MAS8755 : Lie Groups and Lie Algebras
- Offered for Year: 2024/25
- Module Leader(s): Dr Thorsten Heidersdorf
- Owning School: Mathematics, Statistics and Physics
- Teaching Location: Newcastle City Campus
Semesters
Your programme is made up of credits, the total differs on programme to programme.
| Semester 1 Credit Value: | 10 |
| Semester 2 Credit Value: | 10 |
| ECTS Credits: | 10.0 |
| European Credit Transfer System | |
Aims
To introduce the concept of a Lie group and its Lie algebra, study the interplay between their analytical, topological and algebraic properties and develop the representation theory of Lie algebras.
Module summary
This course is an introduction to Lie theory, the study of continuous symmetries. Lie's original vision at the end of the 19th century was to develop a theory of symmetries of differential equations (described by what we call nowadays Lie groups). These Lie groups play a central role in many areas of mathematics and physics. In this course we will focus on matrix Lie groups which are easier to study, but still contain the most important examples such as the general linear, orthogonal and symplectic groups. Any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. The Lie algebra is an algebraic linearization of the Lie group and usually easier to work with. There is a close correspondence between the Lie algebra and the Lie group which we will develop in this course. This correspondence allows one to study the structure of Lie groups in terms of Lie algebras. Often times Lie groups and Lie algebras are studied via their representations, that is, the way they can act (linearly) on vector spaces. We will study the representation theory of classical matrix Lie groups and semisimple Lie algebras and apply it in many examples.
Outline Of Syllabus
Matrix Lie groups. The matrix exponential. The Lie algebra of a Lie group. Representations of Lie groups and Lie algebras. The exponential map and its properties. The Baker-Campbell-Hausdorff formula. Coverings and the fundamental group. Lie's theorems. Representations of sl(2,C). Representations of the classical semisimple Lie algebras of type ABCD. Complete reducibility. Root space decompositions and highest weight theory for classical semisimple Lie algebras
Teaching Methods
Teaching Activities
| Category | Activity | Number | Length | Student Hours | Comment |
|---|---|---|---|---|---|
| Scheduled Learning And Teaching Activities | Lecture | 10 | 1:00 | 10:00 | Problem Classes |
| Scheduled Learning And Teaching Activities | Lecture | 4 | 1:00 | 4:00 | Revision Lectures |
| Scheduled Learning And Teaching Activities | Lecture | 40 | 1:00 | 40:00 | Formal lectures |
| Guided Independent Study | Assessment preparation and completion | 30 | 1:00 | 30:00 | Completion of in course assignments |
| Guided Independent Study | Independent study | 116 | 1:00 | 116:00 | Preparation time for lectures, background reading, coursework review |
| Total | 200:00 |
Teaching Rationale And Relationship
The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.
Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Tutorials (within lectures) are used to discuss the course material, identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work.
Assessment Methods
The format of resits will be determined by the Board of Examiners
Exams
| Description | Length | Semester | When Set | Percentage | Comment |
|---|---|---|---|---|---|
| Written Examination | 90 | 1 | A | 40 | N/A |
| Written Examination | 90 | 2 | A | 40 | N/A |
Other Assessment
| Description | Semester | When Set | Percentage | Comment |
|---|---|---|---|---|
| Prob solv exercises | 1 | M | 3 | Problem-solving exercises assessment |
| Prob solv exercises | 1 | M | 3 | Problem-solving exercises assessment |
| Prob solv exercises | 1 | M | 4 | Problem-solving exercises assessment |
| Prob solv exercises | 2 | M | 3 | Problem-solving exercises assessment |
| Prob solv exercises | 2 | M | 3 | Problem-solving exercises assessment |
| Prob solv exercises | 2 | M | 4 | Problem-solving exercises assessment |
Formative Assessments
Formative Assessment is an assessment which develops your skills in being assessed, allows for you to receive feedback, and prepares you for being assessed. However, it does not count to your final mark.
| Description | Semester | When Set | Comment |
|---|---|---|---|
| Prob solv exercises | 1 | M | Problem Solving Exercises Formative Assessment |
| Prob solv exercises | 2 | M | Problem Solving Exercises Formative Assessment |
Assessment Rationale And Relationship
A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.
Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.
The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.
The written examination is split into two parts due to the wish of the students.
Reading Lists
Timetable
- Timetable Website: www.ncl.ac.uk/timetable/
- MAS8755's Timetable