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