MMath&Phys Mathematics and Physics

The course unit will deal with finite-dimensional Lie algebras, that is, with anticommutative algebras satisfying the Jacobi identity. These algebras have various applications in representation theory, mathematical physics, geometry, engineering and computer graphics. Lie theory is currently a very active area of research and provides many interesting examples and patterns to other branches of mathematics.

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.

To introduce students to some more sophisticated concepts and results of Lie theory as an essential part of general mathematical culture and as a basis for further study of more advanced mathematics.

On successful completion of the course students will have acquired: 

• define what is meant by a (non-associative) algebra over a field and verify whether such an algebra is anti-commutative and satisfies the Jacobi identity;
• define what is meant by a nilpotent and solvable Lie algebra and provide examples of nilpotent and non-solvable Lie algebras;
• state Engel's theorem and Lie theorem in the form involving irreducible representations and flags of subspaces;
• define the Killing form of a finite-dimensional Lie algebra and state Cartan's criterion;
• define the radical of a Lie algebra L, show that the factor-algebra of L by its radical is semisimple, and prove that a finite-dimensional Lie algebra over a field of characteristic zero is semisimple if and only if its Killing form is non-degenerate;
• define what is meant by a Lie-algebra direct sum and use the Killing form to characterise semisimple
• Lie algebras as direct sums of their simple ideals;
• define the Jordan--Chevalley decomposition in a finite-dimensional Lie algebra and prove that any finite-dimensional semisimple Lie algebra over a field of characteristic zero
• admits such a decomposition;
• state Weyl's theorem on complete reducibility and use it to classify all finite dimensional representations of the Lie algebra sl(2).
• define maximal toral subalgebras of semisimple Lie algebras and state the main properties of related Cartan decompositions;
• define the root system and the Weyl group of a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero.

• Definitions and first examples. Ideals and homomorphisms. [4]
• Nilpotent Lie algebras. Engel's theorem. [3]
• Solvable Lie algebras. Lie's theorem. Radical and semisimplicity. [3]
• The Killing form and Cartan's criterion. [2]
• The structure of semisimple Lie algebras. [3]
• Representation theory of the Lie algebras (2). [3]
• Toral subalgebras and root systems. Integrality properties. Simple Lie algebras and irreducible root systems. [4]

For MATH42112 the lectures will be enhanced by additional reading on related topics.

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.

• 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.

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  

Please see Blackboard / course unit related emails for any further updates.