Topics in Geometry: Lie Groups and Lie Algebras

Syllabus for 2023 Lie Groups and Lie Algebras

Course name:

Course number:

MTH 675.

Instructor:

Tevian Dray

Course credits:

Class meeting times:

3 hours of lecture per week

Prerequisites:

Familiarity with linear algebra, including matrix mutiplication, change of basis, determinant, trace, eigenvalues and eigenvectors; Some exposure to abstract algebra, including vector spaces and groups; Comfort with differential calculus; Some acquaintance with complex numbers.

Course description:

This course will discuss the geometry of both Lie groups and Lie algebras, with an emphasis on the exceptional cases and their description in terms of the octonions. Lie groups are groups of continuous symmetries, generalizing the familiar notion of rotation groups; Lie algebras are their infinitesimal versions. Symmetry groups describe many physical phenomena, and Lie groups are widely used in physics, notably in the description of fundamental particles. In the late 1800s, Killing and Cartan classified the simple Lie algebras into 4 infinite classical families and 5 exceptional cases. The classical Lie algebras correspond to matrix groups over the reals, the complexes, and Hamilton's quaternions. In the 1960s, a unified description of the exceptional Lie algebras was given by Freudenthal and Tits in terms of the octonions, the largest of the four division algebras. The goal of this course is to describe the structure of the corresponding exceptional Lie groups, utilizing tools from both geometry and algebra. We will discuss the general structure of Lie algebras and Lie groups, the classification theorem, and the Freudenthal–Tits magic square of Lie algebras, culminating in a treatment of the largest of the exceptional Lie groups, $E_8$, in terms of the octonions.

Learning resources:

Textbook 1 (ebook available through OSU library): Lie Groups: A Problem-Oriented Approach via Matrix Groups (HP);
Textbook 2 (ebook available through OSU library): The Geometry of the Octonions (GO);
Textbook 3 (online only): The Geometry of the Exceptional Lie Groups (GELG).

Learning outcomes:

1) (none yet)

Course content:

1) TBA

See also course schedule.

Evaluation of student performance:

TBA

Course Expectations and Policies

TBA

Academic calendar:

All students are subject to the registration and refund deadlines as stated in the Academic Calendar: https://registrar.oregonstate.edu/osu-academic-calendar

Statement regarding students with disabilities:

Accommodations for students with disabilities are determined and approved by Disability Access Services (DAS). If you, as a student, believe you are eligible for accommodations but have not obtained approval please contact DAS immediately at 541-737-4098 or at http://ds.oregonstate.edu. DAS notifies students and faculty members of approved academic accommodations and coordinates implementation of those accommodations. While not required, students and faculty members are encouraged to discuss details of the implementation of individual accommodations.

Expectations of student conduct:

Students are expected to comply with the University code of conduct, available at https://beav.es/codeofconduct.

Student bill of rights

OSU has twelve established student rights. They include due process in all university disciplinary processes, an equal opportunity to learn, and grading in accordance with the course syllabus: https://asosu.oregonstate.edu/advocacy/rights

Reach out for success:

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