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Zbl 0735.58004
Lu, Jiang-Hua
Momentum mappings and reduction of Poisson actions. (English)
[A] Symplectic geometry, groupoids, and integrable systems, Sémin. Sud- Rhodan. Geom. VI, Berkeley/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 20, 209-226 (1991).

[For the entire collection see Zbl 0722.00026.]\par Poisson Lie groups and dressing transformations have been studied by {\it M. A. Semenov-Tian-Shansky} [Publ. Res. Inst. Math. Sci. 21, 1237-1260 (1985; Zbl 0674.58038)]\ and the author and {\it A. Weinstein} [J. Differ. Geom. 31, No. 2, 501-526 (1990; Zbl 0673.58018)]. An action on a Poisson manifold $P$ is said to be tangential if it leaves the symplectic leaves in $P$ invariant. In the present paper the author gives for such an action a Maurer-Cartan type criterion for them to be Poisson and proves that the dressing actions on a Poisson Lie group are Poisson actions. Afterwards a momentum mapping for a general left (resp. right) Poisson action is defined as a map from $P$ into the dual Poisson Lie group $G\sp*$ with certain properties and is shown that every Poisson action on a simply connected symplectic manifold has a momentum mapping. Finally the author defines the semi-direct product Poisson structure on $P\times G\sp*$, associated with a right Poisson action of $G$ on $P$, which is used in his Ph. D. Thesis (Univ. California, Berkeley) to construct symplectic groupoids for affine Poisson structures on Lie groups.
[F.C.Klepp (Timişoara)]

MSC 2000:: *58B25 Group structures and generalizations on infinite-dim. manifolds
37J99 Finite-dimensional Hamiltonian etc. systems
58H05 Pseudogroups on manifolds

Keywords: Maurer-Cartan type criterion; Poisson Lie group; Poisson actions; momentum mapping

Citations: Zbl 0688.58012; Zbl 0722.00026; Zbl 0674.58038; Zbl 0673.58018

Cited in: Zbl 0877.58025

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