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Momentum mappings and reduction of Poisson actions. (English) Zbl 0735.58004
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.]

Poisson Lie groups and dressing transformations have been studied by M. A. Semenov-Tian-Shansky [Publ. Res. Inst. Math. Sci. 21, 1237-1260 (1985; Zbl 0674.58038)] and the author and 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 ${G}^{*}$ 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 $P×{G}^{*}$, 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.

MSC:
 58B25 Group structures and generalizations on infinite-dimensional manifolds 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 58H05 Pseudogroups and differentiable groupoids on manifolds