Meeting days/times and credit: Lectures Monday and Wednesday, 930-1045. Five weeks, first class May 4. Room: Parker Hall TBA. Office
hour after class Monday and Wednesday 11-12. Credit variable, grade determined by HW, make a decent effort and you'll get an A.
Course Description:
A beautiful and ubiquitous topic in mathematics are simple Lie algebras; there are four infinite families A,B,C,D and five sporadic ones. They arise from simple Lie groups, and the classification requires basic group theory and linear algebra. Remarkably they also play a key role in physics, essentially because they are strongly related to symmetry. In this five week class we will cover a closely related topic: finite reflection groups, which mirror (ha) many of the key ingredients of the Lie algebra classification: we will study finite subgroups of the orthogonal group.
Schedule by week:
Lecture 1: Chapter 2: Finite groups in dimension two
Lecture 2: Chapter 2: Finite groups in dimension three
Lecture 3: Chapter 3: Fundamental domains
Lecture 4: Chapter 4: Coxeter groups: root systems
Lecture 5: Chapter 4: Coxeter groups: fundamental domains
Lecture 6: Chapter 5: Coxeter graphs part 1
Lecture 7: Chapter 5: Coxeter graphs part 2
Lecture 8: Chapter 5: Crystallographic condition
Lecture 9: Connection to Lie algebras
Lecture 10: Physics and the Standard Model
Grading: Your grade will be determined by homework scores; you'll get out of the class what you put in.
Academic Integrity: I encourage group work on the homework problems. This does not include copying each others solutions.
Textbook: Finite Reflection Groups. Copies are also available on the web.
Updated 3/1/26 (hks).