Let $n$ be a maximal nilpotent subalgebra of a complex simple Lie algebra of type A,D,E. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of modules over a preprojective algebra of the same Dynkin type as $n$. We prove a formula for the product of two elements of the dual of this semicanonical basis, and more generally for the product of two evaluation forms associated to arbitrary modules over the preprojective algebra. This formula plays an important role in our work on the relationship between semicanonical bases, representation theory of preprojective algebras, and Fomin and Zelevinsky's theory of cluster algebras. It was inspired by recent results of Caldero and Keller.

We show how all the main results involving acyclic cluster algebras can be proved using the module category of an hereditary algebra. The starting point is the Caldero-Chapoton map from modules to the ambient field $\mathcal F$, defined in terms of Hall numbers. This map immediately implies the denominator formula. We present a new proof of the Caldero-Keller Cluster Multiplication Theorem, using Hall numbers and remaining within the module category. Finally we show how to associate to any tilting module a skew-symmetrisable matrix, determined purely combinatorially from the Euler form, and thus realise the cluster algebra.

Cluster categories are 2-Calabi-Yau triangulated categories which realize cluster algebras in the sense that tilting objects correspond to clusters and their mutation correspond to the exchange rule of cluster variables. In my talk we discuss mutation in a more general framework.

We will recall the notion of a triangulated category and illustrate it on examples. We will explain why the orbit category of a triangulated category under the action of an autoequivalence is not triangulated in general. Finally, we will sketch why this is nevertheless true for cluster categories.

(joint work with Caldero)
We will rapidly review the definition of the Caldero-Chapoton map. We will then outline the proof of the theorem, conjectured by Buan-Marsh-Reineke-Reiten-Todorov, that for acyclic quivers, it establishes a bijection between the exceptional indecomposables of the cluster category and the cluster variables of the cluster algebra. The mutation theorem, due tu Buan-Marsh-Reiten, plays an important role in this proof.

I shall present, in a motivated fashion, the generalized non-crossing partitions of Drew Armstrong, combinatorial objects that are associated to finite reflection groups. They are fascinating in many ways. In particular, they have extremely interesting enumerative properties, some of which I will mention. The main result of the talk will be a surprising relation between the Möbius function of the poset of generalized non-crossing partitions and certain face numbers of the generalized cluster complex of Fomin and Reading. As yet, there is no intrinsic understanding for this relation, my proof being case-by-case (with, in fact, a gap to be filled in type D).

(joint work with Barot and Lenzing)
Let A be a hereditary or a canonical algebra. By a result of B. Keller the cluster category of A admits a triangulated structure containing the induced triangles of the derived category. We describe its Grothendieck group explicitly in the cases where A is canonical or the path algebra of a Dynkin quiver.

(joint work with Aslak Buan, Markus Reineke, Idun Reiten and Gordana Todorov)

This will be an introduction to the cluster category and cover the correspondence between indecomposable objects and cluster variables in finite type. It should also look at the modelling of cluster mutation via approximation theory/quivers of cluster tilted algebras. The emphasis will be on the cluster category itself rather than the cluster tilted algebras.

(joint work with Keller)
We show that the cluster-tilted algebras are Gorenstein of dimension at most 1, and that the stable category of their Cohen-Macaulay modules is 3-Calabi-Yau. Actually we work in a more general setting, replacing cluster categories with Hom-finite 2-Calabi-Yau triangulated categories. Examples include the stable category of finitely generated modules of the preprojective algebra of Dynkin diagrams and of finitely generated Cohen-Macaulay modules over a 3-dimensional commutative local complete isolated Gorenstein singularity.

We end with characterizing the cluster categories amongst the algebraic Hom-finite 2-Calabi-Yau categories,and give an application to Cohen-Macaulay modules.

This talk is on the following realization of cluster-tilted algebras. Let C be a finite dimensional algebra of global dimension at most 2. Its relation-extension is the trivial extension of C by the C-C-bimodule Ext²_C(DC,C). Now, if C is a tilted algebra then its relation-extensions is cluster-tilted and any cluster-tilted algebra is obtained in this way.

(joint work with Geiss and Leclerc)
Berenstein, Fomin and Zelevinsky proved that the algebra C[N] of polynomial functions on a maximal unipotent subgroup of a Lie group G of Dynkin type can be equipped with a cluster algebra structure. This is of interest since certain "canonical bases" live inside C[N], and one can hope that the cluster algebra structure helps to understand these bases. Or vice versa, one can hope that the cluster algebra structure yields a basis with such nice properties that it deserves the name "canonical basis".
One can realize the cluster algebra structure on C[N] inside the category of finite-dimensional modules over a preprojective algebra. This is what we call "categorification".
For type A we will show that the seeds of the cluster algebra C[N] are (in a suitable way) uniquely determined by their exchange matrix. The proof uses the representation theory of preprojective algebras.
Furthermore, we will realize some cluster algebras associated to double Bruhat cells, numerous acyclic cluster algebras and also coordinate algebras of partial flag varieties as cluster subalgebras of C[N].

(joint work with A. Buan and I. Reiten)
It was observed by inspection that simply-laced minimal infinite cluster type quivers are in a one-to-one correspondence with quivers of tame concealed algebras. In this talk, we will explain this correspondence in the set-up of cluster categories. In particular, we will discuss a characterization of tame concealed algebras in terms of a natural class of quadratic forms associated with minimal infinite cluster type quivers.

(joint work with Aslak Buan, Robert Marsh and Idun Reiten)
Cluster algebras are sub-algebras of fields of rational functions. They are generated by cluster variables which are the elements of certain transcendence bases, called clusters. All these clusters are obtained by starting with an initial cluster seed (x,Q), which consists of a transcendence basis x and a quiver Q. New cluster seeds (x',Q') are obtained by finite sequences of mutations, where mutation of the basis x, i.e. cluster x, is defined in terms of the quiver Q.
A conjecture of Fomin and Zelevinsky is, that after any finite sequence of mutations, the cluster seed (x',Q') is determined by its cluster x'.
We prove the conjecture for acyclic cluster algebras with no coefficients. In the proof we use: 1. Representation-theoretic notion of cluster categories, 2. Already known relations between cluster algebras and cluster categories, 3. Prove the existence of quite a strong relation which is actually a mapping from cluster seeds to tilting seeds and 4. Use the fact that the tilting seed is determined by the tilting object.
Furthermore, in the above process, we also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.

(joint work with Igusa, Orr and Todorov)
Recently it was observed that the cluster combinatorics for finite type cluster algebras appears in formulas for differentials of certain complexes calculating homology of torsion free nilpotent groups. These differentials are described by the theory of pictures of Igusa-Orr. This suggests possible generalizations of cluster combinatorics. I will describe the connection between the Igusa-Orr theory and quiver representations.

I will give a general introduction to the subject, presenting main definitions, known structural results, and some open problems and conjectures.

The talk will be based on the joint paper with S. Fomin "Cluster algebras IV: Coefficients" posted on the archive a few months ago. One of the objectives of my talk is to advertise the study of various coefficient systems (somewhat neglected in the developments on cluster categories and cluster-tilted algebras).

(joint work with S. König)
We put cluster tilting in a general framework by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal one-orthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be modules categories of Gorenstein algebras of Gorenstein dimension at most one.