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

Notes

Instead of following Humphreys, Chapter VI for the Weyl character formula, we will be following these notes, bASEd on an improvement in the BGG proof of the Weyl character formula due to Victor Kac. This proof was found after Humphreys' book was written, and avoids the Theorem of Harish-Chandra.

PBW

We will make use of the Poincaré-Birkoff-Witt (PBW) theorem. This is proved in Humphreys, but the notes of Garrett give a useful discussion and another proof.

Sage

Sage has methods for doing calculations with Lie group representations. Typical problems involve things like decomposing tensor products into irreducibles, computing symmetric or exterior powers of representations, branching rules (i.e. restricting to a subgroup). Additionally Sage can do computations for crystals of representations, and for representations of affine Lie algebras. The following tutorial explores the toolkit.

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
 Solutions 3
May 1
Homework 4
Sec 7 (p.34) # 2,6,7a,b*
Sec 8 (p.40) # 5,8**,11
 Solutions 4
*Since Problem #7 in Sec 7 is long you may omit (c).
**Do this just for $\mathfrak{sl}(n)$.
May 8
Homework 5
Sec 9 (p.45) # 2,6
Sec 10 (p.54) # 2,6
 Solutions 5
May 15
Homework 6
Sec 6 (p.30) # 4
Section 10 (p.54) # 9,12*
Section 13 (p.72) # 9
 Solutions 6
*For Section 10 #12, the following stronger statement is true:
If $\lambda\in\mathfrak{C}(\Delta)$ and $w\in W$ such that $w\lambda\in\mathfrak{C}(\Delta)$ then $w=1$.
If you prove this stronger statement it may help with Section 13 #9.
May 22
Homework 7
Sec 12 (p.67) # 4
Sec 13 (p.72) # 7 (explain why this is relevant!)
Sec 22 (p.126) # 7
Sec 24 (p.141) # 1
Solutions 7
Homework 7 will be the last Homework.