MTH 675/679: Topics in Geometry
Winter 2023
Lie Groups and Lie Algebras

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Tevian Dray

Office hours: WF 3:15–3:45 PM in Kidder 298A (and by appt via Zoom)

Phone: 737-5159

Email:

Class Meetings:

WF 4‐5:20 PM in Kidd 238

Texts:

• Lie Groups: A Problem-Oriented Introduction via Matrix Groups, by Harriet Pollatsek
  (available through OSU to read online here; errata available here)
• The Geometry of the Octonions, by Tevian Dray and Corinne A. Manogue
  (available through OSU to read online here; prepublication version available to read online here)
• The Geometry of the Exceptional Lie Groups by Tevian Dray
  (freely available online text)

Website: https://oregonstate.edu/~tevian/onid/MTH675

Check the announcements page frequently.

Course Evaluation:

Based on homework and class participation.

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, E₈, in terms of the octonions.

Prerequisites:

MTH 674 or the equivalent is strongly recommended as a prerequisite, as is a first (undergraduate-level) course in abstract algebra, but the course will be largely self-contained. Students with little or no prior knowledge of differential geometry who wish to take the class are encouraged to contact the instructor to discuss their background.

Students who have previously taken MTH 675 should register for this course as MTH 679.

Undergraduates may wish to consider registering for this course as MTH 405.

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. 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.

Student Conduct:

Student conduct is governed by OSU's Code of Student Conduct.

Academic Calendar:

All students are subject to the registration and refund deadlines as stated in the Academic Calendar.

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