# 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.

129 questions
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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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### 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 ...
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 $$\require{AMScd}... 1answer 73 views ### How to construct free category on a quiver? Let Q be a quiver defined as a covariant functor from this category \chi:$$\left\{Q_1 \overset{h}{\underset{t}{\Large\rightrightarrows}} Q_0 \right\}\overset{\Gamma}\longrightarrow\mathsf {Set}$$... 1answer 46 views ### 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 ... 1answer 102 views ### 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! 1answer 225 views ### 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 ... 1answer 58 views ### 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 ... 1answer 49 views ### 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,... 2answers 70 views ### 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 ... 0answers 47 views ### 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 ... 1answer 188 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 ... 1answer 50 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 ... 1answer 46 views ### 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 ... 1answer 240 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 ... 1answer 81 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 ... 2answers 225 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 \... 1answer 41 views ### 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 ... 1answer 175 views ### 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). ... 1answer 43 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 ... 1answer 204 views ### 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 ... 0answers 54 views ### 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 ... 1answer 174 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} ... 1answer 83 views ### 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 \...
I 've heard that a quiver representation $R_{Q}$ of a quiver $Q$ is a variety (I guess they mean an algebraic variety). Is this is the case we know that an algebraic variety is given by the zero set ...