"Quiver" is the word used for "directed graph" in some parts of representation theory. The main reason to use the term quiver is to indicate an interest in considering representations of the quiver.

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Is there a cotangent bundle of a stable $\infty$-category?

Let $C$ be a stable $\infty$-category. Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following? When $C$ is the ...
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116 views

Ext Quivers and their applications to Representation Theory

I am looking for references that provide an overview of the following two topics (it can be multiple references if necessary): How to compute the Ext-quiver of a (locally finite or finite) ...
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46 views

“embedding” various matrix equivalences into the equivalence of particular linear map

Consider the square matrices over a (local) ring $R$, up to conjugation, $A\rightarrow UAU^{-1}$, where $U$ is an invertible matrix over $R$. Such an equivalence embeds into the "left-right" ...
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1answer
75 views

Bounded algebras of finite global dimension

Let $k$ be a field, $Q$ an acyclic finite quiver and $I$ an admissible ideal of $kQ$. I am looking for a reference for the fact that the bounded algebra $kQ/I$ has finite global dimension.
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1answer
237 views

Quiver representations

I'm wondering how to find indecomposable representations of a given quiver explicitely. In particular, I'm interested in the maximal indecomposable representation of $\mathbb{E}_8$(I'm working over ...
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203 views

How can one show that orbit closures in representations of a linear quiver don't have small resolutions?

Let $1\to \cdots\to n$ be a linear quiver of length $n$. Let $\mathbf{d}=(d_1,\dots,d_n)$ be a dimension vector. It's well known (for example, by Gabriel's theorem, but also by basic linear algebra) ...
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143 views

Cycles in Quivers and Path Algebras

I cannot find anything giving the algebra of a quiver with a single cycle on three or more vertices. In other words if your quiver consists of n vertices (n>2), and e_i is connected to e_{i+1} (taking ...
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943 views

How can classifying irreducible representations be a “wild” problem?

Let $q$ be a prime power and $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$-matrices. I've read and heard in several places (see e.g. this mathoverflow question) that classifying ...
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1answer
211 views

finitely presented representations

Let Q=(V,E) be a direct graph where V is the set of all its vertices and E denotes the set of all its arrows. $X$ is called a representation of Q by modules if it is a functor from Q to R-Mod. i.e. ...
4
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1answer
178 views

When are infinite dimensional path algebras hereditary?

I allready asked this on MO, but did not get any answer. Given a finite quiver with relations. When is the path algebra modulo relations hereditary? If the path algebra is finite dimensional or ...
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1answer
155 views

Pairs of paths with the same source and target

Commutative diagrams usually express path equivalences in a category and thus involve pairs of paths in a category with the same source and target. General diagrams - in categories resp. category ...
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2answers
264 views

Do morphisms of finitely-decomposable Quiver representations map indecomposables nicely?

Consider two quivers $Q$ and $Q'$ of type $A_n$, laid out horizontally like so: Given representations of $Q$ and $Q'$, Gabriel's theorem guarantees the existence of finitely many indecomposables ...
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303 views

Tensor product of quivers

As a special case of a general construction I have constructed "accidentally" a tensor product of quivers aka directed multigraphs (aka directed graphs for category theorists). Probably this is ...
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2answers
637 views

Derived category of varieties and derived category of quiver algebras

I have heard that derived category of coherent sheaves $\mathrm{Coh}(X)$ on any Fano varieties $X$ may be realized as derived category $\mathrm{Coh}(\mathrm{Rep}(Q,W))$ of representation of quiver $Q$ ...
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253 views

Homological dimension of completed path algebras.

Let A = c[Q]/I be a finite dimensional quotient of a path algebra over a quiver Q, with I being the ideal of relations. Is it true that the I-adic completion of A has finite homological dimension?
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1answer
304 views

The Fukaya category of a simple singularity (reference request)

I have heard that for an ADE singularity $f$, $ D^b\mathrm{Fuk}(f) \simeq D^b(\mathrm{Rep}\\ Q)$ where $Q$ is the corresponding Dynkin quiver. (As one would hope, if $\mathrm{Fuk}$ is some kind of ...
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232 views

Vertex embeddings of quantum groups via quivers

Let Uq be a quantised enveloping algebra of type affine ADE (untwisted). By the loop presentation of Uq, we see that for each vertex of the finite Dynkin diagram, there is an inclusion ...
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2answers
660 views

Do fixed point sets in equivariant crepant resolutions have the same cohomology? How about for the specific case of Nakajima quiver varieties?

A crepant resolution $f:Y\to X$ is a resolution of singularities with $f^*(K_X)=K_Y$. Crepant resolutions do not always exist, and when they exist they may not be unique. However, different crepant ...
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368 views

The derived category of integral representations of a Dynkin quiver.

Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write ...
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3answers
335 views

Quiver on tensor product

Let $Q=(Q_{0},Q_{1},h,t)$ be a finite quiver where $Q_{0}$ are the vertices, $Q_{1}$ the arrows and we have two maps $h: Q_{1} \rightarrow Q_{0}$ (head) and $t: Q_{1} \rightarrow Q_{0}$ (tail). Fix a ...
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1answer
183 views

Ring completion of $kQ$

Hello, Let $Q$ be a finite quiver, let $M$ denote the arrow ideal and let $kQ$ denote the path algebra. Endow $kQ$ with the $M$-adic topology. Now let $\mathcal{A}$ be the set of all formal series ...
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1answer
169 views

Indecomposable extensions of regular simple modules by preprojectives

Given four points in general position on $\mathbb{P}^2$ there exists a projection to $\mathbb{P}^1$ collapsing these four pairwise to two points. Its kernel is some fifth point on $\mathbb{P}^2$. In ...
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2answers
478 views

Does anyone recognize this quiver-with-relations?

Below I describe an infinite (but locally finite) quiver with relations. My question is whether anyone recognizes it and can provide appropriate pointers to the literature. I'm mainly interested in ...
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1answer
351 views

An algorithm for constructing the AR-quiver of a path algebra corresponding to a change in the orientation.

Considering the path algebra of the quiver $\mathbb{A}_n$, it is well known its Auslander-Reiten quiver with the canonical orientation of $\mathbb{A}_n$, that is, with all the arrows from, say, left ...
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349 views

Morita equivalence of acyclic categories

(Crossposted from math.SE.) Call a category acyclic if only the identity morphisms are invertible and the endomorphism monoid of every object is trivial. Let $C, D$ be two finite acyclic categories. ...
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488 views

Quivers of selfinjective algebras.

Let's say a quiver $Q$ is covered by cycles if each of it’s arrows can be included in an oriented cycle. It's easy to prove that if a path-algebra with relations $KQ/I$ (where $I$ is an admissible ...
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1answer
458 views

Indecomposable modules over preprojective algebras

Would you please give some references concerning the number of indecomposable modules over preprojective algebras of type $A_n$? More precisely, I need references about the following claim: The ...
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1answer
2k views

Intersection between category theory and graph theory

I'm a graduate student who has been spending a lot of time working with categories (model categories, derived categories, triangulated categories...) but I used to love graph theory and have always ...
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2k views

Why did Gabriel invent the term “quiver”?

A quiver in representation theory is what is called in most other areas a directed graph. Does anybody know why Gabriel felt that a new name was needed for this object? I am more interested in why he ...
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1answer
247 views

Is the F-polynomial of an indecomposable quiver representation irreducible ?

My question is as follows: is the F-polynomial of an indecomposable quiver representation irreducible as a polynomial, i.e. can it have a nontrivial factor? Here the F-polynomial is the generating ...
4
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1answer
582 views

Quiver varieties and the affine Grassmannian

Recently I was watching a talk: http://media.cit.utexas.edu/math-grasp/Ben_Webster.html and at the end the lecturer gave a correspondence (I was having trouble with subscripts so changed the notation ...
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445 views

Are the extra vertices in Nakajima's doubling of a quiver related to Langlands duality?

To define a Nakajima quiver variety associated to a quiver $Q = (Q_0,Q_1)$ (vertices and arrows), one first doubles it to $Q^\heartsuit$ by attaching an extra vertex to every old vertex in $Q_0$. Then ...
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178 views

Invariant Subvarieties of Variety of Quiver Representations

I'd like to understand a special case of the following rather general algebraic geometry question: Given an algebraic group $G$ acting on a variety $V$, can we describe the $G$-invariant subvarieties ...
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1answer
245 views

A question about saturation of quivers

Let $Q$ be an acylic quiver. Let $E$ and $F$ be finite dimensional representations, with $E$ indecomposable. Suppose that, for some positive integer $r$, the representation $F$ injects into $E^{\oplus ...
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quiver mutation

Hello to all, The phrase "quiver mutation has been invented by Fomin and Zelevinsky and has found numerous applications throughout mathematics and physics" is one that some of us encountered on a ...
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4answers
2k views

Deformations of Nakajima quiver varieties

Are deformations of Nakajima quiver varieties also Nakajima quiver varieties ? In case the answer to this is (don't k)no(w), here are some simpler things to ask for. (If you're a differential ...
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1answer
197 views

Hereditary algebras as quotient algebras

This is the first time I post a question on MO, so I'm shy a liite bit. Can you give a "non-trivial" example of a finite dimensional hereditary algebra which is quotient of an infinite dimensional ...
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724 views

Are the underlying undirected graphs of two mutation-equivalent acylic quivers isomorphic?

Quiver mutation, defined by Fomin and Zelevinsky, is a combinatorial process. It is important in the representation theory of quivers, in the theory of cluster algebras, and in physics. We consider ...
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Hirzebruch-Riemann-Roch for quiver varieties?

These days, I attended a workshop at North Carolina State University. The key lecturer is Professor Nakajima. He introduced two types of quiver variety. One of them is affine, another one is ...
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when are algebras quiver algebras ?

Good Morning from Belgium, I'm no stranger to the mantra that quiver-algebras are an extremely powerful tool (see for example the representation theory of finite dimensional algebras). But what is a ...
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879 views

construct scheme from quivers?

I heard from some guys working in noncommutative geometry talking about the idea that one can construct the noncommutative space from quivers. I feel it is rather interesting. However, I can not image ...
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397 views

What's known about the stalks of Lusztig's perverse sheaves on quiver varieties?

Lusztig has defined a category of perverse sheaves on the moduli space of representations of a Dynkin quiver (see his paper) corresponding to canonical basis vectors. I'm interested in the stalks ...
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3answers
489 views

Acyclic quivers differing only in arrow directions: functorial isomorphism of representation categories?

Let $Q$ and $R$ be two acyclic quivers which differ only in the directions of their arrows (i. e., the underlying undirected graphs are the same). 1. Does there exist an isomorphism of additive ...
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2answers
686 views

Are the strata of Nakajima quiver varieties simply-connected? Do they have odd cohomology?

Nakajima defined a while back a nice family of varieties, called "quiver varieties" (sometimes with "Nakajima" appended to the front to avoid confusion with other varieties defined in terms of ...
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855 views

What do the local systems in Lusztig's perverse sheaves on quiver varieties look like?

In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined ...
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1answer
417 views

Global dimenson of quivers with relations

Let Q be a finite quiver without loops. Then its global dimension is 1 if it contains at least one arrow. I'm trying to get some intuition about how much the global dimension can grow when we ...
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995 views

Near Trivial Quiver Varieties

So, today I started learning the definition of a quiver variety, and wanted to make sure I'm understanding things right, so first, my setup: I've been looking at the simplest case that didn't look ...
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ubiquity, importance of path algebras

I work in planar algebras and subfactors, where the idea of path algebras on a graph (alternately known as graph algebras, graph planar algebras, etc.) is quite useful. The particular result I'm ...
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2answers
421 views

Matrices into path algebras

I was thinking about quivers recently, and the following idea came to me. Let ei,j denote the matrix unit in Mn for 1 ≤ i,j ≤ n. Let Γ denote the complete quiver on vertices {1, ...
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When should I expect a quiver with potential to be rigid?

This question is pretty technical, but there are some very smart people here. Fix a quiver Q, WITH oriented cycles. Let k[[Q]] be the completed path algebra. (Like the path algebra, but we allow ...