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6 votes
0 answers
749 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 ...
3 votes
1 answer
102 views

References on coefficient quivers

I would like to study about coefficient quivers, but I cannot find a good reference, as book for example. I could find many papers working with coefficient quivers, but none of them give a book or a &...
13 votes
0 answers
615 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 $\...
4 votes
1 answer
108 views

How much is known about the non-degeneracy of Quiver-with-potential associated to closed punctured surfaces?

The potential of the quiver associated to surfaces is the canonical one given by Labardini-Fragoso's 2009 paper, who proved that the the QP associated to surfaces whose boundary is nonempty is rigid ...
2 votes
0 answers
104 views

$G$-module representations of a profinite quiver

I have a profinite directed graph $\Gamma$, i.e., I can think of $\Gamma$ as the inverse limit of a directed system of finite directed graphs under inclusion. To each vertex of the graph a $G$-module ...
8 votes
1 answer
663 views

References for quivers and derived categories of coherent sheaves for a string theory student

I'm a student mostly from physics knowledge hoping to learn about the math involved the string theory paper Topological Quiver Matrix Models and Quantum Foam. Context: The topological string theory ...
10 votes
1 answer
648 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 ...
3 votes
1 answer
177 views

$M^{ss}_{(2,2)}(K_3,(-1,1))$ is isomorphic to $M_{\mathbb{P}^2}(0,2)$

Suppose $K_3$ is the Kronecker quiver with 3 arrows, and $M^{ss}_{(2,2)}(K_3,(-1,1))$ is the moduli space of semi stable representation of dimension $(2,2)$ wrt the weight $(-1,1)$. It is claim in the ...
5 votes
0 answers
383 views

Kac's theorem for quiver representations over an arbitrary ground field

Let $Q$ be a quiver without loops (cycles of length 1). Kac proved that if $K$ is algebraically closed, the dimension vectors of indecomposable representations of $Q$ over $K$ are exactly the ...
4 votes
2 answers
418 views

Indecomposable representations of euclidean quivers

The classification of indecomposable representations of a Euclidean quiver is well-known over an algebraically closed field. I am interested in an analogous classification, but over an arbitrary field....
0 votes
0 answers
195 views

Automorphisms of weighted quiver

I am reading this paper strongly primitiv species with potentials I : mutations. In page 6, they give the definition of weighted quiver: a weighted quiver is a pair $(Q,d)$, where $Q$ is a loop-...
4 votes
2 answers
288 views

The explicit indecomposable representations of (any) Euclidean quiver of type E

It is known that for any quiver $Q$ that is an orientation of $\tilde{\mathbb{E}}_8$, the hereditary path algebra $KQ$ ($K$ being an algebraically closed field) is tame (but not finite). That is, in ...
5 votes
2 answers
566 views

Dimension of preprojective algebra of Dynkin type

Fix a field $\Bbbk$. Let $Q$ be a Dynkin quiver and let $\Pi(Q)$ be its preprojective algebra. It is well-known that in this case $\Pi(Q)$ is finite-dimensional, but I've been unable to find a ...
6 votes
0 answers
209 views

Classification of representation-finite algebras up to stable equivalence of Morita type

Assume $K$ is an algebraically closed field. I wanted to ask if there is a classification of the representation-finite $K$-algebras up to stable equivalence of Morita type (at least for some small ...
4 votes
0 answers
429 views

Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...
5 votes
1 answer
525 views

what is the injective hull of indecomposable module of preprojective algebra

Let $Q$ be a ADE type quiver and $s_i$ ($i$ runs through the vertices of $Q$) be the simple $\Lambda$-module with 1-dimensional vector space at vertex $i$ and zero-dim at other vertices. Here $\Lambda$...
1 vote
1 answer
172 views

Why are exchange graphs of quivers with the same underlying graph but have different orientations isomorphic?

I know the fact that (undirected) exchange graphs of quivers with the same underlying undirected graph but have different orientations are isomorphic (i.e. quivers that are just finitely many arrow-...
5 votes
1 answer
911 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
200 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 ...
3 votes
0 answers
679 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) $\mathbb{C}...
6 votes
1 answer
413 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 ...
18 votes
1 answer
6k 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 ...
8 votes
1 answer
1k 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 ...