Bogdan Ion
May 12, 1999
11.00 am - 1.00 pm
Committee : Z. Szabo (chair), W. Browder, A. Ram
Topics: Representation theory of Lie groups, Hopf algebras
Algebra (R):
Which is the connection between Hom and the tensor product ? (they are
adjoint functors). How is this called in representation theory ?
(Frobenius reciprocity).
Representations of Z. Groupal algebra of Z. Connection with the
structures of modules over PID's. When is such a representation
completely reducible ? Why not always ? Which are the indecomposable
modules ? Give an example of a finite dimensional C-algebra which is not
semisimple.
What is the center of S_n ? (it was obvious to me that it has trivial
center but they asked for a proof. First I wrote down the decomposition
into conjugacy classes, but meantime I had some idea for a simpler
proof, which didn't work, so I had to get back and finish the first proof)
State the fundamental theorem of invariant theory for S_n. (here I stated
a theorem (Schur's double centralizer thm) which he said it is the
fundamental theorem of invariant theory for GL_n. At this point I got totally
confused and I said that in this case I don't what he is talking about. He
specified that he wants S_n to act on some polynomial ring, so I realized that
he was talking about the fundamental theorem of symmetric polynomials.)
Complex analysis (S):
Riemann mapping thm. Proof. Why is not C conformally equivalent to the
unit disc.
Removable singularities thm. proof.
How many holomorphic structures are on the torus ? (he wanted the
dimension of the space of holomorphic structures. I said dimension two,
but I got stuck in parametrizing. After some hints it worked out.)
Which are Hol(CP^1, CP^1) ? Which are Hol(torus, CP^1) ? Examples.
There where some more questions which I can't remember, but nothing
nonstandard.
Real analysis (B):
What is the Taylor expansion formula with reminder ? What does it mean
that a function is analytic in terms of the reminder ? If an analytic
function has an accumulation point of zeros, what is happening ? proof.
Hopf algebras (R+B):
(B) What is the definition ? Aha, you are not supposing that is graded ?
(I think that we was talking about Hopf algebras in braided categories
(braided groups), but they didn't insisted, so I didn't mentioned anything)
(R) Which is the connection with algebraic groups, group schemes ? Here
Browder asked the proof for some obvious thing, but he didn't seemed to be
convinced.
(R) Which is the connection with monoidal categories ? Where do you use,
the coproduct, counit, antipode ?
(R) The Drinfel'd double. Construction. Quasi-triangular structure. How
this construction comes naturally ? (here I started to tell them some
motivations which I think they didn't knew). What I meant is: giving the
R-matrix and the product, is the coproduct uniquely determined ? (I guess
it is. Is this a theorem ? He said it is. In fact, I realized latter that
is a simple consequence of the fact that the modules over the double form
the center of the category over the original Hopf algebra.)
(R) Maschke's theorem. What does it mean for groups ?
(R) Milnor-Moore thm. (I said I don't know the name, but I might know the
thm. He said that it is sometimes attributed to Konstant, and I realized
what he was talking about.)
Representation theory (R+B):
At this point, they realized that was almost 1.00pm, and they seemed very
eager to end the exam.
(B) The conjugacy of the maximal tori in a compact Lie group. proof.
(When I was stating the Lefschetz fixed point theorem, Szabo said that I
am missing a hypothesis (orientability). I couldn't realize, and then he
said we should go through the proof. I said that I don't know the proof,
and he asked what is the degree of a map. At this point I realized that
orientability was missing from the hypothesis.)
(R) What are the discrete series ? (I started the construction, but he
said that he wants some abstract description, so I told them what they
are in Langland's classification.)
They decided to stop. Browder asked me to leave the room. They where out
after a short time telling me that I passed.
General comments: All the questions (except Szabo's) where vague. It is a
good idea to try to restate their question, and ask if this is what they
meant. Everybody was very friendly and willing to help.