Poisson Groupoid And Its Role On The Manifold

In this paper, we particularly discuss the action of groupoids on manifolds and tlicir reduc- tiouis. As we know well, the actions of Lic groups on manifolds have important applications in physics. Hence it becotncs an important research content of diffcrcntial geometry. In sytnplectic geometry, for the study of the Hamiltonian action and syrnplcctic reduction we solve many prob- lcnis in symmetry Hamiltonian systems. Using Poisson action , sonic problems about quantum groups are solved. For the analog of the Lic group actions, we can consider the actions of the Lie groupoids on manifolds. If the action of a Lie group on a symplectic manifold has a coadjoirit equivariant momentum .1: M ? 鐍, then the action can be regarded to be the symplectic action of the syrnplcctic groupoid T#G on M. The theorem of the synmpLcctic action and its reduction was given by Mikami and Weinstein. In this paper, we find the connection among reduction symplcctic manifolds, and this syrnplcctic action is equivalent to the action of the symplcctic groupoid which derived from primary one on sonic submanifold. Because the groupoids actions are riot defined in the whole, we give the sufficient and necessary conditions of the symplcc- tic groupoids actions and the Poisson groupoids actions by using biscctions in groupoids. The conditions of the theorem are weakened with compare to the conclusion to be given by Liu zhang-jiu, Weinstein and Xu ping. We also get many properties about Poisson action, to unify with the conclusion of actions of Poisson Lie groups on Poisson manifolds in form. For Poissou actions, Lie bialgebroids on Poisson manifolds have connection with the tangent Lie bialgebroids of the Poisson groupoids. We obtained that there is the morphism of Lie bialgebroids between Lie bialgcbroids on Poisson manifolds and the tangent Lie bialgebroids of the Poisson groupoids. This conclusion is generalization of the result to be given by Liu zhang-jiu and Xu ping. We give tire reduction theorem of the actions of Poisson groupoids on Poisson manifolds. In the proof of the theorem we used inherence properties of the Poisson action, it is not the samc as the formalization proof before. It is more easy to hold the essences of the actions. Using the concept of the character pairs of Dirac structurc, we unity many kinds of integrable conditions of Dirac structures in each Lie bialgebroid, which include the Dirac structures on general manifolds, the Dirac structures on Poisson manifolds and the graphs of Hamniltonian operators. We use the idea to give ncw proofs of a. scrics of theorems in the paper ~3] of Liu zhang-jiu. The proofs become simple and clear. Since the actions of groupoids arc different with the actions of groups, thc homogeneous spaces of the groupoids actions are also different with the homogeneous spaces of groups actions. Hence, Liu zhang-jiu gave its new dcflnition. But there are weak spaccs to compare with the homnogcncous spaces, they arc kinds of spaces naturally. Using the connection between Poisson action and time morphism of Lie bialgebroids, and the notion of pull back Dirac structure, we discuss the reduction Poisson actions. The last, we study the deformations of Lie algebra arid Nijcnhuis operators, and discuss the relation between Nijenhuis tensors and the Poisson structures on Poisson manifolds , presym ii plcctic structurcs on presyniplectic structures, i.e., Poisson-Nijcnhuis manifolds and ~2-Nijcnhuis manifolds. Through to give the coniposahLc Nijenhuis opcrator pairs, we unify the two cases i...