Abstract We construct Grassmannian categories of infinite rank, providing an analogue the cluster introduced by Jensen, King, and Su. Each category rank is given as graded maximal Cohen–Macaulay modules over a certain hypersurface singularity. show that generically free $1$ in are bijection with Plücker coordinates appropriate algebra rank. Moreover, this structure preserving, it relates rigidi...

In this note we explain how to obtain cluster algebras from triangulations of (punctured) discs following the approach of [FST06]. Furthermore, we give a description of m-cluster categories via diagonals (arcs) in (punctured) polygons and of m-cluster categories via powers of translation quivers as given in joint work with R. Marsh ([BM08a], [BM07]).

We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective ...

We construct the intermediate coverings of cluster-tilted algebras by defining the generalized cluster categories. These generalized cluster categories are Calabi-Yau triangulated categories with fraction CY-dimension and have also cluster tilting objects (subcategories). Furthermore we study the representations of these intermediate coverings of cluster-tilted algebras.

We study higher cluster tilting objects in generalized higher cluster categories arising from dg algebras of higher Calabi-Yau dimension. Taking advantage of silting mutations of Aihara-Iyama, we obtain a class of m-cluster tilting objects in generalized m-cluster categories. For generalized m-cluster categories arising from strongly (m + 2)-Calabi-Yau dg algebras, by using truncations of minim...

We prove the existence of an m-cluster tilting object in a generalized m-cluster category which is (m+1)-Calabi–Yau andHom-finite, arising froman (m+2)-Calabi–Yau dg algebra. This is a generalization of the result for them = 1 case in Amiot’s Ph.D. thesis. Our results apply in particular to higher cluster categories associated to Ginzburg dg categories coming from suitable graded quivers with s...

We introduce the continuous Frobenius category. This category is constructed using representations of the circle over a discrete valuation ring. We show that it is Krull-Schmidt with one indecomposable object for each pair of not necessarily distinct points on the circle. By putting restrictions on these points we obtain various subquotient categories with good properties. The main purpose of o...

Cyclic posets are generalizations of cyclically ordered sets. In this paper we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The stable category of a Frobenius category is always triangulated and has a cluster structure in many cases. The continuous cluster categories of [14], the infinity-gon of [12], the m-cluster category of type A∞ (m ≥ 3)...

This is an introduction to some aspects of Fomin-Zelevinsky’s cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture fo...