Study of Combinatorial Operator on Poisson Manifolds

Generalized Hamilton system is defined by generalized Poisson bracket, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion,. In the paper, Combinatorial operator η in 1-form space ( ) P Λ on Poisson manifold P is constructed and sufficient and necessary conditions that the vector field which induced by 1-form is symplectic vector field are offered, that is, Poisson bracket is a special type of combinatorial operator. On the basis of the above discussion, some properties about the combinatorial operator in Poisson manifold and groupoid are obtained. All the G -invariant 1-form induced by Poisson structure η construct the Lie subalgebra of ( ) 1 P Λ .