Web page of the mastermath course
Semisimple Lie Algebras

(Spring 2015)


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: F3.04 and F3.03 (Science Park 105-107, 3rd floor, 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. We will treat Chapters 1-14,17,18,20,21,25 about the structure theory and representation theory of semisimple Lie algebras.

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 6-21 (starting date: February 6, 2015 and end date: May 20, 2015).
Time: 10:15-13:00.
Location: Science Park 904 Amsterdam, room C0.110 (directions).

The program will be updated below on a weekly basis. We also provide recommended exercises every week. Homework is given on a regular 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 to j.v.stokman@uva.nl.
Exam and mark: The exam will be a written exam on all the treated material during the course except for Section 25 of Humphrey's book. It is scheduled on Wednesday June 3, 10:00-13:00 at UvA-IWO 4.04A blauw (Meibergdreef 29, Amsterdam Zuid-Oost).The re-exam will take place Wednesday June 24, 10:00-13:00 in room D1.116 (Science Park 904, Amsterdam).

The final marks are known. If you did not get your final mark by email yet then we did not have your email-address (or it was unreadable on your answer sheet). In this case please send an email to j.v.stokman AT uva.nl to inquire about your final mark.

The recommended exercises (see below) give a good idea of the type of questions you can expect on the written exam. From the following results of the book you should be able to reproduce the proof: Prop. 3.1, Prop. 3.2, Lemma 6.1, Lemma 7.1, Lemma & Prop. 8.1, Lemma 10.2B and its corollary, Prop. 11.1, Lemma 13.3A, Prop. 14.1, Prop. 18.1, Lemma 20.1, Cor. 20.2, Lemma 21.2.

If the mark for the written exam is at least a 5, then the written exam contributes 70% to the final mark and the average of the homework mark contributes the remaining 30%. If the mark of the written exam is less than 5, then you failed the course and the mark of the written exam will also be the final mark. For the re-exam the homework does no longer count, so the mark of the written re-exam will also be the final mark.
Program (chapter/section numbers refer to Humphreys' book unless stated explicitly otherwise):

Wednesday February 4: Sections 1.1-1.4 and 2.1, 2.2.
Wednesday February 11: Sections 6.1, 7.1 and 7.2.
Wednesday February 18:. 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. Reading homework: please read carefully I.3.2: Theorem of Engel, and I.3.3: the proof of Engel's theorem.
Wednesday February 25: 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 section 4.3 carefully by yourself.
Wednesday March 4: Killing form, semisimple Lie algebras, Casimir element, Schur's Lemma, Weyl's Theorem (Humphreys II 5,6 and Corollary of II 7.2).
Wednesday March 11: Cartan subalgebras, root space decomposition (Humphreys II 8.1-8.4). Reading homework: read carefully II 8.4.
Wednesday March 18: Root systems (Humphreys II 8.5, III 9.1-9.4 up to Lemma 9.4).
Wednesday March 25: Bases and Weyl chambers (Humphreys III 10.1-10.3 up to and including Theorem 10.3). Reading homework: Humphreys III 10.2 Lemma A and C and their corollaries).
Wednesday April 1: Irreducible root systems, Cartan matrices and Dynkin diagrams (Humphreys III 10.3 Lemma A and B, 10.4 up to Lemma A, 11.1-11.4, 12.1 (classical root systems)).
Wednesday April 8: Construction of the root systems and weights (Sections 10.4 Lemma A, 12.1, 13.2 Lemma A and 13.3 Lemma A). Reading homework: Humphreys III 10.4 Lemma B,C and D.
Wednesday April 15: NO LECTURES. Reading Homework for this week: Reread Humphreys' Chapter II on (the root space decomposition of) semisimple Lie algebras. Next week we will use the classification of the root systems to classify the semisimple Lie algebras.
Wednesday April 22: Humphreys IV Sections 14.1 and 14.2 (isomorphism theorems for simple Lie algebras), Section 18.1, and section 18.3 only the statement of Serre's theorem.
Wednesday April 29: Section 17.1, 17.2 (tensor algebras, symmetric algebras and universal enveloping algebras), Section 17.3, 17.4 (Poincare-Birkhoff-Witt theorem).
Wednesday May 6: NO LECTURES.
Wednesday May 13: Standard cyclic modules and finite dimensional modules (Section 20 and section 21.1).
Wednesday May 20: Irreducible finite dimensional modules, Chevalley algebras and groups (Section 21.2 and Section 25).
Homework (from Humphreys' book unless stated explicitly otherwise):

February 4: Exercises I.1.3, I.1.5 and I.2.5 from Humphreys' book (hand-in: February 11 before 12am).
February 11: Hw11Feb.pdf. (hand-in: February 18 before 12 am).Handwritten sketch of solutions: Hw11FebSol.pdf.
February 18: no homework.
February 25: Hw25Feb.pdf. (hand-in: March 4 before 12 am).
March 4: no homework.
March 11: no homework.
March 18: no homework.
March 25: Hw25mrt.pdf. (hand-in: April 1 before 12 am).
April 1: no homework.
April 8: no homework.
April 22: no homework.
April 29: Hw29apr.pdf. (hand-in: May 13 before 12 am).
May 6: no homework.
May 13: no homework.
May 20: no homework.

The homework has to be handed in at the start of next week's lecture. You can also send the solutions by email to j.v.stokman@uva.nl. The homework will be marked and given back during class. Please note that the homework exercises are of a different nature then the exercises of the written exam: the homework exercises are meant to practice with the notions you have learned during the last lecture.
Recommended exercises (all taken from Humphreys' book):

Wednesday February 11: Exercises 2,6 of Chapter II.6 and exercises 2,3,4,7 of Chapter II.7.
Wednesday February 18: I.3 exercise 1,2,3,4,7,8.
Wednesday February 25: II.4 exercise 5,6,7; II.5 exercise 5,6,7.
Wednesday March 4: II.6 exercise 1,2,4,5.
Wednesday March 11: II.8 exercise 1,2,4,6.
Wednesday March 18: II.8 exercise 9, 10; III.9 exercise 2,4,5.
Wednesday March 25: III.10 exercises 3,6,7 and 8.
Wednesday April 1: III.10 exercise 12; III.11 exercise 2.
Wednesday April 8: III.12 exercise 3; III.13 exercises 1,4.
Wednesday April 22: V.14 exercise 2.
Wednesday April 29: VI.17, exercises 1,3.
Wednesday May 13: VI.20, exercises 1,3,4,6 and 7.
Wednesday May 20: VI.21 exercise 6,11; VII.25, exercise 2.