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Lie Theory - MAT00098M

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  • Department: Mathematics
  • Module co-ordinator: Prof. Maxim Nazarov
  • Credit value: 20 credits
  • Credit level: M
  • Academic year of delivery: 2024-25
    • See module specification for other years: 2023-24

Module summary

This module is an introduction to Lie algebras, Lie groups and their representation theory.

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Additional information

MSc students taking this module should be familiar with basic topological notions such as continuity and compactness. The module uses ideas covered in first courses in abstract linear algebra and group theory.

Module will run

Occurrence Teaching period
A Semester 1 2024-25

Module aims

This module is an introduction to Lie algebras, Lie groups and their representation theory.

Module learning outcomes

By the end of the module, students will be able to:

  1. Work with examples of Lie groups and Lie algebras, and explain the links between the two.

  2. State and use general properties of Lie algebras, including the notions of solvable and semisimple Lie algebras.

  3. Recognise and define properties of classical Lie groups.

  4. Recognise and work with complex representations of Lie algebras and Lie groups

  5. Use Weyl’s complete reducibility theorem

  6. Use highest weight theory for representations and calculate with explicit examples

Module content

Lie algebras and Lie groups are fundamental objects of study in several disciplines of contemporary Mathematics, especially in Algebra and Mathematical Physics. The study of these objects is usually the first encounter with “classification by root data”. In itself, the classification of complex simple Lie algebras was a highlight of Mathematics in the 20th century. Subsequently such classifications have become ubiquitous. It is difficult to overstate the importance of root data in Mathematics. Another major aim of this module is to gain understanding of Representation Theory. Again, this is one of the most important facets of modern research Mathematics.

Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Lie Theory		 3 hours 100

Special assessment rules

None

Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Lie Theory		 3 hours 100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

  • W Fulton and J Harris, Representation theory: a first course, Springer

  • B Hall, Lie groups, Lie algebras, and representations: an elementary introduction, Springer

  • J E Humphreys, Introduction to Lie algebras and representation theory, Springer

  • H Weyl, The classical groups: their invariants and representations, Princeton University Press


The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.