In 2021, the topic of this course is Lie Algebras.

Outline

Lie algebras and Lie groups are fundamental concepts which arise in many areas of mathematics and theoretical physics. Lie groups arise as symmetry groups of continuous structures, just as finite groups arise as symmetries of discrete structures. The group of rotations of the unit sphere in R^3 is an example of a Lie group. Lie algebras and Lie groups arise in algebra, geometry, topology, differential equations and number theory. In physics Lie algebras and Lie groups are important in describing symmetries of physical systems and in gauge theories.

An important aspect of the theory is that to each Lie group there is a corresponding Lie algebra, which is to be thought of as a kind of infinitesimal linearisation of the Lie group. Due to their linear structure, Lie algebras are considerably easier to work with than Lie groups as they can be studied using tools from linear algebra. Furthermore, the structure of the Lie algebra almost completely determines the structure of the corresponding Lie group. In this way the study of Lie groups can largely be reduced to the study of Lie algebras.

The main goal of this course will be to study Lie algebras (and therefore, indirectly, Lie groups), understand their basic structure theory, and to obtain the classification of complex semisimple Lie algebras via root systems and Dynkin diagrams. The main emphasis in this course will be on the structure theory of Lie algebras, however we will also examine the relation between Lie algebras and Lie groups.

Topics

1. Basic concepts of Lie algebras and Lie groups.
2. The relation between Lie algebras and Lie groups.
3. Solvable and nilpotent Lie algebras, Engel's theorem, Lie's theorem.
4. The Killing form, semisimple Lie algebras, Cartan's criterion.
5. Cartan subalgebras, root space decompositions.
6. Root systems and their classification.
7. The classification of complex semisimple Lie algebras.

Learning Outcomes

On successful completion of this course, students will be able to:

1. Understand the definition of Lie algebras and related concepts.
2. Understand how Lie algebras arise from Lie groups.
3. Distinguish between nilpotent, solvable, semisimple and simple Lie algebras using effectively several criteria such as Lie's theorem, Engel's theorem, Cartan's criterion.
4. Recognise Cartan subalgebras and use them to obtain root space decompositions of complex semisimple Lie algebras.
5. Use root systems and Dynkin diagrams in order to classify the complex semisimple Lie algebras.

Prerequisites

The main prerequisite for this course is a good understanding of linear algebra at the level of Algebra II. Basic knowledge of group theory will also be helpful. Pure Math Topic A (Differential Geometry) will be useful in understanding the relationship between Lie groups and Lie algebras, but not mandatory.

This is a reading course. The course will make use of the following textbook, which is available online through the university library.

K. Erdmann and M. J. Wildon, Introduction to Lie algebras.

Supplementary notes on Lie groups and their relation to Lie algebras will also be provided.

There are many excellent references for Lie groups and Lie algebras. The most suitable ones for this course are:

K. Erdmann and M. J. Wildon, Introduction to Lie algebras (this will be our main reference for Lie algebras)

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory (a slightly more advanced reference for Lie algebras)

H. Samelson, Notes on Lie Algebras (covers much the same material as Humphreys)

A. K. Knapp, Lie Groups, Beyond an Introduction (contains much more material than we will cover in this course, but is very well-written)

W. Fulton, J. Harris, Representation Theory, A First Course (covers somewhat different material, but is heavily example based so may be useful as a learning resource for Lie algebras and Lie groups)

The course will have an active MyUni website.

This is a reading course. Students are expected to read and engage with the assigned reading material. There will be weekly face to face meetings in which students can discuss the material with the course coordinator. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.

1. Basic concepts of Lie algebras and Lie groups.
2. The relation between Lie algebras and Lie groups.
3. Solvable and nilpotent Lie algebras, Engel's theorem, Lie's theorem.
4. The Killing form, semisimple Lie algebras, Cartan's criterion.
5. Cartan subalgebras, root space decompositions.
6. Root systems and their classification.
7. The classification of complex semisimple Lie algebras.

An aggregate score of 50% is required to pass the course.

There will be five homework assignments distributed approximately once every two weeks across the semester. There will also be a final examination.

Homework assignments must be submitted on MyUni as pdf files. Failure to meet the deadline without reasonable and verifiable excuse may result in a significant penalty for that assignment.