MSc Pure Mathematics and Mathematical Logic

Lie algebras are abstract algebraic structures, like groups, but they are inherently linear. One way to interpret this is that any Lie algebra can be realised as a vector space of matrices with the algebra structure inherited from matrix multiplication. This means that we can utilise concrete tools from linear algebra when studying Lie algebras, as well as utilising abstract ideas like those applied to groups. Our focus will be on describing the basic structure theory of Lie algebras.

The language we use is the language of weights, which generalises the usual theory of eigenvalues from linear algebra. This is used to introduce the root space decomposition of a Lie algebra. We will compute several explicit examples of this decomposition. To enable you to compute more interesting examples you will learn to use the computer algebra system GAP, which can efficiently perform the necessary multiplication of matrices.

To develop the basic properties of the root space decomposition we will need to utilise two key tools: representation theory and the Killing form. Of upmost importance is the representation theory of the Lie algebra sl2, which will be treated in detail.

Students are not permitted to take more than one of MATH42112 or MATH62112 for credit or in an undergraduate programme and then a postgraduate programme, as the contents of the courses overlap significantly.

Lie algebras are a fundamental algebraic object arising in mathematics and physics. This unit aims to introduce students to the basic structure of Lie algebras and the key techniques involved in their study.

On successful completion of the course students will be able to: 

1. Provide and identify examples of Lie algebras such as abelian, solvable and semisimple Lie algebras.
2. Analyse the structure of a Lie algebra using the adjoint representation.
3. Construct weight space decompositions of representations and show how weight spaces are used in the representation theory of the Lie algebra sl2.
4. Apply the representation theory of sl2 to the root space decomposition of a Lie algebra.
5. Compute the Killing form and apply it to produce ideals of a Lie algebra.
6. Use the computer algebra system GAP to produce examples of Lie algebras and to demonstrate and explore the theory covered in course.

The course will be taught through three in-person contact hours consisting of lectures, tutorials, and computer lab sessions. In some weeks, as part of their independent study, students will engage with additional material provided through written notes and videos. Students will be encouraged to collaboratively solve problems online through the message board system Piazza.

Coursework: weighted 20%

Examination: weighted 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

The course notes will be self-contained. However, the following two books provide good background reading on the subject.

• Karin Erdmann and Mark J. Wildon, Introduction to Lie Algebras, Springer Undergraduate Mathematics Series, Springer-Verlag London Limited, 2006.
• J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer, 1972.

For further reading on linear algebra one can consult the following texts:

• Sheldon Axler, Linear algebra done right (third edition), Undergraduate Texts in Mathematics, Springer, Cham, 2015.
• Thomas S. Blyth and Edmund F. Robertson, Further linear algebra, Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd., London, 2002.
• Thomas S. Blyth and Edmund F. Robertson, Basic linear algebra, Springer Undergraduate Mathematics Series,Springer-Verlag London, Ltd., London, 1998.