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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understanding the root system of a quiver

I am beginning to study quiver representation theory and I am trying to understand the basic set-up of the root system associated to a finite connected graph, as first described by V. Kac. I read (e.g....
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Intuitive explanation of quiver $Q^{(n,s)}$.

On page 6 of this article, the author defined a new quiver. Fix positive integers $n$ and $s$. We define the quiver $Q:=Q^{(n, s)}$ with the set $Q_{0}$ of vertices and the set $Q_{1}$ of arrows by $$ ...
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When does a Path Algebra give a unique Quiver?

I am studying the text An introduction to Quiver Representations by Derksen. Exercise 1.5.4 asks to prove that if the path algebras $\mathbb{C}Q$ and $\mathbb{C}Q'$ are isomorphic $\mathbb{C}$-...
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Simple modules corresponding to vertices of a quiver with potential.

Given a quiver with reduced potential $(Q,W)$. The simple dg module over the complete Jacobian algebra $J(Q,W)$ are in bijection to the set of vertices of Q ? How do the simple modules look like ? And ...
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Kerodon 2.4.4.10.: pushouts of graphs associated to a pushout of simplicial sets

I don't understand a nuance in Theorem 2.4.4.10. of Kerodon. To not burden the question with exposition, I will only provide information for the relevant part. Fix a natural number $m$. For a ...
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36 views

Quiver representation - definition clarification

I'm studying quiver representation on "Introduction to representation theory" by Etingof and others. Often I read on this book about a representation being injective or surjective at some ...
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50 views

Walks on graphs and tensor products

In Qiaochu's article https://qchu.wordpress.com/2010/03/07/walks-on-graphs-and-tensor-products/#more-4807, he gives the following setup: Construct a directed graph $\Gamma(V)$ as follows: its vertices ...
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32 views

How do you talk about representations of a subquiver of $A_n$?

Let $Q$ be a linear Dynkin quiver of type $A_n$, which looks like $$1 \longrightarrow 2 \longrightarrow 3 \longrightarrow \cdots \longrightarrow (n-1) \longrightarrow n$$ Then we know by Gabriel's ...
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46 views

Are small categories necessarily free?

In the same spirit that all vector spaces are free, is any small category a free category?. If not, there is a counterexample? I'm interested to know if diagrams indexed by quivers covers all cases of ...
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Section on category of $K$-representations of a quiver

Given a quiver with $Q_0$ and $Q_1$ the sets of its vertices and edges respectively and a commutative ring $K$, we define a $K$-representation of $Q$ as a pair $$V=\left[ \{V_i\}_{i \in Q_0}, \{V(\...
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33 views

Confusion in module structure of indecomposable injectives over quiver path algebra

I got very confused while working out the explicit module structure on the indecomposable injective modules $I_i$ over a finite-dimensional quiver path algebra $kQ/I$. I know that the right injective ...
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Morita theory and non basic path algebra

I'm learning about representation theory of associative algebras. In my studies arrised less technical questions involving Morita theory and quivers that will exposed. Morita theory give us a way to ...
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54 views

Equivalence between rep$(Q)$ and modules over $KQ$. Does it preserve indecomposability?

I am studying representation theory at the moment and I was wondering the following (Since I found no source discussing it): Let $Q$ be a quiver and let $KQ$ be its path algebra. Let Rep($Q$) be the ...
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30 views

Computing the number of projective-injective modules

Let $A$ be a finite dimensional $\mathbb{K}$-algebra, where $\mathbb{K}$ is an algebraically closed field. How does one compute (homologically) the number of projective-injective $A$-modules? Maybe ...
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71 views

Projective representations of 1-loop quiver

I study (finite) representations (over an algebraically closed field $k$) of the 1-loop quiver $Q$ which is defined by having a single vertex and a single edge. So, representations of $Q$ are just ...
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51 views

Intuition on the Cuntz-Krieger relations for Leavitt path algebras and graph $C^*$-algebras

Let $k$ be a commutative ring with unity and $E$ a quiver with source and target functions $E^1 \xrightarrow{s,t} E^0$. The Leavitt path algebra of $E$ is the quotient of the path algebra of the ...
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Injectivity of quiver representation

I study representations of the quiver Q with two verticies and one arrow (from the 1. to the 2. vertex). In particular, I'm interested to determine wheter or not the representation $V:=(k \overset{id}{...
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42 views

Find two non-isomorphic categories with the same graph

I've started reading Abstract and Concrete Categories: The Joy of Cats to learn about Category Theory, and I've got stuck on one of the first exercises. I have a background in Computer Science, and to ...
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37 views

Extensions of simple modules and indecomposable morphisms of projectives

$\newcommand\rad{\operatorname{rad}}$When reading about the construction of Gabriels Ext-quiver, I often read somethink like the following. Let $A$ be a f.d. algebra over an algebraically closed field....
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Without Gabriel's theorem, how do you classify indecomposable reps of $\mathbb{A}_n$?

Given the Dynkin quiver of type $\mathbb{A}_n$ without orientation, $$ \begin{aligned} 1\!-\!2\!-\dotsb-\!n \end{aligned} $$ let $M(i,j)$ be a representation of $\mathbb{A}_n$, such that $V_l=K$ ...
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40 views

equivalence of Rep$(Q)$ and Mod($KQ$). Does it preserve length/dimension?

I am studying representation theory at the moment and I was wondering the following (Since I found no source discussing it): Let $Q$ be a quiver and let $KQ$ be its path algebra. Let Rep($Q$) be the ...
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58 views

Categories and quivers / directed graphs associated

I am currently working on associating categories and quivers. a) Let $C$ be an arbitraty category. I need to show that one can associate with $C$ a quiver $Q_C$ by sending an object $X$ in $C$ to a ...
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46 views

Paths in a quiver form a $K$-algebra.

I am attending a course in foundations in representation theory and I am struggling to grasp the concept of path algebras, more explicitly why paths in a quiver form a $K$-algebra. Let $Q$ be a ...
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Connected quiver

Let $\mathcal{Q}$ be a quiver. My goal is to show that if $k$ has no zero divisors then: $\mathcal{Q}$ is connected if and only if $k\mathcal{Q}$ is connected $k\mathcal{Q}$ is the algebra generated ...
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58 views

Path in a quiver to a path in path algebra in GAP

Is it possible to obtain a path in the path algebra without explicitly specifying the arrow? More specifically, I have a string with letters which are arrows in the quiver. I need to check whether in ...
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Field elements in quiver and relations

Let $A=KQ/I$ be a quiver algebra such that the admissible ideal $I$ contains only the field elements $0,1$ and $-1$. Question: Is it true for every basic idempotent $e$ that the algebra $eAe$ is ...
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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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54 views

Why is a morphism of quiver representations a subset of this product of morphisms?

Given two representations $M,N$ of a quiver Q, why is the set of all morphisms of representations from $M$ to $N$ a subspace of $\prod_{i\in Q_0} \mathrm{Hom}(M,N)$? I mean why the product of $\mathrm{...
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Extension-algebras of $A_4$

Consider the quiver $Q\colon 1\xrightarrow{\alpha} 2\xrightarrow{\beta} 3\xrightarrow{\gamma} 4$ and the algebra $A=k[Q]/(\gamma\beta\alpha)$. Denote the simple $A$-modules by $L(-)$ and let $M$ be ...
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Are these the projective and injective representations of these quivers?

Find all indecomposable projective and indecomposable injective representations of these two quivers over the field $k$ up to isomorphism. I've drawn my answer in this picture. Can you please check ...
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How do we factorize the path algebra by the relation?

I read Representation Theory of Artin Algebras by Auslander, Reiten and Smalo and I have a question about this example. Can you please explain why does the basis of $\Lambda \bar{e_1}$ contain $\bar{\...
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Showing that two path algebra quotients aren't isomorphic

I've been working through Auslander, Reiten, and Smalø's Representation Theory of Artin Algebras, and I've gotten stuck on Exercise III.8(c). The larger exercise has me consider the following quiver $...
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73 views

Topological Invariance in Data Structures

I need to do a PhD in Pure Mathematics and I am thinking of Topological Data Analysis. I want to use persistent homology and quiver representation to obtain topological features in data structures. ...
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Indecomposble Direct Summand of Nilpotent Endomorphism of Quiver Representation

I was reading the proof of Gabriel's Theorem (Theorem 8.12) from Ralf Schiffler's 'Quiver Representations'. The proof begins by showing that for an indecomposable module $M$, the vector space $Hom(M,M)...
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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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64 views

Topological Group Structure on Endomorphism Group of a Quiver Representation and Relations with Dimensions of Varieties

I have been reading chapter 8 of Ralf Schiffler's 'Quiver Representations'. There he has defined, for a finite quiver $Q=(Q_0,Q_1)$ without oriented cycles and for a (fixed) dimension vector $\textbf{...
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Variety of Quiver Representations

I have been reading chapter 8 of Ralf Schiffler's 'Quiver Representations', where he proves Gabriel's Theorem characterizing connected quivers with finitely many isoclasses of indecomposable ...
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Generating relations of an admissible ideal are not uniquely determined

Let $Q=(Q_0,Q_1)$ be a quiver as shown below, where $Q_0=\{1,2\}$, and $Q_1=\{\alpha,\beta,\gamma\}$, $\alpha:1\to 1$, $\beta:1\to 2$, $\gamma:2\to 1$. I would like to show that two admissible ideals $...
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Bound quiver representation of matrix algebras

I have been trying to write a bound quiver representation of each of the following algebras: $$A=\begin{bmatrix} K&0&0&0&0\\ K&K&0&0&0\\ K&0&K&0&0\\ K&...
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62 views

Every indecomposable representation is a brick if $Q$ has no cycles

I'm currently following a course on quiver representations, and stumbled across the definition of a brick, which is a quiver representation $M$ with the property that $\mathrm{End}(M)\cong k$, where $...
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85 views

Maximal ideals in the path algebra of a quiver

Let $Q = (Q_0,Q_1)$ be a finite quiver and let $kQ$ be its path algebra, that is the algebra of formal linear combinations of paths on $Q$ with multiplication given by concatenation of paths. I think ...
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Decomposition of quiver representation with Jordan cell map

I am reading Persistence Theory: From Quiver Representations to Data Analysis by Steve Y. Oudot and have the following question on Gabriel's theorem. Consider the quiver $$\bullet \longrightarrow \...
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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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An admissible numbering of an acyclic quiver

Let $Q$ be an acyclic quiver. In here, we read 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 $i \to j$, then $j >...
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A Request for Clarification on Quiver Reps and Gabriel's Theorem

I've turned myself around thinking about indecomposable representations of quivers and Gabriel's Theorem and I would be grateful if someone could point out my misunderstanding. Suppose you have a $...
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55 views

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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104 views

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 ...