Related papers: Abstract representation theory via coherent Auslan…

We study the representation theory of Dynkin quivers of type A in abstract stable homotopy theories, including those associated to fields, rings, schemes, differential-graded algebras, and ring spectra. Reflection functors, (partial)…

We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between…

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…

We show that variants of the classical reflection functors from quiver representation theory exist in any abstract stable homotopy theory, making them available for example over arbitrary ground rings, for quasi-coherent modules on schemes,…

This paper concerns preprojective representations of a finite connected valued quiver without oriented cycles. For each such representation, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain…

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…

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…

Let $\mathbb{k}$ be an algebraically closed field. Connections between representations of the generalized Kronecker quivers $K_r$ and vector bundles on $\mathbb{P}^{r-1}$ have been known for quite some time. This article is concerned with a…

The quiver Hecke algebra $R$ can be also understood as a generalization of the affine Hecke algebra of type $A$ in the context of the quantum affine Schur-Weyl duality by the results of Kang, Kashiwara and Kim. On the other hand, it is…

We develop a novel combinatorial perspective on the higher Auslander algebras of type $\mathbb{A}$, a family of algebras arising in the context of Iyama's higher Auslander-Reiten theory. This approach reveals interesting simplicial…

We consider a new correspondence between representations of algebras with radical square zero and representations of species. We show that the stable category of representations of such algebra embeds into the representation category of the…

Representations of equipped graphs were introduced by Gelfand and Ponomarev; they are similar to representation of quivers, but one does not need to choose an orientation of the graph. In a previous article we have shown that, as in Kac's…

This survey offers an overview of an on-going project on uniform symmetries in abstract stable homotopy theories. This project has calculational, foundational, and representation-theoretic aspects, and key features of this emerging field on…

We completely describe the tree classes of the components of the stable Auslander-Reiten quiver of a quantum complete intersection. In particular, we show that the tree class is always $A_{\infty}$ whenever the algebra is of wild…

These are the notes for a course on representations of quivers for second year students in Paderborn in summer 2007. My aim was to provide a basic introduction without using any advanced methods. It turns out that a good knowledge of linear…

For a complex finite-dimensional simple Lie algebra $\mathfrak{g}$, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this…

Given a quiver, a fixed dimension vector, and a positive integer n, we construct a functor from the category of D-modules on the space of representations of the quiver to the category of modules over a corresponding Gan-Ginzburg algebra of…

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…

We introduce a notion of representation for a class of generalised quivers known as Coxeter quivers. These representations are built using fusion categories associated to $U_q(\mathfrak{s}\mathfrak{l}_2)$ at roots of unity and we show that…

For each simple Lie algebra $\mathfrak{g}$ of simply-laced type, Hernandez and Leclerc introduced a certain category $\mathcal{C}_{\mathbb{Z}}$ of finite-dimensional representations of the quantum affine algebra of $\mathfrak{g}$, as well…