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Web page of the mastermath course
Semisimple Lie Algebras

(Spring 2013)


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: C3.115 and C3.116 (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 "Lie-Algebren" of Wolfgang Soergel which you can download from his homepage.

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.

Schedule: Wednesdays, week 6-17 through week 19-21 (starting date: February 6, 2013 and end date: May 22, 2013). No meeting on May 1.
Time: 10:15-13:00.
Location: Science Park Amsterdam, room F1.02 (Science Park 500, Amsterdam directions).

The program will be updated below on a weekly basis. We also provide recommended exercises every week. Homework is given on a weekly basis (the homework exercises will be listed below after each lecture). 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 the solution to Marcelo Goncalves de Martino, email: m.goncalvesdemartino AT uva.nl.

The homework will be marked and given back during class. The homework will contribute 20% to the final mark. The exam will be a takehome exam. It contributes 80% to the final mark. The dates for the takehome exam will determined in April.

Note: Sometimes the website of the course cannot be immediately updated on this location. In that case, the update will be given first on the following mirror website.

Program:

Week 6 (February 6): 1.1, 1.2 (not yet the symplectic and orthogonal algebras), 1.3 (not the notion of derivation), 1.4, 2.1, 2.2.
Read yourself: the last paragraph of section 2.1 about normalizers and centralizers.
Week 7 (February 13): 1.2 (symplectic and orthogonal algebras), 1.3 (derivations), link to Lie groups (see Warner's book for much more on this), 7.1, 7.2 (representation theory of sl(2,F)).
Week 8 (February 20): 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. See also section 1.4 in Soergel's lecture notes.
Week 9 (February 27): Lie's Theorem, Cartan's solvability criterion, Killing form, Jordan decomposition and its functorial properties (Humphreys II, 4 and 5.1 (not yet Lemma and Theorem 5.1); Soergel 1.5). Study Lemma 5.1 and its proof by yourself.
Week 10 (March 6): Killing form, semisimple Lie algebras, Casimir element, Schur's Lemma, Weyl's Theorem (Humphreys II 5,6 and Corollary of II 7.2).
Week 11 (March 13): Cartan subalgebras, root space decomposition (Humphreys II 8.1-8.3).
Week 12 (March 20): Continuation of root space decomposition, definition of root systems (Humphreys II 8.4, 8.5 and Humphreys III 9.1, 9.2 (not yet lemma 9.2)).
Week 13 (March 27): Enjoying the view (Compact Lie groups with trivial center versus semisimple complex Lie algebra's; outlook), Preservation of Jordan decomposition (Humphreys II 6.4), Root systems (Humphreys III.9) (We did not do Lemma III.9.4 or "root strings"; this is a reading assignment (see below)).
Week 14 (April 03): The root system: Bases, the Weyl group and simple reflections. (Humphreys III, chapter 9 and 10.1, 10.2, and the dfn. of reduced expressions and length from 10.3).
Week 15 (April 10): Weyl groups; construction and classification of root systems. (Humphreys III, Sections 10.3, 10.4 (but not yet Lemmas A,C,D), 11.1 (but not yet the Proposition), 11.2, 11.3, 11.4 (statement of the result, and introduction of the Gram matrix G(X) of a finite graph X having edge multiplicities in the set {0,1,2,3}).
We have been proving the following claims:
(Claim 1) If X is a graph underlying one of the diagrams of the list (L) on page 58, then G(X) is positive definite.
(Claim 2) If X~ is an extension of a graph X of the list (L) as given in the class, then det(G(X~))=0.
(Claim 3) If Y is a subgraph of X (i.e. the vertex set of Y is a subset of the vertex set of X, and the edge multiplicities in Y are bounded by those in X between corresponding vertices) and G(X) is positive definite then also G(Y) is positive definite.
(Claim 4) If X is connected and G(X) positive definite, then X appears in the list (L).
We have proved Claim 1 and Claim 2, and will continue to prove Claims 3 and 4 and the rest of the classification proof next week.
Week 16 (April 17): Humphreys Chapter III, Sections 11.4 and 12.1 (recalling and finishing the classification of irreducible root systems), Section 12.2.
Section 10.4 Lemmas A, C, Section 11.1 Proposition 11.1, and Subsections 13.1 and 13.2.
Week 17 (april 24): Humphreys IV Sections 14.1 and 14.2 (isomorphism theorems for simple Lie algebras) and V Sections 17.1 and 17.2 (tensor algebras, symmetric algebras and universal enveloping algebras).
May 1: NO LECTURE.
Week 19 (May 8): Section 17.2, 17.3, 17.5 and general background discussing categories, functors, natural transformations of functors, universal properties and left adjoint functors.
Homework:

The homework has to be handed in at the start of next week's lecture. You can also send the solutions by email. In that case, please email the solution to Marcelo Goncalves de Martino, email: m.goncalvesdemartino AT uva.nl.

The homework will be marked and given back during class. The homework will contribute 20% to the final mark. Please note that the homework exercises are of a different nature then the exercises of the final exam: the homework exercises are meant to practice with the notions you have just learned during the last lecture. The homework exercises are taken from the exercises in Humphreys' book unless indicated otherwise.
Week 6 (February 6): Exercise 3 of Chapter 1 and Exercise 5 of Chapter 2.
Week 7 (February 13): Exercise 1 of Chapter 2.
Week 8 (February 20) 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.
Week 9 (February 27): II.4 exercise 1, 3, and II.5 exercise 1.
Week 10 (March 6): II.6 exercise 6,7.
Week 11 (March 13): II.8 exercise 1 and 2, both only for L=sp(2l,F).
Week 12 (March 20): II.8 exercise 5 and 8, exercise 8 only for L=sp(2l,F).
Week 13 (March 27): III.9 exercise 6 (hand in), and study section III.9.4 (especially the Lemma and the root strings) by yourself
Week 14 (April 03): III.10 exercises 1 and 5.
Week 15 (April 10): III.10 exercises 9, 14; III.11 exercise 3. For the latter exercise you need to read by yourself in section 11.1, the text below the proof of the Proposition, in order to see an algorithm for constructing the root system in terms of the Cartan matrix.
Week 16 (April 17): 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.),
III.13 exercises 1,4.
Week 17 (April 24): IV.14 exercise 2 and V.17 exercise 2.
Week 19 (May 8): V.17 exercise 4 and 5.
Recommended exercises (all taken from Humphreys' book):

Week 6: Exercise 2,4,6,7,12 of Chapter 1 and exercise 9 of Chapter 2.
Week 7: Exercise 11 of Chapter 1 and exercise 3,4,5,7 of Chapter 7.
Week 8: I.3 exercise 1,2,3,4,7,8.
Week 9: II.4 exercise 5,6,7; II.5 exercise 5,6,7.
Week 10: II.6 exercise 1, 2, 4, 5.
Week 11: II.8 exercise 1,2 (case L=sp(2l,F) is the homework exercise), 4,6.
Week 12: II.8 exercise 9,10; III.9 exercise 1.
Week 13 (March 27): II.6 exercise 9; III.9 exercise 2,4,5.
Week 14 (April 03): III.10 exercises 3, 6, 7, and 8.
Week 15 (April 10): III.10 exercises 11,12; III.11 exercise 2,5
Week 16 (April 17): III.13 exercises 7,8,12.
Week 17 (April 24): V.17 exercise 1,3.
Week 19 (May 8): V.17 exercise 6.