Abstract. We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-Calabi-Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [27]. In addition, we prove a general result about relative 3-Calabi-Yau duality over non stable endomorphism rings. This strengthens and generalizes the Ext-group symmetries obtained in [27] for simple modules. Finally, we generalize the results on relative Calabi-Yau duality from 2-Calabi-Yau to d-Calabi-Yau categories. We show how to produce many examples of d-cluster tilted algebras. 1.

Abstract. 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. 1. Introduction and

Abstract. 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 for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences. Contents

Abstract. Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQ-module M (these are certain preinjective CQ-modules), we attach a natural subcategory CM of mod(Λ). We show that CM is a

. A method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself. The method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal of relations which is either homogeneous or admissable (with some additional finiteness restrictions in the latter case). In particular, it applies to any finite dimensional algebra over an algebraically closed field. The method is illustrated by a number of examples, viz. truncated algebras, monomial algebras and Koszul algebras, with the aim of unifying existing treatments of these in the literature. 1991 Mathematics Subject Classification. Primary: 16E99, 18G10. Secondary: 16D20, 16E40, 16G20, 16W50. 1. Introduction A projective resolution of an algebra , considered as a bimodule over itself, is fundamental in governing the homological properties of the algebra. Such a resolution may be used to compute Hochschild homology and cohomology, to ...

Abstract. We prove that mutation of cluster-tilting objects in triangulated 2-Calabi-Yau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2-CY-tilted algebras and Jacobian algebras associated with quivers with potentials. We show that cluster-tilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3-CY algebras. The nearly Morita equivalence for 2-CY-tilted algebras is shown to hold for the finite length modules over Jacobian algebras.

Abstract. This article presents a study of the algebra spanned by the semigroup of faces of a hyperplane arrangement. The quiver with relations of the algebra is determined and the algebra is shown to be a Koszul algebra. A complete set of primitive orthogonal idempotents is constructed, the projective indecomposable modules are described, the Cartan invariants are computed, projective resolutions of the simple modules are constructed, the Hochschild cohomology is determined and the Koszul dual algebra is shown to be anti-isomorphic to the incidence algebra of the intersection lattice of the arrangement. In particular, the algebra depends only on the intersection lattice of the hyperplane arrangement. Connections with poset cohomology are explored. The algebra decomposes into subspaces isomorphic to the order cohomology of intervals of the intersection lattice. A new cohomology construction on posets is introduced and the resulting cohomology algebra of the

We give a geometric construction of the Verma modules of a symmetric Kac-Moody Lie algebra g in terms of constructible functions on the varieties of nilpotent finite-dimensional modules of the corresponding preprojective algebra Λ. 1

Abstract. In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2-CY tilted algebras for simple/minimally elliptic curve singuralities.

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the Ext-algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the “no loop” conjecture.