MATHS 725 : Lie Groups and Lie Algebras
Science
2025 Semester Two (1255) (15 POINTS)
Course Prescription
Course Overview
It explores and develops the interaction between algebra, analysis, geometry, and topology and is extremely useful for those wishing to obtain a conceptual and practical understanding of many of the tools used in other areas of mathematics such as ordinary partial differential equations, partial differential equations as well as the applications of these things to, for example quantum mechanics.
Capabilities Developed in this Course
| Capability 3: | Knowledge and Practice |
| Capability 4: | Critical Thinking |
| Capability 5: | Solution Seeking |
| Capability 6: | Communication |
Learning Outcomes
- Understand that the subject combines ideas from analysis, geometry, topology and algebra. (Capability 3, 4, 5 and 6)
- Know how to design and execute proofs of theorems in this area. (Capability 3, 4, 5 and 6)
- Have a working knowledge of the basic tools of the exponential map and how it connects analytic properties with algebraic properties. (Capability 3, 4, 5 and 6)
- Have knowledge of how to combine tools and techniques from many different areas of mathematics in order to prove new results and obtain a heightened level of understanding. (Capability 3, 4, 5 and 6)
- Display a high level of knowledge of mathematical concepts. (Capability 3, 4, 5 and 6)
Assessments
| Assessment Type | Percentage | Classification |
|---|---|---|
| Assignments | 50% | Individual Coursework |
| Final Exam | 50% | Individual Examination |
| 2 types | 100% |
| Assessment Type | Learning Outcome Addressed | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||||||
| Assignments | ||||||||||
| Final Exam | ||||||||||
A minimum of 35% in the exam is required to pass the course.
Key Topics
- Preliminaries from differential geometry
- Lie groups
- The exponential map
- The adjoint map
- The Campbell–Baker–Hausdorff formula
- Lie subgroups
- Examples of Lie algebras
- Nilpotent and Solvable Lie Algebras. Radical
- Engel’s theorem
- Lie’s theorem
- The Killing Form
- Cartan’s Criteria. The Structure of Semisimple Algebras.
- Simple Lie Algebras
- Basics of Representation Theory
- Associative Enveloping Algebras. Poincare-Birkhoff-Witt Theorem
- Cartan Subalgebras
- Root Decompositions for Semisimple Lie Algebras
- Introduction to Root Systems
- Bases of Root Systems, Cartan Matrices and Dynkin Diagrams
Tuākana
Whanaungatanga and manaakitanga are fundamental principles of Tuākana Maths, a community of learning for Māori and Pasifika students taking mathematics courses. The Tuākana Maths programme provides workshops, drop-in times, and a space where Māori and Pasifika students can work alongside our Tuākana tutors and other Māori and Pasifika mathematics students.
Workload Expectations
Delivery Mode
Campus Experience
- Attendance is expected at scheduled activities.
- Lectures will be available as recordings if room has appropriate recording facilities.
- The activities for the course are scheduled as a standard weekly timetable delivery.
Learning Resources
Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).
Please remember that the recording of any class on a personal device requires the permission of the instructor.
Course books and notes will be provided on canvas.
Student Feedback
During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.
At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.
Your feedback helps to improve the course and its delivery for all students.
The student feedback was positive. The only proposed improvement was to include an indexing or table of content in the lecture notes for the second half of the course, which will be provided.
Academic Integrity
The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework, tests and examinations as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. A student's assessed work may be reviewed against electronic source material using computerised detection mechanisms. Upon reasonable request, students may be required to provide an electronic version of their work for computerised review.
Class Representatives
Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.
Copyright
The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.
You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.
Inclusive Learning
All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.
Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website http://disability.auckland.ac.nz
Special Circumstances
If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.
If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.
This should be done as soon as possible and no later than seven days after the affected test or exam date.
Learning Continuity
In the event of an unexpected disruption we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and your assessment is fair, and not compromised. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator, and if disruption occurs you should refer to the University Website for information about how to proceed.
The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.
Student Charter and Responsibilities
The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html.
Disclaimer
Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.
In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.