# Tagged Questions

"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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### Quotient of triangulated category? (quiver)

This maybe a stupid question, but I really want to know the answer: Background: Given a quiver with potential, one can consider the derived category of the complete Ginzburg algebra of it, then ...
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### Realisation of the preprojective algebras as $F(\Delta)/T$ over some quasi-hereditary algebra

Let $A$ be the Auslander algebra of $K[x]/(x^n)$ for some $n \geq 2$, which is quasi-hereditary with some characteristic tilting module $T$. Dlab and Ringel showed in their paper "The Module ...
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### Auslander-Reiten theory for Gorenstein algebras

In the paper "Cohen-Macauley and Gorenstein artin algebras", Auslander and Reiten have a short section about Auslander-Reiten theory in Gorenstein algebras (I always assume we have an artin algebra ...
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### 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 ...
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### 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 ...
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### 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 $\Lambda$...
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### Is this modified bound quiver algebra necessarily representation-finite?

Suppose that $A = kQ/I$ is a bound quiver algebra for $k$ an algebraically closed field, $Q=(Q_0, Q_1)$ a finite connected quiver with no oriented cycles with no multiple edges or self-loops, and $I$ ...
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### Is a quotient of a bound quiver algebra of finite representation type also representation-finite?

Let $A = kQ/I$ be a bound quiver algebra for some algebraically closed field $k$, $Q$ a finite connected quiver without oriented cycles, and $I$ an admissible ideal. Say that $I'$ is also an ...
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### 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 ...
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### When is a given quiver algebra a hopf algebra?

Given a finite dimensional selfinjective quiver algebra A over a finite field (or more generally an arbitrary field). Whats the best way to check if the algebra A has a Hopf algebra structure or not? ...
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### Two quivers, finitely many nonisomorphic representations of $\mathbb{C}Q$

Consider the following two quivers:                             ...
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### How is the free modular lattice on 3 generators related to 8-dimensional space?

Here are three facts which sound potentially related. What are the actual relationships? In 1900, Dedekind constructed the free modular lattice on 3 generators as a sublattice of the lattice of ...
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### “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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 Uq($\hat{sl_2}$)...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...