Talk Summaries
References of the form [?] refer to the
bibliography.
Topological duality of Hopf algebras (Theo Johnson-Freyd)
The statement that we need at the end of this
talk is that the universal enveloping algebra U(n) is graded dual to
the algebra C[N] of regular functions on N. This is shown, for
example, in [GLS08, 5.1].
Although it is possible to give a quick proof, as in the reference
above, the "right" way to do it is to introduce the notion of duality
of topological Hopf algebras. The main (perhaps only) examples are
U(g) and C[G], which are topologically dual via the pairing "think of
an element of U(g) as a differential operator on G, apply it to a
function, and evaluate at the identity". This is done in [CP,
4.1.17]. In the special case where G is a maximal unipotent subgroup
of a simple Lie group, topological duality translates to graded duality.
Hall algebras over C (A.J. Stewart):
The goal is to state and (at least partially) prove the theorem that
the Hall algebra (appropriately defined) of a C-linear abelian
category (with appropriate hypotheses) is a Hopf algebra.
The day will be structured as follows:
Talk 0: Overview of the whole workshop
Talk 1: Topological duality of Hopf algebras
Talk 2: Hall algebras over C
Talk 3: The path algebra construction of U(n) and the canonical basis
Talk 4: The preprojective algebra construction of U(n) and the
semicanonical basis.
So the idea is to start with a category where you have a decent notion
of a moduli stack of objects (say, the category of finite-dimensional
representations of some algebra, or perhaps the category of
finite-dimensional nilpotent representations of some graded algebra),
define the Hall algebra using constructible functions, prove
associativity, say something about how to define the coproduct and
counit, and say something about how to prove that what you get
satisfies the axioms of a Hopf algebra. You can (and should) allude
to the examples that will be considered in Talks 3 & 4, but it is
obviously not necessary to discuss these examples in any detail.
The path algebra construction of U(n) and the canonical basis (Bruce Fontaine):
Here we will understand Ringel's construction of the universal
enveloping algebra U(n) as a Hall algebra.
More precisely, let g be a finite-dimensional simply laced simple Lie
algebra, and let Q be a quiver obtained by orienting the arrows of the
Dynkin Diagram however you want. Let n be a maximal nilpotent
subalgebra of g. Then the Hall algebra of the category of
representations of Q over the complex numbers is isomorphic to the
universal enveloping algebra U(n).
I should be a little careful of what I mean when I say "Hall algebra".
Usually one talks about the Hall algebra of an abelian category where
all of the Hom and Ext groups are finite. If M and N are objects of
the category, then one defines [M] * [N] to be the sum over all
isomorphism classes [L] of c(M,N;L) [L], where c(M,N;L) is the number
of maps from M to L that are injective with quotient isomorphic to N.
In the previous paragraph, however, the Hom groups are complex vector
spaces, and therefore not at all finite. In this case, you're
supposed to replace "number of maps" with "Euler characteristic of the
set of maps" in the previous sentence.
If you work over a finite field F_q rather than over C, then
everything is finite, and what you get is the quantized enveloping
algebra U_v(n), where v is a square root of q. But we're not planning
to work with quantum groups at all in this workshop, so we'd prefer to
stick to the classical case.
It's a little tricky where to find this stuff in the literature. The
quantum case is worked out very nicely in [Sc99a, Lecture 3.3].
In fact, there it is even proved that you get an isomorphism of Hopf
algebras, where the Hopf algebra structure on the Hall algebra is
defined in Lecture 1. Presumably the proof of the classical case is
pretty much identical. The precise statement that we need in the
workshop (and, in particular, in the next talk) is made in [Lu91, Prop
10.20]. Here Lusztig uses the language of constructible functions
rather than Hall algebras, but it's the same thing. Note that the
subscript 0 in this proposition is unnecessary when g is a
finite-dimensional simple Lie algebra. Annoyingly, Lusztig gives no
proof, and attributes it to an unpublished manuscript of Schofield.
In addition to stating (and at least partially proving) this theorem,
we'd like you to say something about the various bases for U(n) that
you get this way. The most obvious one is the one indexed by
isomorphism classes of representations of Q, but it turns out that
this is not a very good basis; for example, it depends on the
orientations of the arrows in Q. You can obtain a much better basis
using perverse sheaves or intersection cohomology, which is called the
canonical basis. You won't have any time to go into perverse sheaves
in this talk, but a brief discussion on the level of [Lu91-ICM,
Section 19] would be appropriate.
The preprojective algebra construction of U(n) and the semicanonical
basis (Matt Davis)
The topic for this talk is the preprojective algebra and Lusztig's
construction of the semicanonical basis, which basically means Section
12 of [Lu91].
Let me give some more details. First of all, we'll restrict our
attention to the case where g is a finite-dimensional, simply laced,
simple Lie algebra. Let n be a maximal unipotent subalgebra of g, and
let U(n) be its enveloping algebra. Let Q be the quiver obtained by
orienting the edges of the Dynkin diagram for g however you like. In
the previous talk, we will establish Proposition 10.20 from [Lu91],
which says that U(n) is isomorphic to the Hall algebra of Q.
Using this, you would state and prove Proposition 12.13, which says
that U(n) is also isomorphic to a subalgebra of the Hall algebra of
the preprojective algebra of Q. You'd also go over Section 12.14, in
which Lusztig conjectures that U(n) has a natural basis (the
"semicanonical basis") indexed by the components of the variety
Lambda; this conjecture is proven in [Lu00, Theorem 2.7]. (Probably
you wouldn't have time to go through this proof, but maybe you could
at least say a few words about how it works.) Finally, you would go
over Section 12.15, in which Lusztig shows that this basis does not
depend on a certain choice he made (essentially, this is the choice of
orientations of the arrows in Q).
This will be a very important talk, because the semicanonical basis is
the subject of the entire workshop!
Note that Section 12 can be read pretty much independently from the
rest of the paper, with the exception of Proposition 10.20, which is
sort of the starting point.
Cluster algebras: basic definitions (Kevin Dilks)
Please go through the definition of a
cluster algebra in terms of an m x n B-matrix, with frozen
variables. Feel free to restrict yourself to the skew-symmetric case
(meaning d_i=1), but please do include the frozen variables. Describe
the Laurent phenomena, but don't prove it; that will be another
talk. Please do mention upper and lower cluster algebras and the
finite type classification.
Sections 4 and 5 of Fomin and Zelevinsky's notes from the CDM-03
conference http://arxiv.org/abs/math.RT/0311493
are a good reference for the sort of presentation I'm thinking of.
Cluster algebras: first properties (Peter McNamara)
We'd like you to speak about the Laurent phenomena for cluster
algebras. The main thing that I want this talk to do is to prove the
Laurent phenomenon: I think this result is fundamental enough that it
deserves to be proved. I think the best reference is still a
combination of Cluster Algebras I http://arxiv.org/abs/math.RT/0104151
and the Laurent Phenomenon
http://arxiv.org/abs/math.RT/0104151.
To the extent that you have additional time, it might be nice to talk
about the "canonical basis" strategy for proving positivity: Find a
basis for the cluster algebra which contains the cluster monomials,
and in which multiplication is positive, and positivity of the Laurent
polynomials follows. This unfortunately seems to be a bit of folklore
that isn't written down, but you can see it carried out in an example
in Sherman and Zelevinsky's paper
http://arxiv.org/abs/math.RT/0104151
and the website for Lauren Williams' course makes me think that she
covered it. If there isn't time for this, then just do the Laurent
phenomenon.
Double Bruhat cells
(Anna Bertiger)
We would like you to give a talk defining double Bruhat cells. The
ones that we really care about are the G^{e,w}, which
all live in the upper unipotent, so you can focus on single-wiring
diagrams rather than double ones. Please describe them (at least in
type A) in terms of their geometric meaning, how to parameterize them,
and which minors vanish on them. As I recall it, the original paper
http://arxiv.org/abs/math/9802056
is pretty readable.
We don't care about positivity: Feel free to introduce it for
motivation or to omit it.
Cluster structures on double Bruhat cells
(Jenna Rajchgot)
We would like you to speak on cluster structures on double Bruhat cells.
The ones we really care about are the G^{e,w}'s, which mean that you get to think about wiring diagrams rather than double wiring diagrams. And feel free to restrict to the simply laced case, and mostly to restrict to type A. Earlier speakers that day will have already introduced double Bruhat cells, cluster algebras and double cluster algebras, so your job is to tie them together.
The big reference here is cluster algebras III http://arxiv.org/abs/math.RT/0305434 . The first two sections of this paper are more than enough.
The Euler characteristic of a quiver flag manifold and the cluster
structure on C[N] (Pierre-Guy Plamendon)
We would like you to speak about the map M -> phi_M which Geiss, Leclerc and Schroer use to relate the representation theory of the preprojective algebra to the cluster structure on C[N]. This is section 9 of Rigid modules of preprojective algebras. I'd like this talk to do three things (1) Clearly define quiver flag varieties and state the formula in Lemma 9.1 (2) Unpack the terse remark in that paper that "the proof follows easily from the classical description of the duality between U(n) and C[N]" and (3) present basic examples of what these quiver flag manifolds look like.
The Euler characteristic of a quiver Grassmannian and cluster
structures of type Q
(Qin Fan)
We would like you to speak about the Caldero-Chapoton map.
This talk should (1) explain what a quiver grassmannian is (2) state
the CC formula for cluster Laurent polynomials in terms of quiver
grassmannians and (3) work out some examples and basic properties. The
sort of basic things I am thinking of are that the denominator of the
cluster polynomial is the dimension vector, and that the highest and
lowest degree coefficients are 1. It would be nice to work out why
direct sum turns into product, but we can push that later if it is too
much to reasonably cover.
The speaker before you will be Pierre-Guy Plamendon, talking about the
analogous Geiss-Leclerc-Schroer formula in the preprojective
setting.
Representation theory of the preprojective algebra, I
(Greg Muller)
For a quick definition of the preprojective algebra, see sections 1
and 2 of the original
paper
http://www.ams.org/mathscinet-getitem?mr=1781930
or see Lecture 2 of Crawley-Boevey's lecture notes
http://www.amsta.leeds.ac.uk/~pmtwc/dmvlecs.pdf.
For several of the facts I particularly want, a good reference is
section 8 of
http://arxiv.org/abs/math.RT/0509483. Note that, except for a few
pieces of vocabulary from section 1, you don't need to read the other
7 sections!
What I'd like you to talk about: What is the preprojective algebra?
What is Ext and how do you compute it for representations of the
preprojective algebra? Please come as close as you can to proving
Theorem 3 of math.RT/0509483 . Ideally, you could also explain the
"easy exercise" in Remark 8.3 of the same paper.
This is one of the talks that I am nervous about. Theoretically, it is
a self-contained topic which can be understood without learning about
quiver varieties, singularity theory, the representation theory of
Artin algebras, or any of its other applications. Unfortunately, I
can't find a reference which presents it that way, which is why I am
hoping you can extract such a presentation from the sources I have
linked above.
Representation theory of the preprojective algebra, II
(Sarah Scherotzke)
We would like you to give a talk on mutation of rigid modules for
preprojective algebras. The main reference
is "Rigid modules over
preprojective algebras". Your audience will already have a
fair amount of experience working with the preprojective algebra, and
the talk before yours will be on the structure of the Ext groups for
the preprojective algebra.
What I'd like you to talk about is (1) what a rigid module is, ideally
including both the definition as the intuition as a module with no
deformations (2) how to get a B-matrix from a rigid module (3) what
mutation is and (4) why this mutation coincides with the
Fomin-Zelevinsky combinatorial definition.
The multiplicative formula for the dual semicanonical basis (Dylan
Rupel)
We would like you to speak on the proof of the multiplication formula
of Geiss, Leclerc and Schroer, i.e., Theorem 1 of
math.RT/0509483. This would be at 2:45 PM on Thursday. Talks
before yours would have already covered the definition of the
preprojective algebra, the flag varieties Phi_{i, c, x}, the meaning
of Ext^1, the proof of GLS's Theorem 3 (Ext symmetry) and the notion
of mutation. What we'd like you to do is tie the subject together and
actually go through the proof.
Triangulated categories
(Ben Elias)
You will give an introduction to the language of triangulated
categories.
The basic way that this would fit into the workshop is as follows:
Most of the workshop will concentrate on two ways of categorifying
cluster algebras: The category of representations of the preprojective
algebra and the category of representations of the quiver path
algebra. Both of these are abelian categories and we will present them
in a way which makes sense to someone who is comfortable with
homological algebra (lots of Ext and Hom groups), without ever
mentioning the derived perspective.
But at least some people are going to want to know the more general
picture that these two examples fit into. This more general picture
wants to be stated in the language of triangulated categories. So I
would like to have a talk about what triangulated categories are, for
people who are only comfortable with the more conventional homological
language. This would mean talking a fair bit about the homotopy
category of chain complexes, and making sure to explain how to think
about mapping cones, and how to see Ext groups and exact sequences in
the triangulated language. I'll note that I really do need
triangulated categories which are not simply derived categories or
homotopy categories.
My favorite sources for this material are
http://arxiv.org/abs/math/0001045
and Gelfand and Manin's homological algebra textbook.
Representation theory of quivers
(Carl Mautner)
We would like you to speak on the representation theory of the path
algebra, kQ, of an acyclic quiver. Specifically:
What are the projective and injective objects? What are the Ext's (in
particular, that Ext^2 and higher vanish)? What are the reflection
functors? Please do include the statement that Hom(Y, C_{+} X) = D
Ext^1(X,Y) = Hom(C_{-} Y, X). where D is the dual vector space, and
C_{\pm} are the Coxeter functors.
People will have already been thinking a lot about quiver
representations throughout the conference. What they won't have done
is seen a talk on the homological structure of the category of quiver
representations as a whole. They will have seen a similar talk about
the preprojective algebra the previous day. (Where the results are a
little harder, but their application to canonical bases is
significantly simpler.) I'll also be writing problem sets for people
to work on, and I'll try to give them some experience that should
prepare them for some of these definitions.
Most of this is in section 6 of William Crawley-Boevey's lectures on
quiver representations
http://www.amsta.leeds.ac.uk/~pmtwc/quivlecs.pdf. One thing that
is missing is the equivalence between the functor D
Tr and the Coxeter functor defined as a product s_1 s_2 ... s_n of
reflection functors. This is in Brenner and
Butler,
http://www.ams.org/mathscinet-getitem?mr=442031. If you have
not thought about quivers at all before, I find the
Bernstein-Gelfand-Ponomarev paper to be very well
written
http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/BGG-CoxeterF-Usp.pdf.
Cluster mutation via quiver representations (David Stern)
We would like you to talk about mutation of tilting objects in cluster
categories. Basically, the goal is to get people to understand the
definitions underlying the main result (Theorem 1.3) in
Buan-Marsh-Reiten (
http://arxiv.org/abs/math/0412077) and,
secondarily, to prove as much of it as you can.
What people will already know: They will have seen many lectures about
quivers and quiver representations and will be happy with those. They
will also have a fair bit of experience with the combinatorial
definition of mutation (as given by Fomin-Zelevinsky). And the talk
immediately before you should talk about the representation theory of
the path algebra: What the projectives and injectives are, what the
Ext groups are, what the functor tau is, and the duality theorem
Hom(Y, tau_{+} X) = D Ext(X, Y) = Hom(tau_{-} Y, X).
The also will have seen mutation and tilting objects in the setting of
preprojective algebras, where it is easier because you don't have to
pass to the derived category.
What they won't know: What the cluster category is. I'm hoping you can
not only give the definition, but explain them how to think about
it. I will be writing nightly problem sets for the attendees, and I
will try to guide them in this direction.
And, of course, they won't know substance of the talk: How to do
mutation in this setting, and why it corresponds to the combinatorial
operation of quiver mutation.
2-Calabi-Yau categories and cluster algebras, I
(Ben Webster)
The idea for this would be to be the first of two talks on how cluster
algebras arise as categorifications of 2-Calabi-Yau categories.
There will be a lot of preparation for this material in the preceding
days. The workshop will focus on the two main examples of this: The
categorification in terms of the representations of the preprojective
algebra, and the categorification in terms of the "cluster category"
-- a certain quotient of the derived category of representations of
the path algebra. There will also be a talk on Thursday on what
triangulated categories are. However, there will be no talks before
yours which work in a general triangulated category, only in
particular examples, and so most people will probably not be very
comfortable thinking in terms of a general category.
More specifically, I would like to see you talk about Palu's paper
"Grothendieck Group and Generalized Mutation Rule for 2-Calabi--Yau
Triangulated Categories"
http://arxiv.org/abs/0803.3907, getting
through Theorem 12 where he shows how mutation of tilting objects in a
2 CY category categorifies Fomin-Zelevinsky mutation of B-matrices.
Now, there is a lot of terminology there. Caldero and Keller's earlier
papers math.RT/0506018 and math.RT/0510251 were the beginnings of
explicitly describing the cluster algebra story in terms of
categorification; one warning that I will give you is that several
things which they call conjectural have since been proved. Palu's
earlier paper http://arxiv.org/abs/math/0703540 might also be a
helpful introduction to terminology.
2-Calabi-Yau categories and cluster algebras, II
(Philipp Lampe)
What we would like you to talk about is how the cluster character
formula works in a general 2-Calabi-Yau category. There will have been
a lot of talks leading up to this: The whole workshop will be looking
at the examples of the preprojective algebra and of the quiver path
algebra. I think that, towards the end, we should have a few talks on
how to understand this in a general categorical context. So there will
be a talk on Thursday about triangulated categories, a talk early on
Friday on mutation in a general 2-Calabi-Yau category and, the talk
that I hope you will give, a talk on how cluster characters work in
general 2-CY categories.
The main references here are the following: The general strategy of
how to categorify cluster algebras was first laid out by Caldero and
Keller (
http://arxiv.org/abs/math.RT/0506018
and
http://arxiv.org/abs/math.RT/0510251
) and by Fu-Keller
http://arxiv.org/abs/0710.3152. The best presentation of the
theory
in final form that I've seen has been by Palu, in
http://arxiv.org/abs/math/0703540
and more generally in
http://arxiv.org/abs/0903.3281.