The Linear Algebra And Symplectic Geometry Problems With Higher-order Structure

High-order structure theory is one of hot topics of study in the field of mathematics and physics. Many researchers studied the2?-graded linear algebra and n-plectic manifolds, and gratifying results have been achieved. However, few people study the problems of3?-graded structure and vector field on n-plectic manifolds. In order to improve the linear algebra and symplectic geometry theory, this paper studies the problems of3?-graded structure and vector field on n-plectic manifolds. The main contents of this paper are as follows:In the first chapter, the research background and development history about higher order structure theory are introduced, and overseas scholars’ research achievements about the linear algebra and symplectic geometry are analyzed and summarized. The main contents of the paper are introduced.In the second chapter, the definitions and properties of2?-graded vector space and vector field on symplectic manifolds are introduced, which lay a good foundation for theoretical research and practical application in subsequent chapters.In the third chapter, using the theory of G-graded structure, the linear algebra problems with3?-graded structure are discussed. First, the basic definitions of3?-graded vector space,3?-graded algebra,3?-graded Lie algebra and3?-graded subspace are obtained. Then, a method of constructing left-symmetric3?-graded algebra is given according to the known3?-graded algebra, meanwhile, the two methods of constructing3?-graded Lie algebra are proposed. Finally, the two basic properties of3?-graded subspace are given. Furthermore, the homomorphism and isomorphism theorems between3?-graded Lie algebras are obtained by using the homomorphism and isomorphism between3?-graded Lie algebras. The theory of G-graded structure is generalized, and the relevant conclusions are obtained.In the fourth chapter, the symplectic geometry problem with n-plectic structure is presented based on the symplectic structure on symplectic manifolds. The vector fields on n-plectic manifolds are discussed, and the two necessary and sufficient conditions for n-plectic vector field are given according to the property of Lie derivative,meanwhile, the result that two n-Hamiltonian vector field in integrated bracket is stillan n-Hamiltonian vector field is given. Finally, the short exact sequences are obtained by defining linear maps.In the fifth chapter, the summary of this paper is given and the problems for further study are put forward.