NMAK19010U Introduction to Lie Algebras

Volume 2019/2020
Education

MSc Programme in Mathematics

Content

1) Lie algebras

2) Matrix Lie algebras

3) Some representation theory

4) Structure theory of Lie algebras

5) Cartan-Weyl basis

6) Classification of simple Lie algebras

Learning Outcome

At the end of the course the students are expected to have acquired the following knowledge and associated tool box:

  • the mathematical framework of Lie algebras, including basic examples in the form of matrix Lie algebras 
  • basic representation theory of Lie algebras
  • structure theory of Lie algebras, with main emphasis on semi-simple Lie algebras
  • Killing form, roots, and root space decomposition
  • the fundamental classification theorems for simple Lie algebras, including Serre's theorem 

 

Skills:

  • be able to use the fundamental results on Lie algebras to solve concrete mathematical problems
  • be able to work rigorously with representaions of Lie algebras, including decompositions in special cases 
  • be able to formulate and solve certain types of physical problems by applying the theory of Lie algebras and their representations

 

Competences: The course aims at training the students in formulating and handling specific mathematical problems, possibly inspired by physics, by use of the theory of Lie algebras and their representations.

Literature

Lecture notes will be made available through Absalon.

Knowledge of linear algebra, e.g., the course LinAlg.

Academic qualifications equivalent to a BSc degree is recommended.
8 weeks with 2x2 lectures and 2x2 exercise sessions per week.
  • Category
  • Hours
  • Exam
  • 20
  • Lectures
  • 32
  • Preparation
  • 122
  • Theory exercises
  • 32
  • Total
  • 206
Written
Oral
Continuous feedback during the course of the semester
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Two longer written assignments in week 5 and week 9, plus 5 smaller written assignments in weeks 2,3,4,6,7. The assignments will be graded on a scale from 0 to 100% and in order to pass the student must obtain at least 50% for each assignment.
Aid
All aids allowed
Marking scale
passed/not passed
Censorship form
No external censorship
One internal examiner.
Re-exam

30 minutes oral examination without aids. No preparation time. Several internal examiners at the re-exam.

Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.