Math 549: Lie Groups

Spring 2020

Instructor: Kristen Hendricks

Office: Hill Center 515

Office Hours: MW 4-5. Please let me know by email if you are coming. Also, feel free to make appointments for other times.

Email: kristen. hendricks at rutgers.edu


The revised syllabus for our now-remote course is here. A general overview of the rest of the term, including instructions for accessing class meetings via canvas, is here.

Location and Time

MW 1:40-3:00 in Hill 423.

Prerequisites

Real Analysis, Linear Algebra, and Elementary Topology or permission of the instructor.

Topics

This is a first course in Lie groups. We will discuss roughly the following.

This is probably a little too much in total; the last few topics on the list will be selected among based on student interest.


References

Primary references:

Other Books:

Other online resources:

Representation theory overview:

Differential geometry:

Algebraic topology:

Vector bundles:



Assignments

Homework will be posted here approximately weekly. (Because we start on a Wednesday, the first assignment corresponds to the first three lectures.) You should plan to hand in roughly half the homework exercises to be checked for completion. If I see no homework at all from you, you are at risk of getting an A-.

Exercises for Lectures 1, 2, and 3

Exercises for Lectures 4 and 5

Exercises for Lectures 6 and 7

Exercises for Lectures 8 and 9

Exercises for Lectures 10 and 11

Exercises for Lectures 12 and 13

Exercises for Lectures 14 and 15

Exercises for Lectures 16 and 17

Exercises for Lectures 18 and 19

Exercises for Lectures 20 and 21

Exercises for Lectures 22 and 23

Exercises for Lectures 24 and 25

Exercises for Lectures 26, 27, and 28

Notes

Here are the scanned lecture notes from our course. These notes are intended for students in the course to follow along with the lecture.

Lecture 1 Ref: Teleman Sections 1-15, Gruson and Serganova Chapters 1 and 2

Lecture 2 Ref: Teleman Sections 1-15, Gruson and Serganova Chapters 1 and 2

Lecture 3 Ref: Teleman Sections 1-15, Gruson and Serganova Chapters 1 and 2

Lecture 4 Ref: Lee Chapters 1 and 2

Lecture 5 Ref: Lee Chapter 5, Kirillov Chapter 2, Bump Chapter 5 and 7

Lecture 6 Ref: Bump Chapter 13, Hatcher Chapter 1.

Lecture 7 Ref: Lee Chapter 7, Kirillov Chapter 2 Sections 4 and 5

Lecture 8 Ref: Lee Chapter 7, Bump Chapters 6 and 7, Kirillov Chapter 3

Lecture 9 Ref: Lee Chapter 3 and 15, Bump Chapters 6 and 7, Kirillov 3.3-5

Lecture 10 Ref: Bump Chapter 8, Kirillov 3.1-2, Lee Chapter 15

Lecture 11 Ref: Bump Chapter 14 and 15, Kirillov 3.8-9, Lee Chapter 15

Lecture 12 Ref: Kirillov 4.1-4, Bump Chapter 1

Lecture 13 Ref: Kirillov 3.4-6, 4.5-6, Bump Chapter 1-2

Lecture 14 Ref: Bump Chapter 2, Kirillov 4.7

Lecture 15 Ref: Bump Chapter 2-4, 15, Kirillov 4.7

Lecture 16 Ref: Bump Chapter 3, Kirillov 4.7

Lecture 17 Ref: Bump Chapter 4, Kirillov 4.7

Lecture 18 Ref: Bump Chapter 12, Kirillov 4.8

Lecture 19 Ref: Kirillov 4.9

Lecture 20 Ref: Mitchell's notes here

Lecture 21 Ref: As previous lecture

Lecture 22 Ref: Bump Chapter 16, see also Spivak Chapter 9

Lecture 23 Ref: As previous lecture

Lecture 24 Ref: Bump Chapter 17

Lecture 25 Ref: Bump Chapter 18, see also Chapter 7 of Kirillov

Lecture 26 Ref: Bump Chapter 18, see also Chapter 7 of Kirillov