Virasoro代数和A型扩张仿射李代数的结构及表示的相关研究

发布时间：2018-01-04 17:42

本文关键词：Virasoro代数和A型扩张仿射李代数的结构及表示的相关研究　出处：《湖北民族学院》2016年硕士论文　论文类型：学位论文

【摘要】：众所周知,Virasoro代数是结构和表示理论最简单却又极为重要的一类无限维李代数.由于它在李理论和理论物理上的重要应用,因此得到了许多数学家和物理学家的广泛关注.另一类较常研究的无限维李代数是Kac-Moody代数,它分为三类:有限型、仿射型和不定型.扩张仿射李代数是有限型和仿射型Kac-Moody代数的自然推广.Bruce N.Allison,Saeid Azam,Stephen Berman,Yun Gao,Arturo Pianzola等人在刻画扩张仿射李代数的扩张仿射根系时,提出了半格的概念,并由半格出发构造了一类以Jordan环面为坐标代数的A_1型的扩张仿射李代数和后来称之为TKK代数的一类代数.本文对Virasoro代数和1A型的扩张仿射李代数展开了一些讨论,具体来说,全文安排如下:第一章概述了Virasoro代数和扩张仿射李代数的相关背景、研究现状和研究意义.第二章首先回顾了Virasoro代数的定义和主要性质,接下来研究了无中心Virasoro代数的一类表示,最后对Irving kaplansky的文章进行一个细微的修正.本文第三章一开始温习了TKK李代数的一些基本概念和常用的构造方法,然后考虑了最小的单李代数sl_2(?)的极大子代数,最后重点对(?)中的包含一个固定Cartan子代数的极大子代数及其分类问题展开研究.通过分析,我们发现极大子代数的结构与TKK代数内部所含的半格结构有着密切联系,我们希望通过对极大子代数的研究促进对半格结构的进一步的了解,反过来应用于对其他类型的扩张仿射李代数的研究.
[Abstract]:It is well known that Virasoro algebra is one of the simplest but most important infinite dimensional lie algebras in the theory of structure and representation because of its important applications in lie theory and theoretical physics. Therefore, many mathematicians and physicists have paid close attention to it. Another kind of infinite dimensional lie algebra, which is often studied, is Kac-Moody algebra, which is divided into three types: finite type. Extended affine lie algebra is a natural generalization of finite type and affine type Kac-Moody algebra. Bruce N. Allisonn Saeid Azam. Stephen Bermang Yun Gaoke Arturo Pianzola and others proposed the concept of semilattice in characterizing the extended affine roots of extended affine lie algebras. From semilattice, we construct a class of extended affine lie algebras with Jordan torus as coordinate algebras and a class of algebras later called TKK algebras. In this paper, we discuss the Virasoro algebras and 1. The extended affine lie algebras of type A are discussed. In the first chapter, the background of Virasoro algebra and extended affine lie algebra are summarized. The second chapter reviews the definition and main properties of Virasoro algebras, and then studies a class of representations of Virasoro algebras without center. Finally, a minor correction is made to Irving kaplansky's paper. Chapter three introduces some basic concepts and construction methods of TKK lie algebra at the beginning of this paper. Then we consider the smallest simple lie algebra sl2n? Of the maximal subalgebra, the last key point is? The problem of maximal subalgebra containing a fixed Cartan subalgebra and its classification are studied. We find that the structure of maximal subalgebras is closely related to the semilattice structures contained in TKK algebras, and we hope to promote further understanding of semilattice structures through the study of maximal subalgebras. In turn, it is applied to the study of other types of extended affine lie algebras.
【学位授予单位】：湖北民族学院
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O152.5

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