Math 210C: Lie Groups and Lie Algebras

Office Hours

Introduction

The representation theory of Lie groups and Lie algebras has important applications in nearly all fields of mathematics.

Every Lie group $G$ has associated with it a Lie algebra $\mathfrak{g}$, and a representation of $G$ produces a representation of $\mathfrak{g}$.

Thus there are two approaches to Lie theory: to develop the representation theory of Lie groups, using analytic tools such as integration, or to develop the representation theory of Lie algebras, and then to deduce the representation theory of Lie groups as a biproduct. The key formulas such as the Weyl character formula can be proved either way.

But the representation theory of Lie algebras is in some ways simpler to develop, and also more general, so we will take that approach.

This course will use Humphreys' book Introduction to Lie algebras and Representation Theory as its primary reference in the early parts of the course. We will not follow Humphreys for the representation theory (Chapter VI) since simplifications in the proofs were found (by Kac) after the book was written. Lecture notes will be provided for this material.

Lectures

Homework

Homeworks will be due on Wednesdays. The Gradescope will close (for late submissions) on Fridays, after which solutions will be posted.

You can turn your solutions in via the Gradescope Link

April 10
Homework 1
Sec 1 (p.5) # 3,8,10
For #10 $B_2\cong C_2$ only
 Solutions 1
April 17
Homework 2
Sec 2 (p.9) # 1, 7
Sec 3 (p.14) # 1, 2*, 6
Sec 4 (p.20) # 1, 5
 Solutions 2
*For Section 2 #2, Humphreys ask you to determine maximal nilpotent ideals
for particular Lie algebras. You may omit this.
April 24
Homework 3
Sec 4 (p.20) # 7
Sec 5 (p.24) # 1,5
Sec 6 (p.30) # 6,7
May 1
Homework 4
Sec 7 (p.34) # 2,6,7a,b*
Sec 8 (p.40) # 5,8**,11
*Since Problem #7 in Sec 7 is long you may omit (c).
**Do this just for $\mathfrak{sl}(n)$.