Lie algebras.
(Math 534, Term I 2011)
Tuesday, Thursday 9:3011am, at Math 202.
My office: Math 217.
Course description
Announcements:
Current:
 New written homework is due
December 7.
 Next homework problems discussion

On Wednesday December 7, 3:30pm. Next written homework due
December 7!
Older:
 Please fill out the
doodle poll for your avialability for the last four presentations
(instead of the final exam), December 59. It is the easiest to use
"calendar view"
to fill out this poll. Do not forget to check your exam invigilation
assignments and other aftertheendofterm duties before you fill it out.
We need to find two 1.5hour slots. I hope it works.
 Midterm canceled :).
 Makeup lecture: Monday
November 21, 35pm, in our usual room Math 202.

NO CLASS on Tuesday, November 1.

NO CLASS on Tuesday October 18.

Please fill out the
doodle poll
for the scheduling of makeup 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 makeup 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
problems .
 The first written homework is due
Thursday September 22.
Textbook:

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:
(under construction)
Prerequisites
The only prerequisites are: good understanding of linear algebra (including Jordan canonical form of a matrix), and
a first course in group theory.
Marking
There will be a 1hour written midterm exam (in class),
everyone will be expected to do either an inclass 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 inclass homework
discussions will happen roughly once in two weeks, the dates will be
posted here.
Homework
 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
22.
 Problem set 2.
Due Tuesday October 11.
Some useful Maple code can be found at Carmen
Bruni's page.
 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"
helpful: click
here
and scroll down to find it.
Tentative schedule of presentations:
 Monday Nov. 21, 35pm:
 Lance, Classification of the semisimple orbits (Chevalley's
Theorem);
 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, SchurWeyl duality.
 Li Zheng, Lie algebras over the reals.
 Thursday Dec 1:
 Asif, HarishChandra modules (beginnings of representation theory
for gl(2, R) ).
 Saman, The symplectic structure on nilpotent orbits.
Saman's excellent reference list:
 The "halfmarathon" on Tuesday
December 6 .
 The "marathon" on
Wednesday December 7 .
 1:002:00 Heidar, Kirillov's orbit method. (in Math 126)
 2:003:30 Arman and Shuhang, Freudenthal's, Weyl's and Kostant's
formulas. (in Math 126)
 3:304:15 (room TBA) The last homework discussion.
 4:155:00 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 FultonHarris, and numbers are
the section numbers.
 September 68:
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
2dimensional 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 1315:
No class.
Please fill out the
doodle poll
for scheduling the makeup classes.
 September 2022:
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 2729:
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 46:
JordanChevalley decomposition; representations of sl(2).
Root space decomposition. (FH: appendix C.2, H:chapters 6.4, and 7 and
8.1, 8.2)
 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 1720:
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 1012 in
notes by Prof. Casselman for a discussion of the algorithm for
constructing roots.
October 20: homework discussion.
 October 2527:
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 AG. (H, chapter 12)
 No Class on Tuesday November
1.
 November 3:
Automorphisms of Lie algebras, Cartan subalgebras, inner
automorphisms.
(H, 14.4, chapter 15  home reading, skipped; 16.1, also section 2.3).
 Monday November 7  makeup
lecture.
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, 17.117.2).
 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,
online book
by S. Sternberg, Section 11.
 November 1517:
Representation theory: the weight lattice; weight spaces; standard cyclic
modules; classification of the finitedimensional representations.
(H. Chapters 13, 20, and 21).
 November 2124:
Makeup class: November 21. Presentations start.
 November 29Dec 1:
Presentations.
 The remaining presentations and the last homework discussion  the
week of December 5  9.