Studying at York
Lie Theory - MAT00098M
- Department: Mathematics
- Credit value: 20 credits
- Credit level: M
- Academic year of delivery: 2023-24
- See module specification for other years: 2024-25
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 2023-24 |
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:
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Work with examples of Lie groups and Lie algebras, and explain the links between the two.
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State and use general properties of Lie algebras, including the notions of solvable and semisimple Lie algebras.
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Recognise and define properties of classical Lie groups.
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Recognise and work with complex representations of Lie algebras and Lie groups
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Use Weyl’s complete reducibility theorem
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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.
Indicative assessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Special assessment rules
None
Indicative reassessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 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
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W Fulton and J Harris, Representation theory: a first course, Springer
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B Hall, Lie groups, Lie algebras, and representations: an elementary introduction, Springer
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J E Humphreys, Introduction to Lie algebras and representation theory, Springer
- H Weyl, The classical groups: their invariants and representations, Princeton University Press