Picard groups in Poisson geometry

Abstract

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.

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

References in corpus (1)

• The characteristic classes of Morita equivalent star products on symplectic manifolds

Cited by in corpus (7)

• Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey
• Poisson geometry and Morita equivalence
• The Picard groupoid in deformation quantization
• Part I, Free Actions of Compact Abelian Groups on C*-Algebras
• Poisson manifolds and their associated stacks
• Homology of formal deformations of proper etale Lie groupoids
• Lie II theorem for Lie algebroids via higher groupoids