Dedicated to H. Keller on the occasion of his seventy fifth birthday Abstract. These are expanded notes of four introductory talks on A∞-algebras,

. We study the moment map associated to the cotangent bundle of the space of representations of a quiver, determining when it is flat, and giving a stratification of its Marsden-Weinstein reductions. In order to do this we determine the possible dimension vectors of simple representations of deformed preprojective algebras. In an appendix we use deformed preprojective algebras to give a simple proof of much of Kac's Theorem on representations of quivers in characteristic zero. Keywords: Quiver, Representation, Moment map, Preprojective algebra Mathematics Subject Classifications (2000): Primary 16G20, 53D20 1. Introduction Let K be an algebraically closed field and let Q be a quiver with vertex set I . Representations of Q of dimension vector ff 2 N I are given by elements of the space Rep(Q; ff) = M a2Q Mat(ff h(a) \Theta ff t(a) ; K); where h(a) and t(a) are the head and tail vertices of an arrow a 2 Q; isomorphism classes correspond to orbits of the group G(ff) = / Y i...

Abstract. Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretic interpretation of cluster mutation in case of acyclic cluster algebras.

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.

Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a well-known ill-posed problem. We propose that knowledge of the closure relations, i.e., the stratification, of the orbits and bundles of the various forms may be applied in the staircase algorithm. Here we discuss and complete the mathematical theory of these relationships and show how they may be applied to the staircase algorithm. This paper is a continuation of our Part I paper on versal deformations, but may also be read independently.

Abstract. 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 algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related

Abstract. We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) cluster-tilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of cluster-tilted algebras of finite representation type are obtained along the way.

Let k be a field and A a finite-dimensional k-algebra of global dimension ≤ 2. We construct a triangulated category CA associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When CA is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results rely on quivers with potential. Namely, we introduce a cluster category C (Q,W) associated to a quiver with potential (Q, W). When it is Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic

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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