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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Notation for Modules Over Quiver Algebras

Disclaimer: I know that my question has probably answers in textbooks, but I have only found the answer for quiver representations, and not for modules over quiver algebras. I know that these two ...
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Is the connected assumption necessary in the following theorem?

I'm reading about the following facts from the book 'Elements of the Representation Theory of Associative Algebras' by Assem, Simpson, and Skrowronski. Note that the algebras involved are not assumed ...
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Prove that R is an integral domain

I'm studying for my qualifying exam and I came across the following question in one of the old question bank. Consider the affine space given by four $2\times 2$ matrices, i.e., $\mathbb{A}^{16}\cong ...
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When two slices give the same orientation and any orientation can be obtained from a slice.

In the book quiver representation and quiver varieties he lefts Lemma 6.13 to the reader, but I'm having trouble with it. If $Q$ is a tree, any orientation of $Q$ can be obtained from a slice in $\...
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How to read off distinguished triangles and cluster-tilting objects in the cluster category of a Dynkin quiver?

I'm new to triangulated category and tilting theory. To illustrate, in $Q=A_4$ the module $M=kQ$ is cluster-tilting. While I know that $M$ satisfies $\mathrm{Ext}(M,M)=\mathrm{Hom}(M,M[1])=0$ by some ...
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Reduced adapted expression for the longest element given an orientation for a Dynkin graph

In Kirillov's book "Quiver representation and quiver varieties" in page 45 there's a Theorem 3.33 that he say is due to Lusztig that says: Given an orientation $\Omega$ of a Dynkin graph $Q$,...
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Lemma 6.7 from "Quiver representation and quiver varieties"

I'm triyng to work on a proof for this, it is Lemma 6.7 from "Quiver representation and quiver varieties": For any representations $X,Y$ of $\vec{Q}$, $$\text{Ext}^1(X,Y)\simeq \text{Hom}(Y,...
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A doubt in proof of existence of Auslander Reiten Sequences

I am providing the context. You can skip to the actual doubt below. My doubt doesnt need knowledge of quivers or even almost split sequences. I am reading Theorem 5.4.10 from the book An introduction ...
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An application of Snake Lemma

I am stuck at undderstanding a claim made in Derksen's book An Introduction to Quiver Representations. This is Lemma 2.4.3. But the part I am stuck at is purely homological algebra it seems. Let $A$ ...
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Non-empty stable locus in an irreducible component

I have a vague question: Let $X$ be an algebraic pre-scheme and $G$ be a linear reductive group. Consider the G.I.T quotient $X//G$. Is there any result (maybe in some special case) which tells us ...
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Projective resolution of simple module over path algebra

Let $Q$ be an acyclic finite quiver, $CQ$ its path algebra, $P(i)$ the projective module $CQe_i$ and $S(i)$ the simple module (representation) at $i$. I want to prove directly the following exact ...
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Termination of Buchberger's algorithm for path algebras

I'm reading the paper Noncommutative Gröbner Bases, and Projective Resolutions by Edward L. Green, which presents a version of Buchberger's algorithm for path algebras. I'm trying to show that the ...
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Dimensions of homomorphisms determines the module

I am reading this paper about degenerations for representations. I am mentioning some context though all of it will not be needed I think for my question. We basically have a finite dimensional ...
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Viewing Path Algebras as Matrix algebras

Given a Path Algebra $\mathbb{C}Q$ of an acyclic directed graph(also called a Quiver in this theory) $Q$. I am interested in finding a corresponding matrix algebra for it if possible. I am able to do ...
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no. of minimum generating sets of $M_{n}(\mathbb{C})$

Problem: How many $n \times n$ pairs $(A, B)$ of (0 , 1) matrices are there which generate the whole algebra $M_{n \times n}(\mathbb{\mathbb{C}})$? I don't know if this question is open or hopelessly ...
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Cluster algebras from quivers vs. cluster algebras from skew-symmetrizable matrices

In a course on cluster algebras we first defined the cluster algebra arising from a quiver. Next we saw that each quiver gives rise to an exchange matrix that is skew-symmetric. So we generalized to ...
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Derived functor of a reflection (BGP) functor

I'm studying the book "quiver representations and quiver varieties" of Kirillov and I'm in Theorem 3.10: The functor $\Phi_i^+$ is left exact. Moreover, $R^n\Phi_i^+(V)=0$ for all $n>1$, ...
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An example on Hochschild Cohomology Groups

I'm starting to study Hochschild Cohomology from the book of Sarah Witherspoon, "Hochschild Cohomology for Algebras". By a lecture notes of Maria Julia Redondo, "Hochschild cohomology: ...
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What if the semi-invariant ring is a polynomial ring or hypersurface

I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)? I have been studying about semi-...
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Why do people study semi-invariant ring (in general)?

I'm interested in studying about semi-invariant ring in the context of Quiver representations. I started reading about it in some books and few places on the internet. But, nowhere do they mention why ...
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What is the quiver representation of an algebra, and how is it obtained from a decomposition into spaces of morphisms (the opposite algebra)?

I will make everything concrete so I have something to work with. Let $A$ be an algebra, free, and 4-dimensional, and let $A = P_1 \oplus P_2$ be a decomposition into two 2-dimensional free modules. I ...
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Gabriel's theorem

Need some help with Gabriel's theorem, doing the part "if a graph is of finite type, it must be a Dynkin graph" like this: Let $\vec{Q}$ be of finite type. All representations is a direct ...
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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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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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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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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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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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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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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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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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2 votes
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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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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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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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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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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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1 vote
1 answer
108 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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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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3 votes
1 answer
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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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5 votes
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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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2 votes
2 answers
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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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1 vote
2 answers
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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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