Math 210C: Lie Groups and Lie Algebras
This course will have three texts:
- Humphreys, Introduction to Lie Algebras and Representation Theory
- Bump: Lie Groups (Second edition)
- Kac: Infinite-dimensional Lie algebras (Third edition)
All three texts are available on-line through the Stanford Libraries.
Our main text will be Humphreys, which we will read. However, an
important source of Lie theory is Lie groups, so we will also
discuss Lie groups in the Lectures. Many important results such
as the Weyl character formula apply equally well to representations
of Lie groups or Lie algebras, so we will discuss applications
to Lie groups as we study Lie algebras.
Lectures
- Tuesdays and Thursdays, 9:45-11:15 AM in Littlefield 103
Office Hours
- Tuesdays and Thursdays, 12:15-1:15 PM in 383U.
- Wednesdays 12-1 PM by Zoom. I will email you the link.
- Other times by appointment.
Canvas and Gradescope links
Lecture Notes
- Lie Groups and Lie Algebras.
- The Weyl Character Formula. The
proof of the Weyl Character formula in Humphreys followed
a proof due to Bernstein, Gelfand and Gelfand (BGG). Since the
book was written in 1972, the BGG proof was simplified by
Victor Kac. So these notes give an alternative account
which you may read instead of Humphreys chapter VI.
This avoids use of Harish-Chandra's theorem.
The proof in Humphrey's of the Lemma in Section 4.3 (leading to
Cartan's criterion) is a little confusing. An alternative version
of the proof in this web page of Brian Weber's is less general
since it requires the ground field to be $\mathbb{C}$, but I
found it helpful:
Homework
Homework will be due on Mondays, and handled through
Gradescope.
- Homework 1: (Due Monday, April 4) Humphreys Section 1.4 # 3,5,6,10,11
- For HW1, Problem 10 is optional. It is probably too hard to do at this point in the text.
You will have another chance to do it later. If you want to do it now, you may omit $D_3=A_3$ case.
- Homework 2: (Due Wednesday, April 13) Humphreys Section 2.3 (page 9) #1: also state what the
corresponding fact is for groups. #7. Section 3.3 (page 14): # 1,2,5,6. Section 4.3 (page 20):
# 1,5.
- Correction: the problems on page 20 are in Section 4.3, not Section 5.1.
- Homework 2 Solutions
- Homework 3: (Due Friday April 22) Humphreys Section 4.3 (page 20) #7. Section 5.4 (page 24) # 1,5.
Section 6.4 (page 30) # 1,6,7.
- Homework 3 Solutions
- Homework 4: (Due Friday April 29): Humphreys Section 7.2 (page 34): # 2 and 6; Section 8.5 (page 40) # 2, 8 and 11. For #2, you may assume Problem 1, and for #8, you may just do $\mathfrak{sl}_3$.
- Homework 4 Solutions
- Homework 5: (Due Tuesday, May 10): Humphreys Section 9 (page 45): # 8; Section 10 (page 54) # 4,6; Section 12 (page 68) # 5; Section 13 (page 71) # 2, 9.
- Homework 5 Solutions
- Homework 6: (Due Friday, May 20): Humphreys Section 14 (page 77) # 1; Section 20 (page 110) #3; Section 21 (page 116) #3; Section 22 (page 125) #2,7; Section 24 (page 141) #4.