Poisson 2-Groups

J un 2 00 3 Picard groups in Poisson geometry

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. Variants of this construction are also studied, fo...

Poisson Boundary of Discrete Groups

Picard Groups in Poisson Geometry

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. Variants of this construction are also studied, fo...

Picard Groups of Poisson Manifolds

For a Poisson manifold M we develop systematic methods to compute its Picard group Pic(M), i.e., its group of self Morita equivalences. We establish a precise relationship between Pic(M) and the group of gauge transformations up to Poisson diffeomorphisms showing, in particular, that their connected components of the identity coincide; this allows us to introduce the Picard Lie algebra of M and...

Lie-poisson Structure on Some Poisson Lie Groups

Poisson Lie groups appeared in the work of Drinfel'd (see, e.g., [Drl, Dr2]) as classical objects corresponding to quantum groups. Going in the other direction, we may say that a Poisson Lie group is a group of symmetries of a phase space that are allowed to "twist," in a certain sense, the symplectic or Poisson structure. The Poisson structure on the group controls this twisting in a precise w...