*(English)*Zbl 0735.58004

[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 $P\times {G}^{*}$, 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.