Lie Groups
Objectives and Course Description
Lie groups are smooth manifolds, which admit a group structure consisting of diffeomorphisms. They include many groups encountered in mathematics and physics: The group of invertible matrices, the group of isometries of Euclidean space, the group of symplectic matrices, the special linear group, the Lorentz group… Their prevalence is not a coincidence, since a very general Gleason-Montgomery-Zippin theory shows that any locally compact topological group without small subgroups is a Lie group. This resolved the famous Hilbert’s fifth problem. Indeed, this statement can be thought of as saying that a topological property of a group implies the existence of a smooth structure. Thus, the topology and geometry are intricately linked. Such links play a crucial role in the theory of Lie groups and inspire many results.
This course is about developing the basic theory of Lie groups and Lie algebras primarily through examples. We will learn how Lie groups and mappings between them can be fully described in terms of a finite dimensional vector space: the Lie algebra, and we will see a few ways this algebra can be constructed. The tools allowing us to do this are the exponential and logarithmic functions, and the Baker-Capell-Hausdorff formula. These can be defined easily for linear Lie groups (which are groups of matrices), and for general groups they involve the use of vector fields. Our goal will be to state the three main Lie theorems. At the end of the course, we also hope to discuss some Lie group representation theory, although this will be very limited.
HWs:
LEcture times
Lectures and Demo: Sylvester Eriksson-Bique
Monday+Tuesday 2.9-21.10.2024
Monday 12:15-14:00 MaD 380 (Lecture)
Monday 14:15-16:00 MaD 381 (Demo)
Tuesday 10:15-12:00 MaD 380 (Lecture)
Office hours: Sylvester Eriksson-Bique
Tentatively, Friday 15-16 in Ratkomo
Available upon request at other times.
Exam:
The preferred way of completing the course is to do a written project and presentation. If this does not work for you, an exam is organised 8-12 on Wednesday 23.10.
Syllabus
This is the first time I give this course and the first time it is given with the proposed topics. Thus the following schedule is only a rough estimate, and we may move a bit slower or faster. Contact me if the pace is not ok for you.
Week 1: Matrix Lie Groups, Exponential and Logarithm, Nilpotent and Diagonalizable Matrices
Week 2: Derivatives of Determinant, Recall manifold theory, and definition of Lie group.
Week 3: Properties of Lie groups, Compact, Connected and Simply connectedness, Covering Lie group
Week 4: Tangent space, Vector fields, Lie Brackets, Lie Algebra
Week 5: Baker-Campbell-Hausdorff Formula
Week 6: Lie correspondence, Using the covering
Week 7: Actions of Lie Groups, Nilpotent Lie groups, Carnot groups definition
Literature
- Lecture notes will be written during the course and the progress is available here:
- Brian C. Hall, Lie Groups, Lie Algebras, and Representations
- Enrico Le Donne, Lecture notes on Lie groups – from the differential view point
Additional references:
- Terence Tao, Notes for Lie Groups
- John Lee, Introduction to Smooth Manifolds
- Daniel Bump, Lie Groups
- V.S. Varadarajan, Lie Groups, Lie Algebras and Their Representations
Grading
The course can be completed in one of three ways.
- HW (50 %) and a 5-10 page written report (50%). (Minimum to pass: half of the points on all the HW.)
- HW (60 %) and an exam (40%) (Minimum to pass: one problem right.)
- Just the exam. (Minimum to pass: one problem right.)
Out of the above three options, the first is highly recommended.
You must complete either an exam or submit a written report.
If you do the written report, you must get at least 50 % of the HW credit. If you do an exam, you must get one problem substantially correct in the exam to pass.
If for a justified reason you are unable to complete some of the requirements, please contact me, and we may find a compensatory way.
The course is 5 cr, and graded 0-5. As a base level, I will use the following rubric, which however can be adjusted in a way that increases grades.
90 %, 5
80 %, 4
70 %, 3
60 %, 2
50 %, 1
<50 %, 0/Fail
An excellent written report can compensate for lost points in the homework.
HW
The HW is worth 10+bonus points, and consist of a written portion (5pt), a group portion (5pt).
Each student should complete the written portion on their own, and think ahead of time about the group portion. You should attempt to solve the group portion, and write at least a sketch of your ideas – but you can complete the solution in the sessions. In the demo sessions, the group problems are discussed together.
HW should be returned in the demo sessions and to get the credit for the group portion, you should participate in the discussions. If you are unable to participate, you may submit written solutions to all problems which are graded on completeness and correctness.
At the discretion of the grader, only some problems are graded on correctness and the rest on completeness.
Generally, no late assignments will be accepted. If you are sick, or otherwise unable to complete an assignment, please contact the instructor in advance, and discuss alternate options.
The lowest HW score is dropped and ignored in the grading.
Inclusivity
The course should be an encouraging and open space for discussion, learning and the exchange of ideas. Every participant should be able to join fully and comfortably. If you find challenges with this, or if any course policy is difficult for you, please discuss these with the instructor. Also bear in mind, that the university has resources and support in the event of harassment or of personal difficulties. I can assist you in identifying such resources.
If you need a religious exemption, or other compelling exemption, please contact me at least 1 week prior to the requests effect. I may not be able to accommodate requests that arrive late.
Written Project ideas
You should decide on a project by the end of Week 3 and report the project to the professor. I will follow up on this.
You should feel free to contact me for help or discuss the project in office hours. Generally speaking, it should include at least one result, one proof, some reflection/citations, one example and one definition. You can choose the topic to be anything which at least peripherally connects to the course.
1. Frobenius theorem
2. Representations of SO(n)
3. Symmetric spaces
4. Haar Measures and their construction
5. Carnot groups
6. Heisenberg group
7. Curvature of Lie group
8. Semi-simple Lie groups
9. Nilpotent Lie groups
10. Subriemannian manifolds
11. Engel’s theorem
12. Montgomery-Zippin-Gleason theorem and its explanation
13. Jordan’s theorem (https://terrytao.wordpress.com/wp-content/uploads/2012/03/hilbert-book.pdf)
14. Bieberbach’s theorem / Discrete groups / Crystallographic groups
15. Simply connectedness of Lie groups (with some proofs)
16. Example of Lie group which is not a Matrix Lie group
17. Complexification of Lie algebra (and more on Lie algebras)
18. Universal Enveloping Algebra
19. Complex Lie groups and Lie algebras
20. Regular representation of compact Lie group
21. Peter-Weyl theorem
22. Pontryagin duality
23. P-adic numbers and more general abelian Lie groups
24. Shur’s lemma for representations
25. Root system
36. Weyl group
27. Real and complex Hyperbolic space
28. Classification of semisimple Lie groups (some basic results, not the whole thing)
29. An in depth study of some lie groups (e.g. SO(n), SL(n)). E.g. simply connectivity, Lie algebra, definitions.
30. Pansu’s differentiation theorem
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