The Generalization And Application Of Poisson Structure And Affine Group

In this paper, the phase space of generalized Hamilton system- Poisson manifold is studied, the Poisson structure and affine group are generalized and applied. The main contents of the paper are the following two parts:In the first part, Poisson structure is generalized and applied. Among other things, in section 1, some results about Poisson tensor on Poisson manifolds are discussed; In section 2, the differentiation-contraction operatorηin 1-form spaceΛ1 (P) on Poisson manifold P is defined, and the important proposition which is related with the differentiation- contraction operatorηis offered. That is, the necessary and sufficient condition that the vector field #αinduced by the 1-formαis symplectic vector field isηαβ= {α,β}. This proposition shows that the differentiation-contraction operatorηis the generalization of Poisson bracket { , } on 1-form spaceΛ1 (P); In section 3 and section 4, the application of the differentiation-contraction operatorηon Poisson manifold and algebroid is discussed, and the algebroid on manifold is constructed (example1).In the second part, the R n-affine group is generalized, and G -affine group on product manifold is offered. The properties related with G - affine group are obtained. On the basis of the former, Poisson affine group and its properties are induced naturally.