MATH 261: Lie Groups and Lie Algebras
Spring 2025, UC Davis

Lectures:	MWF 12:10-1:00pm in Young 192 CRN 44992
	Office hours: after each class
Instructor:	Anne Schilling, MSB 3222, aschilling@ucdavis.edu
Text:	Brian C. Hall, "Lie Groups, Lie Algebras, and Representations. An Elementary Introduction", Springer Graduate Texts in Mathematics Other very useful text: Willian Fulton, Joe Harris, "Representation Theory", Springer Graduate Texts in Mathematics James K. Humphreys, "Introduction to Lie Algebras and Representation Theory", Springer Graduate Texts in Mathematics
Prerequisites:	MAT 215A; MAT 240A; MAT 250A; MAT 250B; Or the equivalent, or consent of instructor.
Grading:	Homework: There will be bi-weekly homework sets due on Fridays. Registered students are requested to hand in written homework solutions via Gradescope before class on the due date. Presentation: In addition, we will discuss homework problems every second Friday in class. Each student is expected to present at least one problem during the quarter. The final grade is based on the written homeworks and the presentation.
Web:	http://www.math.ucdavis.edu/~anne/SQ2025/261.html

Introduction to Lie groups and Lie algebras at the graduate level, covering the following main topics:

Lecture 1: Matrix Lie groups and examples
Lecture 2: Orthogonal group, compactness
Lecture 3: Unitary group, connectedness
Lecture 4: Double cover: SU(2) and SO(3), simple connectedness, complexification
Lecture 5: Symplectic group, other bilinear forms
Lecture 6: Homework 1 discussion
Lecture 7: The matrix exponential
Lecture 8: One parameter subgroups
Lecture 9: Matrix logarithm and Lie algebras
Semisimple Lie algebras. Roots, Cartan subalgebras, Weyl group. Killing form

Homework 1 due April 11
Homework 2 due April 25