Questions tagged [quivers]
"Quiver" is the word used for "directed graph" in some parts of representation theory. The main reason to use the term quiver is to indicate an interest in considering representations of the quiver.
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$\mathrm{Ext}$ group in representation theory
Let $\mathcal{X}$ be a finite acyclic quiver, and $v_1$ be a source vertex of $\mathcal{Q}$. Let $\mathcal{X}$ be a representation in $\mathrm{Re}(\mathcal{Q},R)$, where $R$ is a commutative ...
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In quiver rep,is it$\mathrm{Ext}^i_{\mathrm{rep}}(\mathcal{X},\mathcal{R})=0 \leftrightarrow \forall v \mathrm{Ext}^i_R(\mathcal{X}_v,R)=0$?
Let $\mathcal{Q}$ be a finite acyclic quiver, and $R$ be a ring Let $\mathcal{X}$ be a representation in $\mathrm{Rep}(\mathcal{Q},R)$. Let $\mathcal{R}$ represent the image of $R\mathcal{Q}$ under ...
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A question about the quivers with potentials
Let $Q=(Q_0,Q_1,h,t)$ be a quiver consisted of a pair of finite sets $Q_0$(vectors),and $Q_1$ (arrows) supplied with two maps $h : Q_1 → Q_0$ (head) and $t : Q_1 → Q_0$ (tail ). This definition allows ...
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Rigid regular objects of path algebras of tame quivers
In the paper On Maximal Green Sequences by Brustle, Dupont and Perotin the authors argued that in a path algebra $\Lambda=kQ$ of a tame quiver $Q$ with $n$ vertices each tilting module contains at ...
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The explicit indecomposable representations of (any) Euclidean quiver of type E
It is known that for any quiver $Q$ that is an orientation of $\tilde{\mathbb{E}}_8$, the hereditary path algebra $KQ$ ($K$ being an algebraically closed field) is tame (but not finite). That is, in ...
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Which cluster algebras where the existence of maximal green sequences is still unknown?
Maximal green sequences are studied in many papers. For example, Maximal Green Sequences for Cluster Algebras Associated to the n-Torus by Eric Bucher, On Maximal Green Sequences by Thomas Brüstle, ...
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Quiver representations and coherent sheaves
I've heard that under certain assumptions on an algebraic variety $X$ there exist a quiver $Q$ for which there is an equivalence $$D^b(\mathsf{Coh}(X))\simeq D^b(\mathsf{Rep}(Q))$$ between the ...
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Quiver invariants as polynomials/algebraic curves
I'm interested in algebraic curves one can associate to gauge or string theories. Examples involve Seiberg-Witten curves or family of A-polynomials which define holomorphic Lagrangian submanifolds for ...
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About irreducible morphisms
I have asked the following question in Mathematics stack: https://math.stackexchange.com/questions/2202032/about-irreducible-morphisms. But there is no response, so I repost it here.
A morphism $f: X\...
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Injectivity of a standard map in quiver representation
Let $X$ be a smooth projective variety, and assume its divisor class group is finite and free. Let $E_1,E_2,\ldots,E_n$ be line bundles on $X$. Define $L_k=E_1+\ldots E_k$, and let $Q$ be the ...
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Can we infer an isomorphism of quivers from an isomorphism of their corresponding path algebras?
Given a pair $\Delta, \Gamma$ of quivers and a field $K$ one can construct the corresponding path algebras $K\Delta, K\Gamma$. I came upon a paper claiming (section 3, 2nd paragraph) that an ...
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bound quiver of section -- the dga version?
Let $X$ be a smooth projective variety, and $\mathcal{L} = \{L_0, \cdots, L_n\}$ be a list of distinct line bundles. The (complete) bound quiver of sections associated with $\mathcal{L}$ is a quiver ...
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Characterisation of certain quiver algebras
Let algebras always be finite dimensional connected non-semisimple quiver algebras. Say an algebra $A$ has property * in case $eAe$ is a Nakayama algebra, when $eA$ denotes the basic version of the ...
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Does unique factorisation hold for quiver algebras?
Given a finite dimensional quiver algebra A=KQ/I. It can be (possibly) written as $A= B_1 \otimes_k B_2 ... \otimes_k B_r$ and the $B_i$ can not be decomposed into smaller algebras. Is this ...
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Reference request: tangent space to moduli space of coherent sheaves is $\operatorname{Ext}^1(E, E)$
Is there a standard reference for the fact that, in an appropriate algebraic-geometrical context, the tangent space at the point $[E]$ to the moduli space $\mathcal M$ is something like $\operatorname{...
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Largest possible dimension-vector of a indecomposable module over a representation-finite algebra
Let $A$ be a representation-finite quiver algebra and $M$ an indecomposable $A$-module and $s$ the dimension of $A$ and $e_i$ the canonical primitive idempotents.
What is the largest possible value (...
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Construction of irreps of path algebra of cyclic quiver, classification of all finite-dimensional irreps
Originally posted here on Mathematics Stack Exchange.
Let $Q$ be a quiver with vertex set $\{1, 2, \ldots, n\}$ such that $Q$ has a single edge $i \to i + 1$, for every $i = 1, 2, \ldots, n - 1$, one ...
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Algorithm for finding quiver algebras
Im looking for an algorithm that does the following in a quick way:
Input: Natural number $r \geq 2$, natural number $s \geq 3$, prime power $q$.
Output:
Finds all two-sided ideals in $J^2/J^s \...
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Softwares which compute all non-isomorphic quivers in a mutation class
Let $Q$ be a quiver. The mutation class of $Q$ consists of all quivers which can be obtained from $Q$ by a sequence of mutations. Are there some softwares which compute all non-isomorphic quivers in a ...
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Mutation equivalence of quivers
Given two orientations $Q, Q'$ of a Dyinkin diagram. Is it always true that after a sequence of mutations, $Q$ becomes $Q'$? Are the some references about this? Thank you very much.
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Mutation of valued quivers
Mutations of valued quivers are defined in cluster algebras II, Proposition 8.1 on page 28. I have a question about the number $c'$. For example, let $a = 2, b=1, c=1$ and consider the quiver $Q$:
$1 ...
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Cluster algebras of finite type
In the webpage, there is a result:
Theorem 1. Coefficient free cluster algebras without frozen variables are in bijection with Dynkin diagrams of type $A_n$, $B_n$, $C_n$, $D_n$, $E_6, E_7, E_8$, $...
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Realisation of the preprojective algebras as $F(\Delta)/T$ over some quasi-hereditary algebra
Let $A$ be the Auslander algebra of $K[x]/(x^n)$ for some $n \geq 2$, which is quasi-hereditary with some characteristic tilting module $T$.
Dlab and Ringel showed in their paper "The Module ...
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Auslander-Reiten theory for Gorenstein algebras
In the paper "Cohen-Macauley and Gorenstein artin algebras", Auslander and Reiten have a short section about Auslander-Reiten theory in Gorenstein algebras (I always assume we have an artin algebra ...
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Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms
I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...
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Dimension of preprojective algebra of Dynkin type
Fix a field $\Bbbk$. Let $Q$ be a Dynkin quiver and let $\Pi(Q)$ be its preprojective algebra. It is well-known that in this case $\Pi(Q)$ is finite-dimensional, but I've been unable to find a ...
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what is the injective hull of indecomposable module of preprojective algebra
Let $Q$ be a ADE type quiver and $s_i$ ($i$ runs through the vertices of $Q$) be the simple $\Lambda$-module with 1-dimensional vector space at vertex $i$ and zero-dim at other vertices. Here $\Lambda$...
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analog of Lusztig nilpotent scheme
Fix a quiver $Q$ without loop. Denote the set of vectices of $Q$ by $I$.
Let $\Lambda_V$ be the Lusztig nilpotent scheme with associated vector space $V$ over $I$. Briefly speaking, when $Q$ is a $ADE$...
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Why are exchange graphs of quivers with the same underlying graph but have different orientations isomorphic?
I know the fact that (undirected) exchange graphs of quivers with the same underlying undirected graph but have different orientations are isomorphic (i.e. quivers that are just finitely many arrow-...
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An equivalence between projective modules over the preprojective algebra and an orbit category
Let $Q$ be a Dynkin quiver, and let $kQ$ be its path algebra over some field k. Let $\Pi$ be the preprojective algebra of $Q$. Then (c.f. Section 7.3 of Keller's On Triangulated Orbit Categories) the ...
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Short proof of the classification of representation-finite symmetric algebras up to stable equivalence
Assume $K$ is an algebraically closed field and $A$ a finite dimensional $K$-algebra. Assume additionally that $A$ is symmetric and representation-finite.
Then one has the following classification of ...
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Connection between representations of different orientations of graph
In 1973 paper about Gabriel's theorem, there is an open question:
Suppose we have a graph $\Gamma$ and two orientations $\Lambda,\Lambda'$ of it. Then for each indecomposable representation of $\...
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Is this modified bound quiver algebra necessarily representation-finite?
Suppose that $A = kQ/I$ is a bound quiver algebra for $k$ an algebraically closed field, $Q=(Q_0, Q_1)$ a finite connected quiver with no oriented cycles with no multiple edges or self-loops, and $I$ ...
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Is a quotient of a bound quiver algebra of finite representation type also representation-finite?
Let $A = kQ/I$ be a bound quiver algebra for some algebraically closed field $k$, $Q$ a finite connected quiver without oriented cycles, and $I$ an admissible ideal. Say that $I'$ is also an ...
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Classification of representation-finite algebras up to stable equivalence of Morita type
Assume $K$ is an algebraically closed field.
I wanted to ask if there is a classification of the representation-finite $K$-algebras up to stable equivalence of Morita type (at least for some small ...
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When is a given quiver algebra a hopf algebra?
Given a finite dimensional selfinjective quiver algebra A over a finite field (or more generally an arbitrary field). Whats the best way to check if the algebra A has a Hopf algebra structure or not? ...
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Two quivers, finitely many nonisomorphic representations of $\mathbb{C}Q$
Consider the following two quivers:
...
user78817
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How is the free modular lattice on 3 generators related to 8-dimensional space?
Here are three facts which sound potentially related. What are the actual relationships?
In 1900, Dedekind constructed the free modular lattice on 3 generators as a sublattice of the lattice of ...
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Is this algebra isomorphic to an incidence algebra?
This question is motivated by trying to establish a converse to Theorem 7.8 of our paper.
I have a finite poset $P$ with the following properties:
$P$ has binary meets (and hence a least element).
$...
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Second cohomology groups of Nakajima quiver varieties
Let $X=M(v,w)$ be a Nakajima quiver variety for a quiver $Q$. Can one calculate the second singular cohomology groups $H^2(X,\mathbb Z)$ or $H^2(X,\mathbb C)$ explicitly, and if not, are there some ...
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Why Jacobson, but not the left (right) maximals individually?
I firstly asked the following question on MathStackExchange a couple of months ago. I did not receive any answers, but a short comment. So, I decided to post it here, hoping to receive answers from ...
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Reference that contains examples of absolutely indecomposable representations of quivers over a finite field
Is there a reference that lists/discusses examples of absolutely indecomposable representations of quivers over a finite field (absolutely indecomposable = does not decompose into a direct sum over ...
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For which quiver varieties is Kirwan surjectivity known?
The cohomology of Nakajima quiver varieties is a quite interesting object. It's equipped with some natural classes given by the Chern classes of the tautological bundles associated to the spaces in ...
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Is there a cotangent bundle of a stable $\infty$-category?
Let $C$ be a stable $\infty$-category. Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following?
When $C$ is the ...
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Ext Quivers and their applications to Representation Theory
I am looking for references that provide an overview of the following two topics (it can be multiple references if necessary):
How to compute the Ext-quiver of a (locally finite or finite) $\mathbb{C}...
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"embedding" various matrix equivalences into the equivalence of particular linear map
Consider the square matrices over a (local) ring $R$, up to conjugation, $A\rightarrow UAU^{-1}$, where $U$ is an invertible matrix over $R$. Such an equivalence embeds into the "left-right" ...
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Bounded algebras of finite global dimension
Let $k$ be a field, $Q$ an acyclic finite quiver and $I$ an admissible ideal of $kQ$.
I am looking for a reference for the fact that the bounded algebra $kQ/I$ has finite global dimension.
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Quiver representations
I'm wondering how to find indecomposable representations of a given quiver explicitely. In particular, I'm interested in the maximal indecomposable representation of $\mathbb{E}_8$(I'm working over $\...
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How can one show that orbit closures in representations of a linear quiver don't have small resolutions?
Let $1\to \cdots\to n$ be a linear quiver of length $n$. Let $\mathbf{d}=(d_1,\dots,d_n)$ be a dimension vector. It's well known (for example, by Gabriel's theorem, but also by basic linear algebra) ...
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How can classifying irreducible representations be a "wild" problem?
Let $q$ be a prime power and $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$-matrices. I've read and heard in several places (see e.g. this mathoverflow question) that classifying ...
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