# 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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### understanding the root system of a quiver

I am beginning to study quiver representation theory and I am trying to understand the basic set-up of the root system associated to a finite connected graph, as first described by V. Kac. I read (e.g....
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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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### 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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### 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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### Are these the projective and injective representations of these quivers?

Find all indecomposable projective and indecomposable injective representations of these two quivers over the field $k$ up to isomorphism. I've drawn my answer in this picture. Can you please check ...
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### Topological Invariance in Data Structures

I need to do a PhD in Pure Mathematics and I am thinking of Topological Data Analysis. I want to use persistent homology and quiver representation to obtain topological features in data structures. ...
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### Variety of Quiver Representations

I have been reading chapter 8 of Ralf Schiffler's 'Quiver Representations', where he proves Gabriel's Theorem characterizing connected quivers with finitely many isoclasses of indecomposable ...
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