李代数上的局部映射

论文总字数：23513字

摘 要

作为一种非结合代数，李代数是数学的重要研究领域。本文讨论李代数在其模上的2-局部导子，大体上可以分成两个部分。第一部分是对李代数的概念以及李代数的结构理论和表示理论的介绍，其中包括李代数理论的发展背景以及相关的研究现状的介绍。在这里，我们介绍了李代数理论中的诸多概念，比如李代数的定义，子代数，理想，同态与同构，可解李代数，幂零李代数，单李代数，半单李代数，导子，局部导子等。第二部分研究李代数上的局部映射。一些作者把李代数在经典意义下的导子推广到了局部导子和2-局部导子。在本文中，经典意义下李代数的导子被推广到了李代数在其模上的导子，局部导子和2-局部导子分别被推广到李代数在其模上的局部导子和2-局部导子。鉴于三维单李代数在李代数研究中的重要性，本文主要研究三维单李代数在其模上的2-局部导子，并且证明了，对于三维单李代数而言，在其任一模上的2-局部导子均是导子。

关键词：李代数；三维单李代数；导子；局部导子；2-局部导子

Abstract

As a sort of non-associative algebra, Lie algebra is an important research field of mathematics. In this paper, we investigate 2-local derivations on modules of 3 dimensional simple Lie algebras. The paper can be roughly divided into two parts. The first part is an introduction to the notions about Lie algebras and to the theory of structure and representation of Lie algebras, including a presentation of the background of the development of the theory of Lie algebras and the situations of related researches. It is presented here various notions about Lie algebras such as the definition of Lie algebra, subalgebra, ideal, homomorphism, isomorphism, solvable Lie algebra, nilpotent Lie algebra, simple Lie algebra, semi-simple Lie algebra, derivation and local derivation etc. The second part is devoted to the study of local mappings on Lie algebras. Some authors extend the derivations of Lie algebras in the classical sense to the local derivations and the 2-local derivations. In this paper, the derivations in the classical sense is further extended to the derivations on their modules, the local derivations and the 2-local derivations are extended to the local derivations and 2-local derivations of Lie algebras on their modules respectively. For the importance of three-dimensional simple Lie algebra in the studies of Lie algebras, we mainly study the 2-local derivations of three-dimensional simple Lie algebra. It is proved that, for the three-dimensional simple Lie algebra, every 2-local derivation on any module is a derivation.

KEY WORDS: Lie algebra, three-dimensional simple Lie algebra, derivation, local derivation, 2-local derivation

目 录

摘要……………………………………………………………………………………………………………………………Ⅰ

Abstract……………………………………………………………………………………………………………………Ⅱ

1. 引言………………………………………………………………………………………………………………1
2. 背景………………………………………………………………………………………………………………2

2.1 李代数背景……………………………………………………………………………………………2

2.2 局部导子背景 ……………………………………………………………………………………3

1. 李代数理论概述……………………………………………………………………………………………4

3.1 李代数 …………………………………………………………………………………………………4

3.2 子代数与理想 ………………………………………………………………………………………6

3.3 李代数的结构理论与表示理论 ……………………………………………………………7

3.4 导子与内导子…………………………………………………………………………………………8

1. 相关研究现状…………………………………………………………………………………………………9

4.1 李代数的发展及研究现状………………………………………………………………………9

4.2 局部导子的发展及研究现状…………………………………………………………………12

1. 三维单李代数的2-局部导子 ………………………………………………………………………16

5.1 一些预备知识…………………………………………………………………………………………16

5.2 李代数在模上的导子 ……………………………………………………………………………17

5.3 局部映射 ………………………………………………………………………………………………17

5.4 三维单李代数 ………………………………………………………………………………………17

5.5 三维单李代数表示的权 …………………………………………………………………………18

5.6 三维单李代数的2-局部导子 …………………………………………………………………18

5.7 的情形………………………………………………………………………………………18

5.8 主要定理的证明………………………………………………………………………………………19

1. 致谢…………………………………………………………………………………………………………………20

参考文献

第一章 引言

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