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DRPS : Course Catalogue : School of Mathematics : Mathematics

Undergraduate Course: Introduction to Lie Groups (MATH11053)

Course Outline
SchoolSchool of Mathematics CollegeCollege of Science and Engineering
Credit level (Normal year taken)SCQF Level 11 (Year 5 Undergraduate) AvailabilityAvailable to all students
SCQF Credits10 ECTS Credits5
SummaryThis 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.
Course description 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.
Weyl groups.
Cartan matrix & Dynkin diagram.
Classification of Dynkin diagrams, classification of semi-simple complex Lie algebras and compact connected Lie groups.
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: Honours Algebra (MATH10069)
Co-requisites Students MUST also take: Differential Geometry (MATH11235)
Prohibited Combinations Other requirements None
Information for Visiting Students
Pre-requisitesVisiting 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? Yes
Course Delivery Information
Academic year 2022/23, Available to all students (SV1) Quota:  None
Course Start Semester 1
Timetable Timetable
Learning and Teaching activities (Further Info) Total Hours: 100 ( Lecture Hours 22, Seminar/Tutorial Hours 5, Summative Assessment Hours 2, Programme Level Learning and Teaching Hours 2, Directed Learning and Independent Learning Hours 69 )
Assessment (Further Info) Written Exam 80 %, Coursework 20 %, Practical Exam 0 %
Additional Information (Assessment) Coursework 20%. Examination 80%
Feedback Not entered
Exam Information
Exam Diet Paper Name Hours & Minutes
Main Exam Diet S1 (December)Introduction to Lie Groups (MATH11053)2:00
Learning Outcomes
On completion of this course, the student will be able to:
  1. Explain the basic structures of Lie groups and Lie algebras, and their various interplays.
  2. Derive the Lie algebra associated to a Lie group, in particular in the context of matrix groups.
  3. Indicate the particular structures arising for compact Lie groups, and illustrate these in basic examples.
  4. Use the classification of semi-simple Lie algebras in terms of Dynkin diagrams.
Reading List
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.
Additional Information
Graduate Attributes and Skills Not entered
KeywordsILG,Lie groups,Lie algebras,symmetry,geometry,Lie
Course organiserDr Pavel Safronov
Course secretaryMr Martin Delaney
Tel: (0131 6)50 6427
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