12/24/15: I submitted the grades on 12/21/2015. You must be able to
see them online. Thank you for a great semester. Happy holidays! See
you next term.
12/24/15: I submitted the grades on 12/21/2015. You must be able to
see them online. Thank you for a great semester. Happy holidays! See
you next term.
12/6/15: I have
posted
Homework 7, and made it due Friday, December 18, at noon to my
mailbox.
11/30/15: Zeshen Gu pointed an error in Problem 5.7, which is on the
homework, to me today: we must assume that A is strictly
upper-triangular. It is an unnoticed typo in the text.
11/26/15: I have removed one problem from Homework 6, as I have not
noticed it was part Theorem 5.29 (2) and (4). The final count of
problems is seven.
11/26/15: I
posted
Homework 6, due Friday, December 4, partially yesterday and
completed it today. There are eight problems altogether.
11/17/15: I have added the word (generalized) to matrix
coefficients in HW Problem 2: working with a basis is not
essential in that problem.
11/16/15: I have corrected a typo in the formula for the action
Rg in HW Problem 2: it should be f(hg) on the right-hand
side.
11/16/15: I have updated the homework very slightly again and
elaborated the hint to Problem 7: Apply the Peter-Weyl theorem to a
function which is 0 at 1 and greater than 1 outside of a neighborhood
of 1. Where will the kernel of a representation whose matrix
coefficient (or a linear combination of such) approximating this
function lie?
11/16/15: I have updated the homework very slightly by specifying that
U(gl) meant the universal enveloping algebra of the Lie algebra
gl.
11/13/15: I have updated the homework, having removed Section 5.2 from
the reading assignment.
11/11/15, 9:16 p.m.: I forgot to add problems on the universal
enveloping algebra to the homework. I am adding just one problem now,
so as the total count of problems is eight in the final version of the
homework.
11/11/15, 5:30 p.m.: There was so much formula writing in Homework 5,
that I ended up writing a LaTeX file and posting a pdf file on
the
Homework page.
11/10/15: I have
posted
Homework 5, due Friday, November 20, partially. There are not
enough suitable problems in the text, and I will add more hand-written
problems on Wednesday.
10/31/15, 6:01
p.m.:
Homework 4, due Friday, November 6, has been posted.
10/31/15: A couple of corrections: When I was talking about the
Kirillov orbit method the other day, I should not have excluded 0 from
the dual space of the Lie algebra. The zero functional corresponds to
the trivial representation, that is to say, the representation
on the standard one-dimensional space C on which the Lie group
acts as identity and the Lie algebra as zero. Another correction
refers to an error, pointed out by one of you, in the argument showing
that the standard n-dimensional representation of
SO(n,R) in Cn is irreducible. We will talk
about this on Monday in class. Do not forget about the time
switch!
10/31/15: Guys, I am still working on the homework. Plan to post it by
6:30 p.m.
10/15/15:
Homework 3, due Friday, October 23, has been posted.
10/14/15: Correction: the differential equation for Z(t) = μ(tx,y),
where μ(x,y) is the group law in the logarithmic coordinate:
exp(μ(x,y)) = exp(x) exp(y) for x, y in the Lie algebra of a Lie
group is not exp (ad x), but rather a Todd series, namely
| Z'(t) = | log (etad x ead y) | (x), |
| etad x ead y - 1 |
10/8/15: Hint to Problem 2.8: Show that Ad X: g → g for G a
subgroup of GL(n,K) is just (Ad X)y = XyX-1 by taking the
directional derivative of X exp(y) X-1 in the direction y.
10/8/15: Hint to Problem 3.2: The intuitive reason for this to work is
that when you want to compare how two elements of G multiply in
different coordinate charts, you use the change of coordinate map and
its Taylor expansion, which is useful to write through its linear,
quadratic, etc., parts, as in the proof of Lemma 3.11, to compare the
bilinear parts of the group law. Since the quadratic part of the
Taylor expansion is symmetric in all the variables, it can mess up
only the symmetric part of the group law when changing
coordinates. Thus, it is easier to solve a more general problem: if
you write the group law in two different coordinate charts
f1 and f2: g → G near 1 in G, both mapping
0 to 1, and identify their bilinear parts B1 and
B2, then the antisymmetrizations of the bilinear parts are
related by the linear part T1 = T* of the Taylor
series of the change of coordinates map T = T1 +
T2 + ... = f2-1f1 as
follows:
10/2/15: In doing the Homework, you may assume that
πi(Sn) = 0 for i
< n, a standard fact of Algebraic Topology, which follows from
Sard's theorem.
10/1/15:
Homework 2, due Friday, October 9, has been posted.
9/23/15: Starting from this Friday, I have decided to modify my office
hours slightly and start them at 11:15 on Wednesday and at 10:10 on
Friday. Thus, my office hours from now on will be Mon 11:15-12:05, Wed
11:15-12:05, Fri 10:10-11:00, or by appointment.
9/18/15: I have just posted the
first
Homework, due Friday, September 25. I am sorry for the delay: I am
still terribly jet-lagged after Japan and fell asleep on your homework
a few hours ago. ☺ Now it is just past midnight on September
19. The homework will be collected or graded. Getting together with
other students (that is, in study groups) is a very effective way to
do homework. However, you have to write solutions on your own.
9/16/15: I have added a link to the list
of errata
for the text to our main course page.
9/6/15: I am out of town, participating in a workshop
on Braids,
Configuration Spaces and Quantum Topology at the University of
Tokyo, Japan, during the first week of classes at the U, through
September 12. Thus, the first class of the semester will be on Monday,
September 14. No office hours during the first week, either: if you
need to contact me, please write me at voronov@umn.edu . I apologize
for the delay — I know you are all eager to start learning the
amazing Lie theory. I look forward to meeting you and starting the
class, too!
9/6/15: I recommend the following way to study for this class. Attend
each class, take notes, participate in class actively. After each
class review your notes and study the corresponding part of the
text. You can find out which part of the text at
the
Class Outlines page. Then do the assigned homework problems
pertinent to that material. Some students find it helpful to read the
material before it is covered in class, some prefer to do reading
after class.
9/6/15: If you need to register for this class, please, send a message
to Ms. Lawson at ugrad@math.umn.edu and ask for a permission number. If
permission is granted, go to OneStop and register.
Last modified: (2016-01-19 17:26:02 CST)