MMath&Phys Mathematics and Physics

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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.

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 have acquired: 

Provide and identify examples of Lie algebras such as classical, abstract, abelian, solvable and semi-simple Lie algebras.

Analyse the structure of a Lie algebra using the adjoint representation.

Summarise and explain the proofs of fundamental results covered in the course, such as, Lie’s theorem and Cartan’s Criterion.

Construct weight space decompositions of representations and summarise how weight spaces are used in the classification of the irreducible representations of the complex Lie algebra sl2.

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 and tutorials. 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 two books
• 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.
 

Provide good background on the subject. 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.