Extended Mirror Courant Algebroids And Dirac Structures

In recent years,Courant algebroids germs and Dirac structure have been active research subjects in the two university subjects of Poisson geometry and mathematical physics.They have not only excellent mathematical properties but also wide applications in the fields of mathematics and physics such as differential geometry in theoretical research and practical application.On the basis of Courant algebroids and Dirac structure,many scholars have studied Poisson Lie groups and lie bialgebroidss,and obtained rich achievements.However,few people have studied the problem of image Courant germs and Dirac structure expansion.So in this paper,we study some problems of image Courant algebroids and Dirac structure expansion.This paper consists of five chapters,the main contents are as follows:The first chapter is the introduction part,which mainly introduces the research background and development process of Courant algebroids embryo and Dirac structure.The second chapter is the preparatory knowledge,which introduces the basic theoretical knowledge needed by the research content of this paper.Including the definition of Poisson Lie group,Lie algebroids,lie bialgebroids embryo,manin triple,Courant algebroids embryo,Dirac structure and other related concepts,which laid a foundation for the theoretical research and algorithm implementation in the following chapters.In Chapter 3,we introduce extended mirror Courant Algebroids and their mirror Dirac structures on the vector bundle ?1(M).The extended mirror Courant bracket is the bracket corresponding to the standard extended Courant bracket on the vector bundle ?1(M)via the vector bundle automorphism exchanging R to R.This paper is to study the algebraic geometry structures under the extend mirror Courant algebroids.We have found that the morphisms of the extended mirror Courant algebroids are the twisted and extennded mirror Courant algebroids,which are their respective twisted morphisms.In Chapter 4,we give some interesting examples of mirror Dirac structures.In particular,we establish a one-to-one correspondence between Dirac-Jacobi structures coming from the extended Courant bracket[[·,·]]and the mirror Dirac structures coming from the extended mirror Courant bracket[[·,·]]m.which solves the problem in differential geometry The mysteries of integrable subgroup.The last chapter is suggested for the conclusions and some promising future researches.