Integrability and reduction of Poisson group actions

Details Integrability and reduction of Poisson group actions Integrability and reduction of Poisson group actions

2007-10-30

arxiv.org/abs/0710.5753v2

Mathematics

Symplectic Geometry

20 pages, corrected misspellt preposition in the title

Scientific paper

In this paper we study Poisson actions of complete Poisson groups, without any connectivity assumption or requiring the existence of a momentum map. For any complete Poisson group $G$ with dual $G^\star$ we obtain a suitably connected integrating symplectic double groupoid $\calS$. As a consequence, the cotangent lift of a Poisson action on an integrable Poisson manifold $P$ can be integrated to a Poisson action of the symplectic groupoid $\poidd{\calS}{G^\star}$ on the symplectic groupoid for $P$. Finally, we show that the quotient Poisson manifold $P/G$ is also integrable, giving an explicit construction of a symplectic groupoid for it, by a reduction procedure on an associated morphism of double Lie groupoids.

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