These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generators of c...

The goals of this expository article are on one hand to describe how to construct (m-) cluster categories from triangulations (resp. from m+2angulations) of polygons. On the other hand, we explain how to use translation quivers and their powers to obtain the m-cluster categories directly from the diagonals of a polygon.

We show that the quotients of the continuous cluster category Cπ and the rational subcategory X modulo the additive subcategory generated by any cluster are abelian categories and we show that they are isomorphic to categories of infinite and finite length modules, respectively, over the endomorphism ring of the cluster. These theorems extend the theorems of Caldero-ChapotonSchiffler and Buan-M...

Abstract Given a negatively graded Calabi-Yau algebra, we regard it as DG algebra with vanishing differentials and study its cluster category. We show that this is sign-twisted realise category triangulated hull of an orbit derived the singularity finite-dimensional Iwanaga-Gorenstein algebra. Along way, give two results stand on their own. First, coherent sheaves over has natural tilting subca...

Let k be a field and Q a finite quiver without oriented cycles. Let kQ be the path algebra of Q and mod kQ the category of k-finite-dimensional right kQ-modules. The cluster category CQ was introduced in [1] (for general Q) and, independently, in [4] (for Q of type An). It is defined as the orbit category of the bounded derived category D(mod kQ) under the action of the automorphism Σ−1 ◦ S, wh...

We show that the quotients of the continuous cluster category Cπ and the rational subcategory X modulo the additive subcategory generated by any cluster are abelian categories and we show that they are isomorphic to categories of infinite and finite length modules, respectively, over the endomorphism ring of the cluster. These theorems extend the theorems of Caldero-ChapotonSchiffler and Buan-M...

We show how certain u-cluster categories of Dynkin types D and E can be realised as stable module categories of selfinjective algebras. Together with our earlier paper on type A, this completes the classification of those u-cluster categories of Dynkin type which can be realised as stable module categories. We also complete here with types D and E the explicit calculation of the stable Calabi-Y...