辛群與辛代數

Abstract

李群是一種附有群結構的流型G，使得其群運算為平滑的。G在單位元素上的切空間稱為G的李代數。在本論文，我們將運用線性代數來探討矩陣群。這些是由方陣所組成的群，所以某些李群的特徵可以做更直接的計算。利用這些結果，我們將探討一種稱為辛群的矩陣群的結構理論。內容編排如下：第一章，我們介紹矩陣群以及它的一些性質。如同在李理論中常見的例子，我們在第二章探討它的李代數。接著在第三章考慮矩陣群的拓樸與代數性質。在第四章介紹辛群。然後討論在第五章的卡當子代數以及它的根系統在第六章。

A Lie group is a manifold G along with a group structure, such that the group operations are smooth. The tangent space of G at its identity is called the Lie algebra of G. In this thesis, we use linear algebra to study the matrix groups. These are groups formed by square matrices, and so certain feature of Lie groups can be computed more directly. Using these results, we study the structure theory of a type of matrix groups called the symplectic group. The contents of this thesis are arranged as follows : In Chapter 1, we introduce the matrix groups and some of their properties. As is often the case in Lie theory, we study their Lie algebras in Chapter 2. Then we consider the topological and algebraic properties of the matrix groups in Chapter 3. In Chapter 4, we introduce the symplectic group. This is followed by the study of its Cartan subalgebra in apter 5, and its root system in Chapter 6.