**2**

votes

**1**answer

170 views

### Second cohomology groups of Nakajima quiver varieties

Let $X=M(v,w)$ be a Nakajima quiver variety for a quiver $Q$. Can one calculate the second singular cohomology groups $H^2(X,\mathbb Z)$ or $H^2(X,\mathbb C)$ explicitly, and if not, are there some ...

**5**

votes

**1**answer

404 views

### Why Jacobson, but not the left (right) maximals individually?

I firstly asked the following question on MathStackExchange a couple of months ago. I did not receive any answers, but a short comment. So, I decided to post it here, hoping to receive answers from ...

**2**

votes

**1**answer

70 views

### Reference that contains examples of absolutely indecomposable representations of quivers over a finite field

Is there a reference that lists/discusses examples of absolutely indecomposable representations of quivers over a finite field (absolutely indecomposable = does not decompose into a direct sum over ...

**5**

votes

**0**answers

189 views

### For which quiver varieties is Kirwan surjectivity known?

The cohomology of Nakajima quiver varieties is a quite interesting object. It's equipped with some natural classes given by the Chern classes of the tautological bundles associated to the spaces in ...

**16**

votes

**0**answers

275 views

### 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 ...

**2**

votes

**0**answers

166 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) ...

**1**

vote

**0**answers

49 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" ...

**1**

vote

**1**answer

90 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.

**2**

votes

**1**answer

279 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 ...

**7**

votes

**1**answer

217 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) ...

**22**

votes

**3**answers

1k 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 ...

**1**

vote

**1**answer

220 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**

votes

**1**answer

198 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 ...

**3**

votes

**1**answer

160 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 ...

**3**

votes

**2**answers

277 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 ...

**4**

votes

**0**answers

330 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 ...

**6**

votes

**2**answers

728 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 $Q$ ...

**5**

votes

**0**answers

270 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?

**3**

votes

**0**answers

99 views

### Description of modules over self-injective algebras of finite representation type

Is there any description of indecomposable modules and irreducible morphisms over self-injective algebras of finite representation type? I am interested mainly in such a description for nonstandard ...

**5**

votes

**1**answer

324 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 ...

**4**

votes

**1**answer

243 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 ...

**5**

votes

**2**answers

680 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 ...

**12**

votes

**0**answers

390 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 ...

**2**

votes

**3**answers

358 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 ...

**3**

votes

**1**answer

185 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 ...

**2**

votes

**1**answer

178 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 ...

**6**

votes

**2**answers

510 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 ...

**1**

vote

**1**answer

410 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 ...

**5**

votes

**1**answer

357 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. ...

**8**

votes

**1**answer

520 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 ...

**3**

votes

**1**answer

576 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 ...

**8**

votes

**1**answer

3k 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 ...

**19**

votes

**3**answers

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 ...

**4**

votes

**1**answer

254 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**

votes

**1**answer

615 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 ...

**11**

votes

**0**answers

450 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 ...

**3**

votes

**0**answers

186 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 ...

**6**

votes

**1**answer

248 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 ...

**3**

votes

**2**answers

1k views

### 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 ...

**17**

votes

**4**answers

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 ...

**1**

vote

**1**answer

201 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 ...

**7**

votes

**3**answers

803 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 ...

**6**

votes

**0**answers

512 views

### 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 ...

**11**

votes

**2**answers

1k views

### 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 ...

**8**

votes

**3**answers

912 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 ...

**6**

votes

**0**answers

413 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 ...

**5**

votes

**3**answers

508 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 ...

**8**

votes

**2**answers

738 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 ...

**7**

votes

**2**answers

901 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 ...

**6**

votes

**1**answer

455 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 ...