References of the form [?] refer to the bibliography.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.