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.


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