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Web page of the course
Semisimple Lie Algebras

(Spring 2011)


Teachers: Eric M. Opdam and Jasper V. Stokman
Emails: e.m.opdam (AT) uva.nl and j.v.stokman (AT) uva.nl
Tel.: 020-5255205 and 020-5255202
Room numbers: C4.161 and C4.162 (Science Park 904, Amsterdam).

This mastermath course (8 ECTS) is based on the book Introduction to Lie Algebras and Representation Theory of James E. Humphreys, Graduate Texts in Mathematics, 9, Springer Verlag.

Good additional material is the syllabus of Wolfgang Soergel which you can download from his homepage.

A short exposition on the relationship between Lie algebras and Lie groups is given in the lectures of week 6. For more background on Lie groups, you can for instance consult Chapter 3 of the book Foundations of Differentiable Manifolds and Lie Groups by Frank Warner, Graduate Texts in Math. 94, Springer Verlag.
Each wednesday afternoon we will update the homepage by adding the material treated in class and adding the homework exercises of the week.
NOTE: occasionally it may happen that the website is first updated through this mirror site of the webpage.
Schedule: Wednesday, 10:15-13:00, lecture room G4.15 (Science Park 904, Amsterdam), weeks 6-16, 18-21 (so first meeting is on february 9). In week 17 (April 27) there will be no lecture.
Homework: Homework exercises are given on a weekly basis (they will be listed on the webpage on wednesday afternoon). The homework has to be handed in at latest during next week's lecture. You can also send the solutions by email. In that case, please email it to both teachers. The homework will be marked and given back during class. The average of all the homework marks will determine a bonus of at most 2 to the final mark, provided that the mark for the takehome exam is larger or equal to 5.5.
Exam: The exam will be a takehome exam. The take home exam is now available. It should be handed in at latest on monday, June 13, 2011, before 6pm, either electronically (mail it to both teachers), or otherwise put it in the mailbox of one of the teacher (fourth floor C-building).
The takehome exam is corrected, if you would like to know your final grade please send an email to Jasper Stokman (email: j.v.stokman@uva.nl) before friday july 22.
The re-exam of the course will be an oral exam with one of the teachers. In that case the homework mark will be discarded.
Program (we refer below to Humphreys' book):

Week 6, february 9.
Treated: subsections I 1.1 - I 2.1.
Homework: I.1 exercise 3 and I.2 exercises 1 and 2 ( deadline: february 16).

Week 7, february 16.
Treated: subsection I 2.2 and subsections II 7.1 and 7.2 (not the corollary of 7.2).
Homework: I.2 exercise 4 and II.7 exercises 2 and 5.
REMARK: for exercise 5 you may assume that F is algebraically closed (although it is not necessary). In addition, you may use the explicit construction of the representations V(m) from the lecture on wednesday, instead of the construction from exercise 3 or 4 in II.7 as suggested in exercise 5 (deadline: february 23).

Week 8, February 23.
Treated: Semisimple and nilpotent endomorphisms (subsection II 4.2, text above Proposition; semisimplicity and complete reducibility of modules (essentially II 6 exercise 2)); Nilpotent and solvable Lie algebras: subsection I 3.1, I 3.2, I 3.3.
Homework: I.3 exercise 6.
HINT: One way of solving this exercise is as follows. Let J,K be nilpotent ideals in L and let J^0, J^1, J^2, ... and K^0, K^1, K^2, ... denote their lower central series. Prove that J^k and K^l are ideals of L for all k and l. Let I(k,l) be the intersection of J^k and K^l and put I(-1,l)=K^l and I(k,-1)=J^k. What can you say about [J,I(k,l)] and [K,I(k,l)]? Conclude that (J+K)^N=0 for sufficiently large N (deadline: March 2).

Week 9, March 2.
Treated: Lie's Theorem, Cartan's solvability criterion, Killing form, Jordan decomposition and its functorial properties (Humphreys II, 4 and 5.1 (not yet Theorem 5.1); Soergel 1.5).
Homework: II.4 exercise 1, 3, and II.5 exercise 1 (deadline: March 9). In addition: Study the proof of Lemma 4.3 (also see Soergel's Lemma 1.5.11) by yourself.

Week 10, March 9.
Treated: Killing form, semisimple Lie algebras, Casimir element, Schur's Lemma, Weyl's Theorem (Humphreys II 5,6 and Corollary of II 7.2).
Homework: II.6 exercise 6,7 (deadline: March 16).

Week 11, March 16.
Treated: Preservation of Jordan decompositions, root space decomposition (Humphreys II 6.4, II 8.1 up to proposition and Proposition 8.3 (a)).
Homework: II.6 exercise 4 and II.8 exercise 1 for L=sp(2l,F) (deadline: March 23).

Week 12, March 23.
Treated: II.6 exercise 5 and continuation of root space decomposition (Humphreys II 8.1-8.4 up to and including Proposition 8.4 (b)).
Homework: II.8 exercise 2 for L=o(2l,F) and II.8 exercise 4.
REMARK: for exercise 2 you may take the conclusion of II.8 exercise 1 for granted. You may interpret the second part of exercise 2 as the question how the h_alpha are expressed in terms of the standard matrix units (deadline: March 30).

Week 13, March 30.
Treated: Root space decomposition (Humphreys II, Proposition 8.4 (c-f) and section 8.5), Root systems (Humphreys III, section 9.1 (not the lemma), section 9.3 and section 9.4 (not the lemma).
Homework: II.8 exercise 11 and III.9 exercise 1 (deadline: April 6).
REMARK: for exercise 11 (section II.8) the root alpha is not equal to beta.
HINT: for exercise 1 (section III.9) analyze the minimal polynomial of the reflection.

Week 14, April 6.
Treated: Root systems and bases (Humphreys III, chapter 9 and section 10.1).
Homework: III.9 exercise 6, III.10 exercise 1. (deadline: April 13).

Week 15, April 13.
Treated: Weyl groups; simple reflections. (Humphreys III, chapter 10).
Homework: Read by yourself 10.4, Lemma's B--D; III.10 exercises 5, 6, 9. (deadline: April 20).

Week 16, April 20.
Treated: Classification and automorphisms of root systems. (Humphreys III, chapters 11, 12).
Homework: 1. III.11 exercise 3;
2. Read sections 12.1 and 12.2;
3. III.12 exercise 6, but you are only requested to decide if $-1\in W$ (by whatever means necessary) for the irreducible $\Phi$ of type other than $E_6$;
(Hint: If there are no nontrivial diagram automorphisms, use the theory of Section 12.2; for the remaining cases, use the explicit description of $\Phi$ and $W$ of Section 12.1.)
(deadline: May 4).

In week 17 (April 27) there will be no lecture.

Week 18, May 4.
Treated: correspondence semisimple Lie algebras and root systems, tensor algebras, symmetric algebras, universal enveloping algebras (Humphreys IV 14.1, V 17.1,17.2).
Homework: IV.14 exercise 2 and V.17 exercise 3 (deadline: May 11).

Week 19, May 11.
Treated: Poincare-Birkhoff-Witt Theorem, Serre's theorem, existence and uniqueness theorems for semisimple Lie algebras (Humphreys V 17.3-17.5, 18.1-18.4).
Homework: Read by yourself sections 17.4, 17.5 and 18.1; V.18 exercise 4 (deadline: May 18).

Week 20, May 18.
Treated: Standard cyclic modules and finite dimensional modules (Humphreys IV 13.1, VI 20, 21.1).
Homework: VI.20 exercises 6 and 7 (deadline: May 25).

Week 21, May 25.
Treated: Finite dimensional modules and Weyl character formula (Humphreys IV 13.2, VI 21.2, 22.5 and 24.1-24.3).