construction for rings (the origin of the notion of Picard group), Lie groupoids, and symplectic groupoids.

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