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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Cluster algebra of finite type

It is proved in the paper that a cluster algebra is of finite type if its Cartan counter part of the principal part of its seeds is a Cartan matrix of finite type. If the initial quiver of a cluster ...
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Name for quiver representation

Let $Q = (Q_0, Q_1)$ be a quiver, and pick some $i \in Q_0$. Define the quiver representation $M$ by $$M_j = \begin{cases} k & \text{ if there is a path from $i$ to $j$,} \\ 0 & \text{ ...
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Auslander-Reiten theory: exercise $23.b$ of 'Elements of the Representation Theory of Associative Algebras'

I am solving exercise $23.b$ of chapter IV of 'Elements of the representation theory of associative algebras' by Assem, Simson and Skowronski. The question is the following: Consider the following ...
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Is there a name for this acyclic quiver?

Sorry for the trivial question, but I don't know much about the subject and don't seem to be able to come up with much by Googling. Is there an established name for quivers of the form $$\require{...
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Theorem $2.15$ of Quiver Representations by Ralf Schiffler.

I'm reading the proof of theorem $2.15$ of Quiver Representations by Karl Schiffler. The author states the following: Let $Q$ be a finite acyclic quiver, $M=(\left\{M_i\right\}_{i\in Q_0}, \left\{\...
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Proposition 2.3 of Quiver Representations by Ralf Schiffler

I'm trying to prove proposition 2.3 of Quiver Representations by Ralf Schiffler. To any vertex $i$ finite acyclic quiver $Q$ we can associate the indecomposable projective $P(i)$, this proposition ...
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derived equivalences and finite representation type

Given two finite dimensional connected quiver algebras $A$ and $B$. Is there an easy example such that $A$ and $B$ are derived equivalent and $A$ has finite representation type but $B$ has infinite ...
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How does having a cycle in a quiver change the simple objects in the category of representations?

In theorem 1.12 on page 5 of http://www.math.utah.edu/dc/tilting.pdf, which states: Given a bounded acyclic quiver $(Q,R)$, the K-theory of it's representations is given by $\mathbb{Z}^{Q_0}$ why is ...
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How to define a quiver of basic and non-connected associative algebras

Recently, I am reading a book "Elements of the Representation Theory of Associative Algebras". Let $A$ be a basic and connected finite dimensional $~\mathbb{K}$-algebra and $\{e_1, e_2, \cdots, e_n\}$...
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Connection between quivers and representations of Lie algebras

Can anyone recommend a reference to study the connection between quiver theory and representation theory of Lie algebras? Supposedly those two things have something to do with each other, with the ...
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Some questions about standard $K$-duality

Let $A$ be a finite dimensional $K$-algebra, where $K$ is an algebraically closed field. We define a funtor $D: mod A \to mod A^{op}$ called standard $K$-duality. Suppose that $M$ is an arbitrary ...
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How to recognize a monomial quiver algebra

Given a basic split finite dimensional algebra $A$ over a field K, A is isomorphic to $KQ/I_1$, for some quiver Q and a minimal(meaning it is generated by relations $x_i$, such that no relation is ...
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Finding all admissible Ideals of a given quiver with gap/qpa.

Let Q be a given finite quiver (with 1 point to make things easier for a start if necessary, for definitions see https://en.wikipedia.org/wiki/Quiver_%28mathematics%29 ) and fix a finite field K. Let ...
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Reference request: bounded derived categories and their Auslander-Reiten quivers

I have some knowledge of Auslander-Reiten theory, tilting theory, derived categories and triangulated categories though I still find most proofs using derived categories in "Tilting Theory and Cluster ...
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The motivation for quivers? [duplicate]

I would like to know about the reasons (I mean, methodological reasons, not just a penchant for innovation in terminology) for Pierre Gabriel to make use of quivers. Is it fair to say he wanted to ...
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Projective Indecomposable modules of quiver algebra

I am trying to prove that the module $kQe_i$ of paths starting from a node $i$ in an acyclic finite quiver $Q$ is a projective indecomposable module for $kQ$. The projectivity is clear since I can ...
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Help constructing a specific quotient algebra

So I'm a post-graduate maths person and self-studying quiver representations. I came across a problem, and before I even lay out the entire problem (which aims to construct a functor $T$ such that $...
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116 views

Auslander–Reiten quiver

Consider the quiver $1 \overset{\alpha}{\underset{\alpha'}{\leftleftarrows}} 2$. I am trying to find the form of the Auslander–Reiten quiver. So I got : $P(1)= \begin{matrix} 1 \end{matrix}$, $...
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Mutations of an $A_n$ quiver to reach $A_n$ straight orientation

(1)For a quiver $A_n$ of arbitrary orientation does there exist a finite sequence of quiver mutations that can mutate it to $A_n$ straight orientation $1\rightarrow 2\rightarrow\cdots\rightarrow n$ or ...
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Quiver algebra as a wreath product?

I'm having trouble understanding a definition of a quiver Hecke algebra. Suppose $k$ is a commutative ring, and $\Omega$ a finite set. We build a quiver $Q_{\Omega,n}$ with vertex set $\Omega^n$. ...
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Why is the arrow ideal $R_Q$ of a finite, connected, acyclic quiver $Q$ equal to the Jacobson radical?

If $Q$ is a finite, connected, acyclic quiver, why does the arrow ideal $R_Q$ equal the Jacboson radical $J$ of the quiver algebra $k(Q)$? It comes up in showing that the quotient $k(Q)/R_Q$ is a ...
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What can we do about the indecomposable representations of wild quivers and wild algebras?

I know that using finite number of parameters we can not describe indecomposable modules of wild quivers, but is it possible for us to describe them using infinitely many parameters for at least some ...
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How many projectives and injectives exist in a path algebra?

I do not know an efficient way to determine whether a quiver representation is projective or injective. The definitions and properties such as "Projectives are summands of free modules", etc do not ...
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1-loop quiver and the classification of quivers

Gabriel's theorem states that finite type quivers are exactly the ones whose underlying graphs are ADE type Dynkin diagrams. Furthermore, the quivers whose underlying diagrams are ADE type affine ...
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Representations of a quiver and sheaves on P^1

We know from Beilinson that there's an equivalence of derived categories $D^b Rep(Q) \simeq D^b Coh(\mathbb{P}^1)$ where the lefthandside is the derived category of bounded complexes of ...
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Affine variety and dimension

I'm working on a paper about representation of quivers and Gabriel's theorems. See this .pdf if you're interested ; but I guess you can answer my question without knowing anything about quivers, or at ...
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How to show that the ordinary quiver of a semisimple algebra is a quiver consisting of isolated points?

It is said that the ordinary quiver of a semisimple algebra is a quiver consisting of isolated points? How to prove this result? Thank you very much. Edit: the ordinary quiver is the quiver defined ...
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Why Jacobson, but not the left (right) maximals individually?

When we are working with Path Algebras, it does not need very sophisticated tools to prove that for a finite, connected, acyclic quiver $Q$, the Jacobson Radical of $KQ$ is nothing but the arrow ideal....
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Characters of Linear Algebraic Groups

Reading about the semi-invariants of quivers, I see a fact which is frequently referred to in the literature, and is assumed to be trivial. However, I don't see that very easily. So, I was wondering ...
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Questions about subrepresentations of a representation of a quiver.

Let $Q$ be the quiver $\cdot \to \cdot \to \cdot$. Then $$ \mathbb{C} \to^{f} \mathbb{C} \to^g \mathbb{C} \quad (1) \\ 0 \to^{0} \mathbb{C} \to^0 0 \quad (2) $$ are two representations of $Q$, ...
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Should Ext-quiver be a full sub-quiver of its AR-quiver for a basic hereditary algebra A over algebraic closed field K?

For a basic hereditary algebra A over algebraic closed field K, prove its Ext-quiver $\Gamma_{A}$ is a full sub-quiver of its AR-quiver $\Delta_{A}$. I have no clue for this.
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Endomorphism ring of indecomposable representations

Let $Q$ be the quiver given by an $n\times n$ grid where every square commutes and let $F:Q\to {\rm vec_k}$ be an indecomposable (finitely dimensional) representation of $Q$. I am interested in ...
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The indecomposable projective A-modules

Let Q be the quiver bound by $αβ = 0$, $γδ = 0$. The indecomposable projective A-modules are given by where $A=KQ/I$. This an example in Assem-Simson-Skowronski book (Elements of the ...
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Computing quotient representations and Hom set fort wo representations

Consider the representation $M$ defined by We want to find all subrepresentations quotient representations of $M$, and $\mathrm{Hom}(M,N)$, where $N$ is a representation with $N \cong M$. I put B ...
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Computing Path Algebra of a Quiver

Let $Q$ be a quiver over defined as follows Then $KQ\cong$ $\begin{pmatrix}K&K&K\\0&K&K\\0&0&K\end{pmatrix}$, where $KQ$ is just the path algebra. What the professor did was ...
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isomorphism classes of representations of a quiver

Classify all isomorphism classes of representations of dimension vector 1 and 2 of the following quiver The professor briefly did the solution, but I could not understand what was going on. What he ...
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Question for recommending a good textbook in representation of quivers

I am taking representation of quivers, and the lecture notes seems not enough. So could you recommend a good textbook for this course. There is a new book "Quiver Representations, by Ralf Schiffler" ...
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Multidegree semi-invariants for quiver representations

Given a quiver Q=($Q_0,Q_1$) ($Q_0$ is the set of vertices and $Q_1$ is the set of arrows) and a dimension vector $\alpha$, the coordinate ring may be written as $\bigotimes_{a \in Q_1}k[Hom(k^{\alpha(...
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Indecomposable quiver representations

Is there are any way to found indecomposable representation of a given quiver explicitely if it's dimention vector is given?
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Find radical of a quiver representation

How to find the radical of an acyclic quiver and without relations? what is the recipe? For example suppose $Q$ is a quiver with two vertices and two arrows from vertex $2$ to vertex $1$. Now suppose ...
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the real roots of a connected tame quiver

Consider a tame quiver $Q$ whose underlying undirected graph is connected. So that undirected graph is one of the extended, simply laced Dynkin graphs; it's either $\tilde A_n$ for some $n\ge 1$ or $\...
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When are infinite dimensional path algebras hereditary

The title says mostly everything. Suppose we have a quiver, maybe with relations and cycles. Is it known when the path algebra modulo relations is hereditary? Especially in the case that the path ...
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330 views

When are path algebras of quivers hereditary.

Suppose we have a finite quiver with relations, possibly with oriented cycles. Is it known when the path algebra of this quiver (with relations) is hereditary?
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How to compute the ordinary quiver of $B = \operatorname{End}_A(T_{A})$?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory by Ibrahim Assem, Daniel Simson, Andrzej Skowronski. Let $A$ be a $K$...
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A finite-type quiver has no self-loops

I am reading through Etingof et al's notes on representation theory, and they assert in Exercise 5.4(c) on page 80 that a finite-type quiver has no self-loops. I think the way to show this is to ...
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Specific projective dimension of a module over bound quiver

Suppose $K$ is an algebraically closed field, and $A$ is the algebra presented by the quiver $$\require{AMScd} \begin{CD} 1 @>>> 2\\ @V{}VV @V{}VV \\ 3 @>>> 4 @>>> 5 \end{CD}...
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Path Algebra for Categories

For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", ...
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Why are (representations of ) quivers such a big deal?

Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, ...
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What are all representations of the quiver $Q$?

Let a quiver be $Q=(Q_0, Q_1, s, t)$, where $Q_0$ is $\{1, 2\}$. The quiver has only one arrow: $\alpha: 2 \to 2$. What are all representations of $Q$? Thank you very much. In my original question,...
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What softwares are convenient to draw a quiver?

I would like to draw some quivers in a mathematical paper. What softwares are convenient to draw a quiver? Thank you very much.