# 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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### Notation for Modules Over Quiver Algebras

Disclaimer: I know that my question has probably answers in textbooks, but I have only found the answer for quiver representations, and not for modules over quiver algebras. I know that these two ...
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### Is the connected assumption necessary in the following theorem?

I'm reading about the following facts from the book 'Elements of the Representation Theory of Associative Algebras' by Assem, Simpson, and Skrowronski. Note that the algebras involved are not assumed ...
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### How to read off distinguished triangles and cluster-tilting objects in the cluster category of a Dynkin quiver?

I'm new to triangulated category and tilting theory. To illustrate, in $Q=A_4$ the module $M=kQ$ is cluster-tilting. While I know that $M$ satisfies $\mathrm{Ext}(M,M)=\mathrm{Hom}(M,M)=0$ by some ...
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### Reduced adapted expression for the longest element given an orientation for a Dynkin graph

In Kirillov's book "Quiver representation and quiver varieties" in page 45 there's a Theorem 3.33 that he say is due to Lusztig that says: Given an orientation $\Omega$ of a Dynkin graph $Q$,...
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### When does a Path Algebra give a unique Quiver?

I am studying the text An introduction to Quiver Representations by Derksen. Exercise 1.5.4 asks to prove that if the path algebras $\mathbb{C}Q$ and $\mathbb{C}Q'$ are isomorphic $\mathbb{C}$-...
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### Simple modules corresponding to vertices of a quiver with potential.

Given a quiver with reduced potential $(Q,W)$. The simple dg module over the complete Jacobian algebra $J(Q,W)$ are in bijection to the set of vertices of Q ? How do the simple modules look like ? And ...
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### Kerodon 2.4.4.10.: pushouts of graphs associated to a pushout of simplicial sets

I don't understand a nuance in Theorem 2.4.4.10. of Kerodon. To not burden the question with exposition, I will only provide information for the relevant part. Fix a natural number $m$. For a ...
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### Quiver representation - definition clarification

I'm studying quiver representation on "Introduction to representation theory" by Etingof and others. Often I read on this book about a representation being injective or surjective at some ...
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### Walks on graphs and tensor products

In Qiaochu's article https://qchu.wordpress.com/2010/03/07/walks-on-graphs-and-tensor-products/#more-4807, he gives the following setup: Construct a directed graph $\Gamma(V)$ as follows: its vertices ...
1 vote
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### How do you talk about representations of a subquiver of $A_n$?

Let $Q$ be a linear Dynkin quiver of type $A_n$, which looks like $$1 \longrightarrow 2 \longrightarrow 3 \longrightarrow \cdots \longrightarrow (n-1) \longrightarrow n$$ Then we know by Gabriel's ...
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### Are small categories necessarily free?

In the same spirit that all vector spaces are free, is any small category a free category?. If not, there is a counterexample? I'm interested to know if diagrams indexed by quivers covers all cases of ...
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### equivalence of Rep$(Q)$ and Mod($KQ$). Does it preserve length/dimension?

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 ...
1 vote
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### Categories and quivers / directed graphs associated

I am currently working on associating categories and quivers. a) Let $C$ be an arbitraty category. I need to show that one can associate with $C$ a quiver $Q_C$ by sending an object $X$ in $C$ to a ...
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### Paths in a quiver form a $K$-algebra.

I am attending a course in foundations in representation theory and I am struggling to grasp the concept of path algebras, more explicitly why paths in a quiver form a $K$-algebra. Let $Q$ be a ...
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### Path in a quiver to a path in path algebra in GAP

Is it possible to obtain a path in the path algebra without explicitly specifying the arrow? More specifically, I have a string with letters which are arrows in the quiver. I need to check whether in ...
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### Field elements in quiver and relations

Let $A=KQ/I$ be a quiver algebra such that the admissible ideal $I$ contains only the field elements $0,1$ and $-1$. Question: Is it true for every basic idempotent $e$ that the algebra $eAe$ is ...
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### The category of finite dimensional right $KG$-modules is equivalent to the category of finite dimensional representations of a quiver $Q$

Let $K$ be an algebraically closed field and $G$ be a finite group such that $|G|$ is not divisible by the characteristic of $K$ (so that Maschke's theorem can be applied). Let $Q$ be the quiver ...
I'm trying to build a quiver in QPA which has 42 nodes, labeled by some of the signvectors in $\{\pm 1\}^6$. I want to build a rule that assigns an arrow $a\to b$ if $a$ and $b$ differ in exactly one ...