Undergraduate Course: Introduction to Lie Groups (MATH11053)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 11 (Year 5 Undergraduate)
||Availability||Available to all students
|Summary||This course provides an introduction to Lie groups (the general object responsible for smooth symmetries) and Lie algebras (their infinitesimal counterpart). A particular focus will be on compact Lie groups, including a discussion of their structure theory and classification.
The concept of symmetry is omnipresent in modern mathematics. Lie groups are the abstract generators of continuous symmetries, which arise in many contexts in geometry and physics. Important examples are provided by matrix groups, but the subject is strictly larger. Lie groups also always have Lie algebras associated with them, which encode infinitesimal symmetries. This course begins with a broad introduction to Lie groups and Lie algebras, starting from classical matrix groups. It then focuses on compact Lie groups, discussing their structure and ending with a classification.
The course will cover:
Matrix groups, Matrix Lie algebras. Matrix exponentiation and Baker-Campbell-Hausdorff formula.
SU(2) and SO (3)
Lie groups and associated Lie algebras. Adjoint actions. Lie subgroups and subalgebras. Coverings & quotients. Spin groups.
Semi-simple Lie algebras.
Compact Lie groups and their complexification.
Maximal tori, roots, weight lattices. Center & fundamental group.
Haar measure for (compact) Lie group. Killing form.
Cartan matrix & Dynkin diagram.
Classification of Dynkin diagrams, classification of semi-simple complex Lie algebras and compact connected Lie groups.
Information for Visiting Students
|Pre-requisites||Visiting students are advised to check that they have studied the material covered in the syllabus of each pre-requisite course before enrolling.
|High Demand Course?
Course Delivery Information
|Academic year 2023/24, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 20%. Examination 80%
||Hours & Minutes
|Main Exam Diet S1 (December)||Introduction to Lie Groups (MATH11053)||2:00|
On completion of this course, the student will be able to:
- Explain the basic structures of Lie groups and Lie algebras, and their various interplays.
- Derive the Lie algebra associated to a Lie group, in particular in the context of matrix groups.
- Indicate the particular structures arising for compact Lie groups, and illustrate these in basic examples.
- Use the classification of semi-simple Lie algebras in terms of Dynkin diagrams.
|Mark R. Sepanski - Compact Lie Groups, Springer, Graduate Texts in Mathematics Volume 235. (available online through library).|
Wulf Rossman - Lie Groups: An Introduction Through Linear Groups, Oxford.
Anthony W. Knapp - Lie Groups Beyond an Introduction, Birkhauser.
|Graduate Attributes and Skills
|Keywords||ILG,Lie groups,Lie algebras,symmetry,geometry,Lie
|Course organiser||Dr Pavel Safronov
|Course secretary||Mr Martin Delaney
Tel: (0131 6)50 6427