# Questions tagged [quiver]

A quiver is an oriented graph which might contain multiple edges and loops. The terminology is used in representation-theory of finite dimensional algebras, where one considers functors from this graph, viewed as a category, to the category of vector spaces.

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### The category of finite dimensional right $KG$-modules is equivalent to the category of finite dimensional representations of a quiver $Q$

Let $K$ be an algebraically closed field and $G$ be a finite group such that $|G|$ is not divisible by the characteristic of $K$ (so that Maschke's theorem can be applied). Let $Q$ be the quiver ...
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### Systematically labeling arrows in GAP's QPA

I'm trying to build a quiver in QPA which has 42 nodes, labeled by some of the signvectors in $\{\pm 1\}^6$. I want to build a rule that assigns an arrow $a\to b$ if $a$ and $b$ differ in exactly one ...
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### On the Variety of Representations of a Fixed Dimension Vector

I have been trying to understand the proof of Gabriel's Theorem from Ralf Schiffler's 'Quiver Representations'. In chapter 8, he has defined, for a finite quiver $Q=(Q_0,Q_1)$ without oriented cycles ...
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### $KQ$ acylic connected finite dimension path algebra. $Cok(\oplus_{x\in Q_0}Hom_k(M_x,M'_x)\to\oplus_{a\in Q_1}Hom_k(M_{s(a)},M'_{t(a)}))=Ext^1(M,M')$

Let $Q=(Q_0,Q_1,s,t)$ be an acylic connected finite quiver where $Q_0$ are vertices, $Q_1$ are arrows, and $s,t:Q_1\to Q_0$ are source and target respectively. Let $K$ be a field. Let $KQ$ be the ...
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### Looking for an example of finite acyclic quiver path algebra with a simple module of infinite projective dimension

I am looking for an example of finite acyclic quiver path algebra with a simple module of infinite projective dimension. I have considered the following. Consider quiver $1\to 2\to 3\to 4$. Form path ...
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Let $Q$ be a finite, connected, and acyclic quiver. We have that: there exists a bijection between $Q_0$ (set of vertices of $Q$) and the set $\{1,\dots,n\}$ such that if we have an arrow $i \to j$,...
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### Path algebra $A=kQ$ is not isomorphic to $A^{op}$

This is an exercise from An Introduction to Quiver Representations by Harm Derksen and Jerzy Weyman. Exercise 1.6.3. Let $Q$ be the quiver \begin{equation*} \circ \longrightarrow \circ \...
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### Are diagrams of a quiver the same as small diagrams

I've just started to wrap my head around category theory, and came across two (from my perspective not obviously equivalent) definitions of a (small) diagram in a category $\mathcal{C}$: Definition 1 ...
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### Morphisms from injective to preprojective modules for non-Dynkin quivers

Let $Q$ be a quiver without oriented cycles. Suppose in addition that $Q$ is non-Dynkin. I am looking for the reference or proof of the following fact: Let $I$ be an injective $kQ$-module and $M$ be ...
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### Generating arbitrarily large indecomposable quiver representations

Consider a quiver $Q$ which is not of (euclidean or affine) ADE-type, i.e., $Q$ has wild representation type. Question: How can I generate an indecomposable $Q$-representation $M$ of arbitrarily ...
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### Stability conditions for a quiver

Let $Q$ be the quiver with vertices $I = \{i_1,i_2,i_3,i_4,j\}$ and for each $k$ one arrow $i_k \to j$. Let $\Theta$ the function on $\Bbb Z I$ defined by $\Theta(i_k) = 0, \Theta(j) = 1$. ...
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### Explanation of $Theta$ semi stable points (or representation)

I try to understand the precise meaning of a $\Theta$-semi-stable set of points. Say in the context of quiver representations: Let $Q = (Q_0, Q_1)$ be a quiver $(V_i, \alpha_{i,j})$ a representation ...
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### Reflection Functor of a Quiver

Fix a Quiver $Q=(Q_{0},Q_{1})$, where $Q_{0}$ is the set of vertices and $Q_{1}$ is the set of edges. For $j\in Q_{0}$ define $\sigma_{j}Q$ to be the quiver in which all that include the vertex $j$ ...
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### Computing Hochschild cohomology of an algebra in GAP

I would like to calculate Hochschild cohomology of a path algebra of a quiver (modulo some relations) using GAP. Creating the quiver and its path algebra modulo relations is explained in detail in ...
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### Choosing different term orders for Groebner basis calculations in GAP

I am using the QPA and GBNP packages in GAP to analyze path algebra quotients. I use the GBNP package for computing Groebner bases for the ideals in the path algebras, and the term orders for these ...
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### admissible ideal of affine rank 3 acyclic quiver $\tilde{A}_{12}$

Does affine rank 3 acyclic quiver $\tilde{A}_{12}$ with arrows $2 \to 1$, $3\to 2$ and $3 \to 1$ has admissible ideal? For any finite quiver $Q$, a two-sided ideal $I$ of $KQ$ is said to be ...
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### Problem with hom-spaces and their dimensions in GAP

I am running the following example in GAP using the QPA package: ...
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### Mistake in the proof of Theorem 2.24 of Quiver Representations by Ralf Schiffler?

I’m currently reading through Ralf Schiffler’s Quiver Representations, but I’m having trouble with the proof of the theorem below. In the following, $Q$ denotes a finite, acyclic quiver with set of ...
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For categories $\mathcal{C}$ and $\mathcal{D}$, a pair of functors $\mathcal{C} \xrightarrow{\;R\;} \mathcal{D}$ and $\mathcal{C} \xleftarrow{\;L\;} \mathcal{D}\,$ are an adjoint pair if for any ...
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### Is this functor a left adjoint?

I have a functor $sGrph \to Quiv$ which replaces each edge in a given simple graph with a cospan (and leaves the vertices alone). This functor clearly preserves colimits. Is it a left adjoint?
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### idempotent in quiver theory

I am studyng quivers and algebras of the text "Elements of the representation theory of associative algebras", author Assem. In the page 46 says: "any idempotent $ε$ of $ε_a(KQ)ε_a$ can be written in ...
Definitions A linear representation of a finite group $G$ is a group homomorphism: $$\rho:G\to\text{GL}(V),$$ where we shall assume that $V$ is a finite dimensional vector space. We call $V$ the ...
For the quiver $1 \rightarrow 2$, we can consider the category of representations $Rep \;(Q)$. The objects are representations of this quiver, and the morphisms are such that we have \require{AMScd}...