Pullback Dirac Structure Left Invariant Dirac Structure

The Dirac stracture for Lie bialgebroid (A, A*) is a subbundle L C A + A* , which is maximally isotropic with respect to symmetric bilinear form (,)+, whose section is closed under the bracket [, ]. The dual characteristic pairs of maximal isotropic subbundle is an important conception which is used to describe maximal isotropic subbundle. The discussion of pullback Dirac structures for Lie bialgebroid and lefe invariant Dirac structures on poisson groupoid is our purpose in this paper.In chapter one we introduce lie bialgebriod and related conception, and special situation do detailedly.In chapter two the Dirac structure is discussed clearly. We introduce the conception of the linear Dirac structure and similar Dirac structure in the first section. At the same time, we try to analysis the relations between the special cases of dirac structure and the poisson structure or symplectic structure, moreover the sufficiency and necessary condition in which the Dirac structure is integrable is given. The content of the section two is about the Dirac structure for Lie algebroid. In this section the maximal isotropic subbundle,the characteristic pairs and the dual characteristic pairs are introduced. And some related conclusions are quoted directly.The content in chapter three is main of this paper. At the first all we try to discuss the Lie algebroid morphism and Lie bialgbroicl morphism whose operations are analyzed and discussed. On the basis of this we discuss pullback Dirac structure for Lie bialgebroid clearly. We not only obtain the old conclusions but also generalize some of them. In the section two we study the left invariant Dirac structure on poisson groupoids.