(Math 534, Term I 2011)
Tuesday, Thursday 9:30-11am, at Math 202.
My office: Math 217.
- New written homework is due
- Next homework problems discussion
On Wednesday December 7, 3:30pm. Next written homework due
- Please fill out the
doodle poll for your avialability for the last four presentations
(instead of the final exam), December 5-9. It is the easiest to use
to fill out this poll. Do not forget to check your exam invigilation
assignments and other after-the-end-of-term duties before you fill it out.
We need to find two 1.5-hour slots. I hope it works.
- Midterm canceled :-).
- Make-up lecture: Monday
November 21, 3-5pm, in our usual room Math 202.
NO CLASS on Tuesday, November 1.
NO CLASS on Tuesday October 18.
Please fill out the
for the scheduling of make-up lectures.
(Please ignore the dates -- the poll is about your availability at
pretty much all possible times on Mondays, Wednesdays, Thursdays and Fridays. If we find a slot that works for everyone, we will have to meetings on two
different weeks at that time). When filling out Thursdays, please assume that it applies to a Thursday when there is no Number Theory Seminar.
No class on September 13 and 15. Two make-up classes will be
sheduled later. Next class - Tuesday Sept. 20. Please note that it will
start with the discussion of the problems from the growing list of
- The first written homework is due
Thursday September 22.
James E. Humphreys, "Introduction to Lie algebras and representation theory".
- Also useful (but you do not have to buy it) : W. Fulton and J. Harris "Representation theory. A first course".
Some interesting links:
The only prerequisites are: good understanding of linear algebra (including Jordan canonical form of a matrix), and
a first course in group theory.
There will be a 1-hour written midterm exam (in class),
everyone will be expected to do either an in-class presentation or a short essay (in place of the final exam) -
whether it will be essays of presentations will depend on class size and everyone's interests, we'll decide during
the first week of class;
there will be some written homework problems,
but most homework will be dicussed in class. The in-class homework
discussions will happen roughly once in two weeks, the dates will be
- To be discussed in class: a growing
list of problems . The list of problems will
grow, and they will be discussed as
necessary. The ones discussed already will be marked with a check.
The next discussion date: the last day.
To be handed in:
- Problem set 1 . Due Thursday September
- Problem set 2.
Due Tuesday October 11.
Some useful Maple code can be found at Carmen
- Problem set 3.
Due Tuesday October 25.
- Note: for information about tensor products and symmetric powers, see
Appendix A in FH; also useful: Section 1 of the
notes by P. Etingof et al.
, or a chapter in the book "Linear algebra and geometry" by Kostrikin
and Manin (let me know if you'd like to borrow the book).
- Problem set 4.
You might find Professor Casselman's "Graph paper for Coxeter groups"
and scroll down to find it.
Tentative schedule of presentations:
- Monday Nov. 21, 3-5pm:
- Lance, Classification of the semisimple orbits (Chevalley's
- XinYu, Classification of nilpotent orbits.
- Tuesday Nov 22:
- Thursday Nov 24:
- Ehsan, Lie groups and the exponential map.
- Pang, Topology of the classical Lie groups.
- Tuesday Nov 29:
- Vasu, Schur-Weyl duality.
- Li Zheng, Lie algebras over the reals.
- Thursday Dec 1:
- Asif, Harish-Chandra modules (beginnings of representation theory
for gl(2, R) ).
- Saman, The symplectic structure on nilpotent orbits.
Saman's excellent reference list:
- The "half-marathon" on Tuesday
December 6 .
- The "marathon" on
Wednesday December 7 .
- 1:00-2:00 Heidar, Kirillov's orbit method. (in Math 126)
- 2:00-3:30 Arman and Shuhang, Freudenthal's, Weyl's and Kostant's
formulas. (in Math 126)
- 3:30-4:15 (room TBA) The last homework discussion.
- 4:15-5:00 Hang Yu, Chevalley groups. (room TBA)
Approximate detailed syllabus:
Descriptions of lectures, homework problems, and interesting links will appear here as we go along.
In references, "H" is Humphreys, "FH" is Fulton-Harris, and numbers are
the section numbers.
- September 6-8:
Lie algebras: an overview; the basic definitions (including ideals,
homomorphisms, centre, radical, ...) The notions of nilpotent, solvable,
simple and semisimple; examples (the classical Lie algebras); the
2-dimensional Lie algebras; adjoint
representation; proof that sl(2) is simple.
References: H: 1.1, 1.2, 2.1, 3.1, 3.2; 1.4, 2.2. FH: 9.1, some of 10.1
- September 13-15:
Please fill out the
for scheduling the make-up classes.
- September 20-22:
The radical. Nilpotent and solvable Lie algebras.
Engel's Theorem, Lie's theorem. Cartan's criterion. References FH: 9.2, 9.3,
H: 3.1, 3.2, 3.3, 4.1, 4.3.
- September 27-29:
Discussion of Jordan decomposition (however,
we have not yet proved that the semisimple and nilpotent parts of
every element of g are in g). Complete reducibility of representations: the Killing form, Casimir element, Weyl's theorem.
References: FH 9.3, H: 5.1, 5.2, 6.1, 6.2, 6.3
- October 4-6:
Jordan-Chevalley decomposition; representations of sl(2).
Root space decomposition. (FH: appendix C.2, H:chapters 6.4, and 7 and
- October 11:
Root space decomposition continued (H: 8.3, 8.4, 8.5).
- October 13:
Root systems; the definition of the Dynkin diagram, statement of
the classification theorem. (we did not discuss simple roots
though, will come back to it next class) (H: chapter 9, also sections
11.2, 11.3, statement of the Theorem in 11.4)
- October 17-20:
Bases, action of the Weyl group; (H 10.1, 10.3, some of 10.2)
Automorphisms of root systems.
Construction of the root system from simple roots.
See Sections 10-12 in
notes by Prof. Casselman for a discussion of the algorithm for
October 20: homework discussion.
- October 25-27:
Proof of the classification theorem for irreducible root systems;
Reducible root systems.
Proof of the classification theorem for the semisimple Lie algebras.
(H: 10.4, 11.4, 14.1, 14.2)
Home reading: Construction of the root systems of types A-G. (H, chapter 12)
- No Class on Tuesday November
- November 3:
Automorphisms of Lie algebras, Cartan subalgebras, inner
(H, 14.4, chapter 15 -- home reading, skipped; 16.1, also section 2.3).
- Monday November 7 -- make-up
Borel subalgebras; conjugacy of Borel subalgebras. Automorphisms of Lie
algebras.(Chapter 16, except for 16.4, which we replaced by the "proof"
using the adjoint group.).
Sketch of: The universal enveloping algebra,
PBW theorem (H, 17.1-17.2).
- November 8:
Homework discussion; Continuation fo PBW theorem. H, 17.3.
- November 10:
Existence theorem, Serre's relations (quick summary of Chapter 18).
For free Lie algebras, see for example,
by S. Sternberg, Section 11.
- November 15-17:
Representation theory: the weight lattice; weight spaces; standard cyclic
modules; classification of the finite-dimensional representations.
(H. Chapters 13, 20, and 21).
- November 21-24:
Make-up class: November 21. Presentations start.
- November 29-Dec 1:
- The remaining presentations and the last homework discussion -- the
week of December 5 -- 9.