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Lie Algebras & Lie Groups - MAT00065M

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  • Department: Mathematics
  • Module co-ordinator: Prof. Maxim Nazarov
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2022-23
    • See module specification for other years: 2021-22

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Additional information

Pre-requisite knowledge for MSc students: Knowledge of basic Linear algebra and properties of matrices. A first course in some abstract algebra (rings, fields, groups).

Module will run

Occurrence Teaching period
A Autumn Term 2022-23

Module aims

This module is designed as an “exit level” module for Masters-level undergraduate students and also as an introduction to objects which we be very important to many of the students who go on to further study in Algebra or Mathematical Physics. Lie algebras and Lie groups are fundamental objects of study across many disciplines of mathematics, especially the two just mentioned. Not only that, but the study of these objects is usually the first exposure students get to a “classification by root data”. In itself, the classification of complex simple Lie algebras was a highlight of mathematics in the 20th century, but subsequently such classifications have become ubiquitous, and it is difficult to overstate the importance of root data, Dynkin diagrams, etc. in modern mathematics. A final strand of this module is to develop the representation theoretic understanding of students. Again, representation theory is one of the most important facets of modern research mathematics.

Module learning outcomes

Subject content

  • Be familiar with basic examples of Lie groups (as groups of matrices) and Lie algebras
  • Be familiar with the link between a Lie group and its Lie algebra
  • Understand the classification of the finite-dimensional simple Lie algebras over the complex numbers
  • Understand the basics of the representation theory of some examples of Lie algebras (and Lie groups), including the Lie algebra sl_2(C).
  • Understand how Lie algebras appear in other branches of mathematics, particularly mathematical physics.

 

Academic and graduate skills

  • Students at this stage of the degree will have highly developed analytical and reasoning skills, which will be thoroughly tested by some of the advanced material in this module. The module will draw together ideas from many previous modules in a coherent body of material, allowing students to use their skills to synthesise and apply a large amount of what they already know, as well as furthering their knowledge of an important areas of mathematics.
  • For students going on to further study (e.g., a PhD in algebra or mathematical physics), exposure to this material will be of great help. For all students, by the end of the module they will be able to understand some of the open problems and active areas of research undertaken in the department (and all over the world). This “contact with the research frontier” is what the MMath/MSc course should provide students with.

Module content

 

Outline syllabus:

  • Basic examples of Lie groups and Lie algebras and the link between the two

  • Basic properties of solvable Lie algebras and Lie Theorem

  • Basic properties of semisimple Lie algebras

  • Classification of simple Lie algebras over the complex numbers by Dynkin diagrams

  • Representation theory of Lie algebras and Lie groups (mainly by examples)

Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Lie Algebras & Lie Groups		 2 hours 100

Special assessment rules

None

Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Lie Algebras & Lie Groups		 2 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 Harris, Representation theory: a first course, Springer, 1991 (S 2.86 FUL).
  • JE Humphreys, Introduction to Lie algebras and representation theory, Springer, 1972/1978 (S 2.89 HUM).
  • B. Hall, Lie groups, Lie algebras, and representations: an elementary introduction, Springer, 2003 (Electronic resource)


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.