## 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 $$P\times G^*$$, 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.

### MSC:

 58B25 Group structures and generalizations on infinite-dimensional manifolds 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 58H05 Pseudogroups and differentiable groupoids

### Citations:

Zbl 0688.58012; Zbl 0722.00026; Zbl 0674.58038; Zbl 0673.58018