Teaching

Math 222

Meets: Tuesday and Thursday during the Spring 2026 semester from 12pm to 1:15pm at Science Ctr 309a.
Office hour: Tuesday 1:30-2:30 and by appointment (SC 238).

Course Assistants:

• Preston Bushnell (pbushnell@college.harvard.edu). Office hour: Thursday 7:00PM-8:30PM (Science Center 4th floor Math Lounge).
• Ari Krishna (akrishna@college.harvard.edu).

Course

This is a graduate course on Lie groups, Lie algebras and their representation. We will start with generalities about Lie groups and Lie algebras. We will then cover the classification of semisimple complex Lie algebras and their representations.

References

The main references will be:

• W. Fulton and J. Harris: Representation theory, A first course
• B. Hall: Lie groups, Lie algebras, and representations

Tentative schedule

• Lecture 1 (Tue, Jan. 27): Outline of the course
• Lecture 2 (Thu, Jan. 29): Lie groups, first examples
• Lecture 3 (Tue, Feb. 3): More examples
• Lecture 4 (Thu, Feb. 5): Generalities about Lie groups
• Lecture 5 (Tue, Feb. 10): Lie algebras
• Lecture 6 (Thu, Feb. 12): The Exponential map
• Lecture 7 (Tue, Feb. 17): Applications of the exponential map
• Lecture 8 (Thu, Feb. 19): The Baker–Campbell–Hausdorff formula
• Lecture 9 (Tue, Feb. 24): Preliminaries on representations of Lie groups and Lie algebras
• Lecture 10 (Thu, Feb. 26): Representations of \(\mathfrak{sl}_2\mathbb{C}\)
• Lecture 11 (Tue, Mar. 3): Semisimple Lie algebras, the Killing form
• Lecture 12 (Thu, Mar. 5): Complete reducibility
• Lecture 13 (Tue, Mar. 10): Representations of \(\mathfrak{sl}_3\mathbb{C}\)
• Lecture 14 (Thu, Mar. 12): Representations of \(\mathfrak{sl}_3\mathbb{C}\), continued
• Lecture 15 (Tue, Mar. 24): Midterm
• Lecture 16 (Thu, Mar. 26): Cartan subalgebras
• Lecture 17 (Tue, Mar. 31): Dynkin diagrams
• Lecture 18 (Thu, Apr. 2): The classification of Complex simple Lie algebras
• Lecture 19 (Tue, Apr. 7): Highest weight theorem
• Lecture 20 (Thu, Apr. 9): Verma modules
• Lecture 21 (Tue, Apr. 14): Universal enveloping algebras and the Poincaré-Birkhoff-Witt theorem
• Lecture 22 (Thu, Apr. 16): Proof of the highest weight theorem
• Lecture 23 (Tue, Apr. 21): Weyl character formula
• Lecture 24 (Thu, Apr. 22): Weyl character formula, continued
• Lecture 25 (Tue, Apr. 29):