Realise acyclic cluster algebras via Hall algebras. Original idea was from Caldero-Chapoton, just for A,D,E case. Then Caldero-Keller gave a cluster multiplication theorem, but only in the simply-laced case. Using the correspondence due to BMRRT between clusters and tilting modules,

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.

Abstract Let A be a hereditary algebra over an algebraically closed field k and A (m) be the m-replicated algebra of A. Given an A (m)-module T, we denote by δ(T) the number of non isomorphic indecomposable summands of T. In this paper, we prove that a partial tilting A (m)-module T is a tilting A (m)-module if and only if δ(T) = δ(A (m)), and that every partial tilting A (m)-module has complements. As an application, we deduce that the tilting quiver KA (m) of A (m) is connected. Moreover, we investigate the number of complements to almost tilting modules over duplicated algebras. Key words: contravariantly finite categories, partial tilting modules, m-replicated algebras, tilting quivers

Abstract. Associated to any acyclic cluster algebra is a corresponding triangulated

Abstract. Cluster algebras were introduced by Fomin and Zelevinsky in order to understand the dual canonical basis of the quantized enveloping algebra of a quantum group and total positivity for algebraic groups. A cluster category is obtained by forming an appropriate quotient of the derived category of representations of a quiver. In this survey article, we describe the connections between cluster categories and cluster algebras, and we survey the representation-theoretic applications of cluster categories, in particular how they provide an extended version of classical tilting theory. We also describe a number of interesting new developments linking cluster algebras, cluster categories, representation theory and the canonical basis.

Abstract. Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. We construct many Frobenius subcategories of mod(Λ), which yield categorifications of large classes of cluster algebras. This includes all acyclic cluster algebras. We show that all cluster monomials can be realized as elements of the dual of Lusztig’s semicanonical basis of a universal enveloping algebra U(n), where n is a maximal nilpotent subalgebra of the symmetric Kac-Moody Lie algebra g associated to the quiver Q. Contents

Abstract. We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their cluster tilting objects are related by a simple combinatorial construction, which we call a flip-flop. We deduce that the posets of cluster tilting objects of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent. In particular, all Cambrian lattices corresponding to the various orientations of the same Dynkin diagram are universally derived equivalent. 1.

Abstract. We define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem. In the special case of Dynkin quivers with n vertices this gives the fundamental interrelationship between supports of the semi-invariants and the

Abstract. We define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem. In the special case of Dynkin quivers with n vertices this gives the fundamental interrelationship between supports of the semi-invariants and the Tilting Triangulation of the (n − 1)-sphere. This paper initiates a project to apply quiver representations and their semiinvariants to expose compatible combinatorial underpinnings for the tilting objects of cluster categories (and hence, clusters for cluster algebras), and for the homology of nilpotent groups. Here we focus on semi-invariants and tilting objects in cluster