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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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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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 $M_n(Z)$...
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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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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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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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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 $(KQ)^{op} \cong KQ^{op}$ ...