具有高阶结构的线性代数及辛几何问题
发布时间：2018-08-06 15:59
【摘要】：高阶结构理论是当前数学和物理学中的热点课题之一,它在理论研究和实际应用中都有着非常重要的意义。有许多学者针对2?-分次线性代数、n-辛流形等问题展开了研究,并且取得了丰富成果。然而很少有人研究3?-分次结构和n-辛流形上的向量场问题。因此为完善线性代数和辛几何理论,本文主要针对3?-分次结构和n-辛流形上的向量场问题进行了研究。主要内容如下:第一章是绪论部分,主要介绍了高阶结构理论的研究背影及历史进程,并分析和总结了关于线性代数和辛几何方面国内外学者的研究结果。第二章是预备知识,分别介绍了2?-分次向量空间的定义与性质,以及辛流形上的向量场及其性质,从而为后续章节的理论研究和实际应用奠定了一个良好的基础。第三章利用G-分次结构理论,讨论了具有3?-分次结构的线性代数问题。首先得到了3?-分次向量空间、3?-分次代数、3?-分次李代数和3?-分次子空间的基本概念。其次给出一种由已知的3?-分次代数构造左对称的3?-分次代数的方法,同时提出两种构造3?-分次李代数的方法。最后给出3?-分次子空间的两个基本性质,并利用3?-分次李代数之间的同态与同构映射,得到了关于3?-分次李代数的同态与同构定理。第四章基于辛流形上的辛结构,提出了具有n-辛结构的辛几何问题。讨论了n-辛流形上的向量场,并结合李导数的性质,给出判定向量场为n-辛向量场的两个充分必要条件,得到了两个n-哈米顿向量场在括号积下仍为n-哈米顿向量场的结论。最后通过定义线性映射,得到了相应的短正合序列。第五章是对全文的总结,并提出了今后需要进一步研究的问题。
[Abstract]:High-order structure theory is one of the hot topics in mathematics and physics. It is of great significance in both theoretical research and practical application. Many scholars have studied the problems of 2-order linear algebras n- symplectic manifolds and obtained rich results. However, few people study the problem of 3-order structure and vector field on n-symplectic manifold. Therefore, in order to perfect the theory of linear algebra and symplectic geometry, this paper mainly studies the problem of 3-order structure and vector field on n-symplectic manifold. The main contents are as follows: the first chapter is the introduction, which mainly introduces the research background and historical process of higher-order structure theory, and analyzes and summarizes the research results of domestic and foreign scholars on linear algebra and symplectic geometry. The second chapter is the preparatory knowledge, which introduces the definition and properties of 2-order graded vector space, vector fields on symplectic manifolds and their properties, which lays a good foundation for the theoretical research and practical application of the following chapters. In chapter 3, we discuss the problem of linear algebra with 3-order structure by using the theory of G-graded structure. In this paper, we first obtain the basic concepts of the 3I-graded vector space and the 3H-graded algebras, and the basic concepts of the 3H-graded lie algebras and the 3I-graded subspaces. Secondly, a method of constructing left symmetric 3H-graded algebras from known 3H-graded algebras is given. At the same time, two methods of constructing 3G-graded lie algebras are proposed. In the end, two basic properties of the subspace of the 3-order graded lie are given, and the homomorphism and isomorphism theorems of the 3-order graded lie algebras are obtained by using the homomorphism and isomorphism mapping of the 3H-graded lie algebras. In chapter 4, based on the symplectic structure of symplectic manifold, the symplectic geometry problem with n- symplectic structure is presented. In this paper, the vector fields on n-symplectic manifolds are discussed. Combined with the properties of lie derivatives, two necessary and sufficient conditions for determining vector fields to be n-symplectic vector fields are given, and the conclusion that two n-Hamiltonian vector fields remain n-Hamiltonian vector fields in parentheses is obtained. Finally, by defining linear mappings, the corresponding short exact sequences are obtained. The fifth chapter is a summary of the full text, and puts forward the problems that need further study in the future.
【学位授予单位】：南昌航空大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O151.2
本文编号：2168230
[Abstract]:High-order structure theory is one of the hot topics in mathematics and physics. It is of great significance in both theoretical research and practical application. Many scholars have studied the problems of 2-order linear algebras n- symplectic manifolds and obtained rich results. However, few people study the problem of 3-order structure and vector field on n-symplectic manifold. Therefore, in order to perfect the theory of linear algebra and symplectic geometry, this paper mainly studies the problem of 3-order structure and vector field on n-symplectic manifold. The main contents are as follows: the first chapter is the introduction, which mainly introduces the research background and historical process of higher-order structure theory, and analyzes and summarizes the research results of domestic and foreign scholars on linear algebra and symplectic geometry. The second chapter is the preparatory knowledge, which introduces the definition and properties of 2-order graded vector space, vector fields on symplectic manifolds and their properties, which lays a good foundation for the theoretical research and practical application of the following chapters. In chapter 3, we discuss the problem of linear algebra with 3-order structure by using the theory of G-graded structure. In this paper, we first obtain the basic concepts of the 3I-graded vector space and the 3H-graded algebras, and the basic concepts of the 3H-graded lie algebras and the 3I-graded subspaces. Secondly, a method of constructing left symmetric 3H-graded algebras from known 3H-graded algebras is given. At the same time, two methods of constructing 3G-graded lie algebras are proposed. In the end, two basic properties of the subspace of the 3-order graded lie are given, and the homomorphism and isomorphism theorems of the 3-order graded lie algebras are obtained by using the homomorphism and isomorphism mapping of the 3H-graded lie algebras. In chapter 4, based on the symplectic structure of symplectic manifold, the symplectic geometry problem with n- symplectic structure is presented. In this paper, the vector fields on n-symplectic manifolds are discussed. Combined with the properties of lie derivatives, two necessary and sufficient conditions for determining vector fields to be n-symplectic vector fields are given, and the conclusion that two n-Hamiltonian vector fields remain n-Hamiltonian vector fields in parentheses is obtained. Finally, by defining linear mappings, the corresponding short exact sequences are obtained. The fifth chapter is a summary of the full text, and puts forward the problems that need further study in the future.
【学位授予单位】：南昌航空大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O151.2
【共引文献】
相关期刊论文 前1条
1 ;On higher analogues of Courant algebroids[J];Science China(Mathematics);2011年03期
相关博士学位论文 前2条
1 陈伟;范畴化理论中几个问题的研究[D];华南理工大学;2014年
2 王春月;Hom-超代数的结构[D];大连理工大学;2014年
相关硕士学位论文 前2条
1 陈丹华;Hom-李 2-代数[D];吉林大学;2013年
2 毕海波;关于作用群胚的几点讨论[D];吉林大学;2014年
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