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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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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Differences between quiver 1→2 and quiver 1→3←2

Could anyone tell me about the main differences between the quivers: $$ 1 \to 2 $$ and $$ 1 \to 3 \leftarrow 2 ? $$ Thanks in advance.
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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.
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Question about the connectivity of the ordinary quiver of a connected algebra.

I am reading the book Elements of Representation Theory of Associative Algebras, Volume 1. I have some questions about the connectivity of the ordinary quiver of a connected algebra. On page 61, ...
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Projective modules of the path algebra

Let $Q$ be a finite and acyclic quiver. Let $a,b$ be vertices then $P(a)_{b}$, the projective indecomposable $kQ$-module is equal to the vector space having as a basis the set of all paths from $a$ to ...
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Projective indecomposables versus general indecomposables

Given a finite dimensional algebra, what is the exact relation between the indecomposable projective modules, and a general indecomposable module? In the case of an oriented quiver without cycles for ...
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Is every finitely dimensional basic $K$-algebra isomorphic to a bound quiver algebra?

Given a positive integer $n$, how to classify $n$-dimensional basic $K$-algebras?, where $K$ is algebraically closed. For $n=3$, Let $A=\left[ \begin{array}{ccc} ...
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135 views

Cartan or Coxeter matrix of an algebra of infinite global dimension

Let $(Q, I)$ be a bound quiver such that $A=KQ/I$ has infinite global dimension. I want to ask the following questionss: (1) Is the Cartan matrix $C_A$ of $A$ invertible in the matrix ring ...
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how to find all simple modules for the given path algebra

Let $A = KQ$, where $Q$ is the quiver $$\begin{array}{ccc} & \alpha & \\ 1 & \rightleftarrows & 2 \\ & \beta& \end{array}$$ are there simple right $A$-modules with dimension ...
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how to get the injective envelope and projective cover of a given module

Given a bound quiver $(Q, I)$ and a representation $M$ of $Q$, how to get the injective envelope and projective cover of $M$? how to give the corresponding essential monomorphism and superfluous ...
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268 views

Given a quiver, how do you get the indecomposable injective modules from indecomposable projective modules?

Given a quiver, we know that it is easy to get the indecomposable projective modules, but the indecomposable injective modules are not easy to get. How do you get the indecomposable injective ...
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finite dimensional algebras without bound quiver representation

(1) For the $\mathbb{R}$-algebra $\mathbb{C}$, ($\mathbb{R}$ real number field, $\mathbb{C}$ complex number field) there is no quiver $Q $ such that $\mathbb{C}\cong \mathbb RQ/\mathcal{I}$ with ...
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admissible ideals

How to prove the following conclusion : For any finite quiver $Q$, an ideal $I$ of $KQ$, contained in $R^2_Q$, is admissible if and only if, for each cycle $\sigma$ in $Q$, there exists $s \geq 1$ ...
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Is an abstract simplicial complex a quiver?

Let $\Delta$ be an abstract simplicial complex. Then for $B\in \Delta$ and $A\subseteq B$ we have that $A\in\Delta$. If we define $V$ to be the set of faces of $\Delta$, construct a directed edge from ...
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Question about the definition of cluster algebras.

I have a question about the definition of a seed of a cluster algebra. It is said that a seed is a pair $(R, u)$, where $R$ is a quiver with $n$ vertices, $u = \{u_1, \ldots, u_n\}$ is a free ...
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The sum of trivial paths for a finite quiver is 1?

let $Q=(E_0, E_1)$ be a quiver and let $P_Q$ be a path algebra of $Q$. Let $p_i$ be the trivial path associated to each vertex $i$ in $E_0$. Then why is $\sum_{i\in E_0} p_i=1$ for a finite quiver? ...
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81 views

Stationary paths

Let $Q$ be a finite quiver and denote the stationary parts of $Q$ by $e_{i}$. Suppose we have two arrows $f,g$ such that their composition $f \circ g$ is equal to $e_{i}$. Does this always implies ...
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310 views

The projective module in the quiver representation

I study the quiver just some weeks and I cannot understand the projective module in the quiver representation well.Here are some questions: Suppose $Q$ is a quiver, $a\in Q_{0}$. 1)Show that the ...
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Opposite and connected quivers problem

Here are two problems from Elements of the Representation Theory of Associative Algebras by D. Simson, et. al (Page $65$). $1$. Let $Q=(Q_{0},Q_{1},s,t)$ be a quiver. Prove $(KQ)^{op} \cong KQ^{op}$ ...