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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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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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Are valued-quiver species and combinatorial species related?

In this paper, Joel Lemay defines a species as a generalization of a valued quiver $Q$, where to each node $i \in Q_0$ we assign a division ring $\mathbf{k}_i$, and to each arrow $(a \colon i \to j) \...
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What's the term to describe a morphism that isn't a composite of morphisms?

Is there a term to describe a morphism in a category that cannot be written as a composition of two (non-identity) morphisms? Naively Googling around I haven't found anyone talking about this, so I ...
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Is there a natural way to regard a valued quiver as a category?

You can think of a quiver, a collection of vertices $Q_0$ and arrows $Q_1$, as a free category with objects $Q_0$ and morphisms generated by the morphisms of $Q_1$. More precisely this is a functor ...
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What kind of Categorical object is an RDF Model?

RDF1.1 semantics defines an RDF Interpretation and mixes up the model and the mapping from the syntax to the model. I want to work out independently what its category of all models is. (In order to ...
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How should one model RDF semantics in terms of category theory? [closed]

There is something very striking about the similarity in structure between RDF and Category Theory. The RDF Abstract syntax is a graph, the type known as a Quiver in Category Theory. Every Quiver ...
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What is the (co)limit of a Quiver?

Given that a Quiver (also known as directed multigraphs) is so important in category theory, being that every Quiver produces a free Category, and every Category has an underlying Quiver, I was ...
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77 views

List of path arrows in GAP

I am using the QPA package with GAP and have the following problem: I want to, given a path $p$, obtain a list (preferably ordered) of the arrows which make up $p$. For example, if I have arrows $...
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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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Usage and Realization of a Quiver Representation.

From my understanding, a quiver is basically a finite directed graph (a multidigraph). And a representation assigns a vector space to each vertex, and a linear map to each arrow. Formally, $Q = (Q_0, ...
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Can this puzzle be solved using the representation theory of quivers?

This riddle originates in the youtube video here. It's mathematical content was summarised here as follows: There's a $5\times 5$ grid of nodes, all nodes are (bidirectionally) connected to their ...
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Full projective resolutions for path algebras in GAP

I am trying to do various computations on path algebras using GAP. One such computation is obtaining projective resolutions of simples modules (and possibly other modules). The command in the GAP ...
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Indecomposable representation of an acyclic quiver on 3 vertices

I have a question: Suppose we have the quiver \begin{matrix} & 1 & \overset{}{\longrightarrow} && 2 \\ && \searrow_{} & & \downarrow_{} \\ & & &&...
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From a lower triangular matrix to its quiver representation

I have a question about quivers: Suppose we have an arbitrary lower triangular matrix algebra $A= \{\begin{pmatrix} a & 0 & 0\\ c & b & 0\\ e&d&a \end{pmatrix}: a , b, c, d \...
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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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Significance of adjoint relationship with Ext instead of Hom

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 ...
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All linear representations of a finite group are quiver representations?

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 ...
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62 views

Meaning of “fully decomposable”

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}...
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How to construct free category on a quiver?

Let $Q$ be a quiver defined as a covariant functor from this category $\chi$: $$\left\{Q_1 \overset{h}{\underset{t}{\Large\rightrightarrows}} Q_0 \right\}\overset{\Gamma}\longrightarrow\mathsf {Set}$$...
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Algebras with the same Auslander -Reiten quiver have isomorphic categories of modules?

Maybe this is a stupid question but suppose you have two quivers $Q$ and $Q'$ (maybe we can suppose some nice property, like not having oriented cycles, so that kQ has finite dimension over $k$) whose ...
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Radical and socle of the path algebra

Let $Q$ be an infinite quiver without oriented cycle. Is it true that the radical of $KQ$ is generated by all the arrows? What can we say about its socle? Thank you!
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Kronecker quiver and tensor product

Good day, based on representation theory of Assem, and the definition of tensor product I need to find $\varphi^1_{21}$ and its domain in the kronecker quiver. I did the next: but Im confused to ...
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path algebra and quiver

Im studyng quivers with the text of Assem: My question is: In "the words on {$\alpha, \beta$}", refers it in particular to the free monoide kleene star? I dont uderstand the relation with the free ...
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How do the minimal right almost split maps ending in an injective module of a path algebra look like?

Let $Q$ be a finite quiver without oriented cycles and let $k$ be a field. Then $A = kQ$ is a finite-dimensional path algebra. Let $e_1, \dots, e_n$ be the idempotents corresponding to the vertices $1,...
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Citable reference for structure of extension quiver of path algebra

Let $Q$ be a finite quiver such that $A = kQ$ is a finite dimensional path algebra, where $k$ is a field. Then let $1, \dots n$ be the vertices of $Q$ and $S_1, \dots, S_n$ the corresponding simple ...
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Quiver pullback (question about notation)

I am working through the book Quiver Representations by Schiffler and I am confused about some notation regarding pullbacks. The following is example 2.13 (pp. 65): Let $k$ be a field and $Q$ the ...
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193 views

quivers and tensor product

I am studyng this lemma of Assem: Good, now let $Q$ the kronecker quiver : then there is algebra isomorphism $KQ\cong \begin{bmatrix} K &0 \\ K^2 & K \end{bmatrix}$ where $K^2$ is ...
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52 views

How can I tell these three representations from the Auslander Reiten quiver are not isomorphic?

So I was studying the local shape (around the simple representation $S(3)$) of the Auslander Reiten quiver of this $Q$: And so far I concluded that the neighborhood of the representation $S(3)$ looks ...
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One parameter family of non-isomorphic indecomposable quiver representations

Consider a quiver $Q$ that is not of finite type. Can you construct a free $k[t]$-representation $M$ of $Q$ such that the $k$-representations $M\otimes_{k[t]}k[t]/(t-\lambda)$ are indecomposable and ...
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267 views

How to understand the two definitions of path algebra?

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 ...
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82 views

Are identity morphisms on the underlying quiver of a category loops or empty paths?

While thinking about free categories generated from the underlying graph of a category, I got myself a bit puzzled over whether free categories always generate new identity morphisms, distinct from ...
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2answers
249 views

How do I show this isomorphism between an opposite endomorphism ring and a module over path algebras?

So I'm working with quivers with relations. Let $\Gamma$ be a quiver with vertices $\Gamma_0 = \{1,2,...,n\}$ and let $\rho$ be a set of relations in $\Gamma$ with $J^t \subset \langle\rho\rangle \...
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Modules in parabolic $\mathcal O^\mathfrak p$ in terms of path algebras

As already brought up in this question, I have some difficulties understanding the the modules in the parabolic category $\mathcal O_0^\mathfrak p$. Although I got a lot of comments that helped me ...
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Endomorphisms of a quiver

I know that when $k$ is algebraically closed any indecomposable representation of a quiver over $k$ has endomorphism ring isomorphic to $k$ (this is because by Schur's lemma it is a division algebra). ...
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44 views

Special multiserial algebra and monomial special multiserial algebra

Let $K$ be an algebraically closed field. $Q=(Q_0,Q_1)$ is a quiver($Q_0$ is a finite set of vertices and $Q_1$ is a finite set of arrows). An algebra $KQ/I$ is called monomial if $I$ is a ideal ...
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About irreducible morphisms

A morphism $f: X\to Y$ in mod A is called irreducible if f is not a section, f is not a retraction, and whenever $f = gh $ for some morphisms $h: X \to Z$ and $g: Z \to Y$, then either $h$ is a ...
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Classify 2-dim representation of the path algebra of quivers without oriented cycle

This is problem 2.23 from Etingof's Introduction to representation theory. Let Q be a quiver without oriented cycles, and $P_Q$ the path algebra of $Q$. Find irreducible representations of $P_Q$ ...
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179 views

Why is a connected finite dimensional wild path algebra is isomorphic to a special matrix algebra?

Let $H$ be an associative $k$-algebra, where $k$ is a field. Let $1 = e + f$ be a decomposition of $1$ into a sum of two orthogonal idempotents. Question: When is $H \cong \left[ {\begin{array}{cc} ...
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How to actually find almost split sequences?

I was trying to construct the AR-quiver, or at least a part of it, for the quiver $$Q = \quad\begin{array}{ccccc} & & 1 & & \\ & ^\delta \...