I.The Basic Information of the Course
Course Number:202012420011
The English Name of the Course：Lie Algebras and Representation Theory
The Chinese Name of the Course：李代数及其表示理论
In-class Hours and Allocation(total class hours:32, classroom teaching: 28classhours, classroom discussion:4)
Credit(s):2
Semester:2
Applied Discipline(ProfessionalDegreeCategory)：Math
Course Object Oriented：AcademicDoctor
Evaluation Mode:Sample:Closed-bookExamination
Teaching Method:Seminar-styleTeaching
Course Opening Department:School of Mathematical Sciences
II.Prerequisite Course( Time New Rome 12 points bold font)
Linearalgebra、Abstract algebra.
III. The Objectives and Requirements of the Course
The course is abasiccoursefor studying Lie algebra theory, the aim of the course is tointroducethe studentsto the theory of semisimple Lie algebras over an algebrically closed field of characteristic 0 with emphasis on representations.By learning this course, the students should reach the following basic requirements:students should be familiar with the basic structure theory through the classicfication by Dynkin diagrams; use the theory they learn for further studying such as the classification over non-algebraically closed field, Lie algebra in prime characteristic, etc.
IV. The Content of the Course
Chapter 1Basic concepts(4 classhours)
1.1Definitions and first examples
1.2Ideals and homomorphisms
Course for Ideological and Political Education:Introduce the background of Lie algebras and works for mathematicians such as Killing、Cartan and Zhexian Wan.
Chapter 2Solvable and nilpotent Lie algebras（4 classhours）
2.1 Solvability
2.2 Nilpotency
2.3Jordan-chevalley decomposition
Course for Ideological and Political Education:The Chinese Remainder Theorem is a useful tool in number theory and also has proved useful in the study and development of modern cryptographic systems.TheChineseRemainderTheoremoriginated in the book “Sun Zi Suan Jing”, or Sun Tzu’sArithmeticClassic, by the Chinese mathematician Sun Zi, which has moved ahead of the other western countries.
Chapter 3 Semisimple Lie algebras (6 class hours)
3.1 Complete reducibility repersentationsA
3.2 Representations of sl(2)
3.3 Root space decomposition
Chapter 4 Root systems (6 class hours)
4.1 Simple roots and weyl groups
4.2 Classification theory
Chapter 5 Classification of semisimple Lie algebras(6 class hours)
5.1 Isomorphism theroem
5.2 Classification theory
Chapter 6 Representations (6 class hours)
6.1 Finite dimensional modules
6.2 Characters
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1. James Humphreys. Introduction to Lie algebras and representations theory(GTM9). Springer, New York,1973.
2.苏育才，卢才辉，崔一敏.有限维半单李代数简明教程.北京：科学出版社，2008.
3.万哲先. 李代数. 北京：高等教育出版社，2013.
B. Learning Resources (Time New Rome 12 points)
1.https://www.bilibili.com/video/av19977893/?redirectFrom=h5；
2.https://www.aiimooc.com/mall/preshow-htm-itemid-381.html
Outline Writer (Signature):
Leader in charge of teaching at the College (Signature)：
Date：
中文
