# 18.745 Introduction to Lie Algebras

• Meeting time: Tuesday/Thursday 2:30-4, Room E17-139
• Text: Alexander Kirillov, Jr., An Introduction to Lie Groups and Lie Algebras.

David Vogan, E17-442

dav@math.mit.edu

x3-4991

Office hours Monday 3:00-4:00 and Tuesday 12:00-1:30 (or any time).

Very approximate syllabus:

• 2/3-2/12 Introduction
• 2/19-3/10 General Lie algebras; Chapter 5 of text
• 3/12-3/19 Semisimple Lie algebras; Chapter 6
• 3/31-4/16 Classification; Chapter 7
• 4/23-5/14 Representations; Chapter 8
• Grading will be based on the work done on the problems assigned in class and posted here. Approximate plan is that problems assigned in class on Tuesday and Thursday will be due in class the following Tuesday; the total number of problem sets will be approximately ten. You can talk with others about the problems, but writing of solutions needs to be done entirely on your own.

Late problem sets will not be accepted in part because of the plan to post solutions immediately after they are due. If you have a reasonable excuse, we can work out an arrangement not to count a problem set.

Electronic submission is OK, but needs to happen half an hour before class (so that I can get a copy to the grader).

First problem set due in class Thursday, February 12.
First problem set solutions (posted after class 2/12).

Second problem set due in class Thursday, February 19. The definition of Sp(V) in this problem set was misstated (thank you Mr. Pace!); there is a correction here.
Second problem set solutions (posted after class 2/19).

Third problem set due in class Tuesday, February 24.
Third problem set solutions (posted after class 2/24).

Fourth problem set due in class Tuesday, March 3.
Fourth problem set solutions (posted after class 3/3).

Fifth problem set due in class Tuesday, March 10.
Fifth problem set solutions (posted after class 3/10).

Sixth problem set due in class Tuesday, March 17. Two problems added 5 pm Thursday 3/12, so there are now four.
Sixth problem set solutions (posted after class 3/17).

No homework over spring break.

Seventh problem set due in class Tuesday, April 7. This version (posted 5:00 p.m. Sunday March 29) corrects an error in the formulation of the first problem. (The conditions given originally did not characterize a representation uniquely.)
Seventh problem set solutions (to be posted after class 4/7).

Eighth problem set due in class Tuesday, May 5. This version (posted 11 a.m. Wednesday April 29) corrects three small things: the number on top is changed to 8; an unnecessary hypothesis that \alpha be positive is removed from Problem 2; and Problem 4 is made non-trivial by removing zero from the purported root system.
Eighth problem set solutions (to be posted after class 5/5).

Various supplementary notes may be posted here as well.

Here are notes on the representations of sl(2) completing what I started in class 3/19. These are meant also to be helpful for the seventh problem set.

Here is a set of notes on the definition of the free associative algebra generated by a vector space V.

Here is a set of notes on the proof of Cartan's criterion for solvability, as presented (sloppily) in class February 26.

Here is a set of notes on root systems, supplementing Chapter 7 of the text.

Here is an article by Helgason "A centennial: Wilhelm Killing and the exceptional groups," Math. Intelligencer 12 (1990), no. 3, 54-57; and notes from Helgason's IAP lecture on Lie theory. Here is another set of notes by Helgason on Lie theory.