【摘要】：高階結(jié)構(gòu)理論是當前數(shù)學和物理學中的熱點課題之一,它在理論研究和實際應用中都有著非常重要的意義。有許多學者針對2?-分次線性代數(shù)、n-辛流形等問題展開了研究,并且取得了豐富成果。然而很少有人研究3?-分次結(jié)構(gòu)和n-辛流形上的向量場問題。因此為完善線性代數(shù)和辛幾何理論,本文主要針對3?-分次結(jié)構(gòu)和n-辛流形上的向量場問題進行了研究。主要內(nèi)容如下:第一章是緒論部分,主要介紹了高階結(jié)構(gòu)理論的研究背影及歷史進程,并分析和總結(jié)了關(guān)于線性代數(shù)和辛幾何方面國內(nèi)外學者的研究結(jié)果。第二章是預備知識,分別介紹了2?-分次向量空間的定義與性質(zhì),以及辛流形上的向量場及其性質(zhì),從而為后續(xù)章節(jié)的理論研究和實際應用奠定了一個良好的基礎(chǔ)。第三章利用G-分次結(jié)構(gòu)理論,討論了具有3?-分次結(jié)構(gòu)的線性代數(shù)問題。首先得到了3?-分次向量空間、3?-分次代數(shù)、3?-分次李代數(shù)和3?-分次子空間的基本概念。其次給出一種由已知的3?-分次代數(shù)構(gòu)造左對稱的3?-分次代數(shù)的方法,同時提出兩種構(gòu)造3?-分次李代數(shù)的方法。最后給出3?-分次子空間的兩個基本性質(zhì),并利用3?-分次李代數(shù)之間的同態(tài)與同構(gòu)映射,得到了關(guān)于3?-分次李代數(shù)的同態(tài)與同構(gòu)定理。第四章基于辛流形上的辛結(jié)構(gòu),提出了具有n-辛結(jié)構(gòu)的辛幾何問題。討論了n-辛流形上的向量場,并結(jié)合李導數(shù)的性質(zhì),給出判定向量場為n-辛向量場的兩個充分必要條件,得到了兩個n-哈米頓向量場在括號積下仍為n-哈米頓向量場的結(jié)論。最后通過定義線性映射,得到了相應的短正合序列。第五章是對全文的總結(jié),并提出了今后需要進一步研究的問題。
[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.
【學位授予單位】：南昌航空大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O151.2

【共引文獻】

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相關(guān)博士學位論文 前2條

1 陳偉;范疇化理論中幾個問題的研究[D];華南理工大學;2014年

2 王春月;Hom-超代數(shù)的結(jié)構(gòu)[D];大連理工大學;2014年

相關(guān)碩士學位論文 前2條

1 陳丹華;Hom-李 2-代數(shù)[D];吉林大學;2013年

2 畢海波;關(guān)于作用群胚的幾點討論[D];吉林大學;2014年

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本文編號：2168230