Related papers: Auslander-Reiten triangles and quivers over topolo…

In a previous paper, math.AT/0304079, Auslander-Reiten triangles and quivers were introduced into algebraic topology. This paper shows that over a Poincare duality space, each component of the Auslander-Reiten quiver is isomorphic to…

Peter Jorgensen introduced the Auslander-Reiten quiver of a simply connected Poincare duality space. He showed that its components are of the form ZA_infty and that the Auslander-Reiten quiver of a d-dimensional sphere consists of d-1 such…

We give a new characterization of tilted algebras by the existence of certain special subquivers in their Auslander-Reiten quiver. This result includes the existent characterizations of this kind and yields a way to obtain more tilted…

For an indecomposable module $M$ over a path algebra of a quiver of type $\widetilde{\mathbb A}_n$, the Gabriel-Roiter measure gives rise to four new numerical invariants; we call them the multiplicity, and the initial, periodic and final…

In a k-linear triangulated category (where k is a field) we show that the existence of Auslander-Reiten triangles implies that objects are determined, up to shift, by knowing dimensions of homomorphisms between them. In most cases the…

We analyze Auslander-Reiten quivers of functorially finite resolving subcategories. Chapter 1 gives a short introduction into the basic definitions and theorems of Auslander-Reiten theory in A-mod. We generalize these definitions and…

This work is the sequel to Continuous Quivers of Type A (I). In this paper we define the Auslander-Reiten space of a continuous type $A$ quiver, which generalizes the Auslander-Reiten quiver of type $A_n$ quivers. We prove that extensions,…

Let Q be a quiver of type ADE. We construct the corresponding Auslander-Reiten quiver as a topological complex inside the Coxeter complex associated with the underlying Dynkin diagram. We use the notion of chamber weights coming from the…

We first provide an explicit combinatorial description of the Auslander-Reiten quiver $\Gamma^Q$ of finite type $D$. Then we can investigate the categories of finite dimensional representations over the quantum affine algebra…

This paper deals with the representation theory of a locally finite quiver in which the number of paths between any two given vertices is finite. We first study some properties of the finitely presented or co-presented representations, and…

This paper introduces Auslander-Reiten triangles in subcategories of triangulated categories. The main theorem of the paper shows that there is a close connection with covers and envelopes, also known as minimal right- and…

Let Q be a strongly locally finite quiver and denote by rep(Q) the category of locally finite dimensional representations of Q over some fixed field k. The main purpose of this paper is to get a better understanding of rep(Q) by means of…

This paper surveys recent contructions in higher Auslander--Reiten theory. We focus on those which, due to their combinatorial properties, can be regarded as higher dimensional analogues of path algebras of linearly oriented type…

Our main theorem classifies the Auslander-Reiten triangles according to properties of the morphisms involved. As a consequence, we are able to compute the mapping cone of an irreducible morphism. We finish by showing a technique for…

In a paper by Holm and Jorgensen, the cluster category $\mathscr{D}$ of type $A_\infty$, with Auslander-Reiten quiver $\mathbb{Z} A_\infty$, is introduced. Slices in the Auslander-Reiten quiver of $\mathscr{D}$ give rise to direct systems;…

We analyze Auslander-Reiten components for the bounded derived category of a finite-dimensional algebra. We classify derived categories whose Auslander-Reiten quiver has either a finite stable component or a stable component with finite…

We introduce the notion of filtered representations of quivers, which is related to usual quiver representations, but is a systematic generalization of conjugacy classes of $n\times n$ matrices to (block) upper triangular matrices up to…

Let X be a smooth projective scheme of dimension d >= 1 over the field k, and let C be an indecomposable coherent sheaf on X. Then there is an Auslander-Reiten sequence in the category of quasi-coherent sheaves on X, with C as right hand…

We give a survey on Auslander-Gorenstein algebras with a focus on finite-dimensional algebras. We put an emphasis on recent classification results for special classes of algebras and the newly discovered interactions of the Auslander-Reiten…

We recall several results in Auslander-Reiten theory for finite-dimensional algebras over fields and orders over complete local rings. Then we introduce $n$-cluster tilting subcategories and higher theory of almost split sequences and…