【作者】 孙超；

【导师】 景乃桓；

【作者基本信息】 华南理工大学 ， 基础数学， 2022， 硕士

【摘要】 李代数理论是十九世纪七十年代兴起的理论,自引入以来发展迅速,并已在许多数学领域得到应用,其中包括群论、组合学、微分方程和不变理论等。本学位论文主要的研究的是关于型李代数的迹方程及相关组合恒等式。在论文的第一部分,我们给出了一些论文研究需要用到的基础知识。这一部分主要介绍了本文中所涉及到李代数知识,主要包括李代数的概念,一般线性李代数及其子代数的内容,Killing型,Casimir元素及Weyl定理,₇)的基本模等基本概念和一些需要用到的定理和命题等。论文第二部分是本论文的主要结果,我们基于矩阵理论推导出的关于7)（9）)的迹方程,导出了一些组合恒等式。并在后面给出了例子来展示我们是如何使用它们的。主要结果是证明了关于7)（9）)的伴随表示、基本表示和所有的7)（2）的不可约表示的迹方程。我们希望以此为例,能够基于现在的结果进行推广得到更一般的迹方程。更多还原

【Abstract】 The theory of Lie algebra,which was newly developed in the 1870 s,has developed rapidly since it was introduced and has been applied to many mathematical fields,including group theory,combinatorics,differential equation theory and invariance theory.The main focus of this dissertation is trace equation of Lie algebra of type and related combinatorial identities.In the first part of the paper,we give some basic knowledge used in the research.This part mainly introduces the knowledge of Lie algebra in this paper,including the concept of Lie algebra,the contents of general linear Lie algebra and its subalgebra,Killing type,Casimir element,Weyl theorem,the fundamental modules for Lie algebra of type and some theorems and propositions used in this paper.The second part is the main result of this paper.The trace equation of 7)(9))based on the matrix theory and some combinatorial identities were derived.As an application,we give an example.The main result is to prove the trace equation of the adjoint representation,the fundamental representation of 7)(9))and all irreducible representations of7)(2).We hope that the example can be used to obtain a more general trace equation based on the present result.更多还原