# 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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### References for quivers and derived categories of coherent sheaves for a string theory student

I'm a student mostly from physics knowledge hoping to learn about the math involved the string theory paper Topological Quiver Matrix Models and Quantum Foam.
Context: The topological string theory ...

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### Finite dimensional algebras over $\mathbb{Q}$

It is known that a finite dimensional basic algebra over an algebraically closed field is isomorphic to the path algebra of a finite quiver modulo an admissible ideal.
Question 1: Is the same true ...

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### Meaning of an algebra having “sufficiently many primitive idempotents”?

This is a phrase Ringel uses a few times in his writing, and I'm not sure exactly what he means by it. The context is that we have a quiver $Q$ with path algebra $\mathbf{k}Q$. If $Q$ is not a finite ...

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### What's an illustrative example of a tame algebra?

A finite-dimensional associative $\mathbf{k}$-algebra $\mathbf{k}Q/I$ is of tame representation type if for each dimension vector $d\geq 0$, with the exception of maybe finitely many dimension vectors ...

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### Intuition for the McGerty-Nevins compactification of quiver varieties

In Section 4 of the paper Kirwan surjectivity for quiver varieties (Inventiones Math. 2018) McGerty and Nevins define a compactification of the moduli space of representations
of the preprojective ...

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### indecomposable modules of gentle algebras

Let $A = \mathcal{k}Q/I$ be a gentle algebra (where $\mathcal{k}$ is algebraically closed). In the paper Auslander-Reiten Sequences with Few Middle Terms and Application to String Algebras, Butler and ...

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### Injective morphisms between preprojective representations

Let $Q$ be an acyclic quiver. Is it true that if $P$ is a preprojective representation of $Q$ and $r\geq 0$, there exists $s\geq r$ and a preprojective $P'$ with an injective morphism
$$
P\rightarrow \...

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### Closures of orbits in the space of representations of a quiver

Let $Q$ be a quiver, and let $d=(d_i)$ be a dimension vector. We can consider Rep($Q,d$), the affine space consisting of representations of $Q$ with dimension vector $d$. The general linear $GL(d)= \...

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### Obtaining the reduced incidence algebra in QPA

Given a finite poset $P$ (we can assume it is connected), the reduced incidence algebra of $P$ is the subalgebra of the incidence algebra of $P$ consisting of functions constant on isomorphic ...

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### Are there non-trivial automorphisms of stable framed quiver representations?

Let $Q=(Q_0,Q_1)$ be a quiver and $q\in Q_0$ a chosen vertex. Let $d$ be a dimension vector with $d_q=1$ and let $\theta\in \mathbb R^{Q_0}$ be a $d$-generic stability parameter. Let $M$ be a $\theta$-...

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### Intuition for the Euler form in a finitary category

Suppose that $\mathcal{C}$ is a finitary category, so for any two objects $A$ and $B$ we have that $|\mathrm{Ext}^i(A,B)| < \infty$ for $i\geq 0$, suppose $\mathcal{C}$ has finite global dimension, ...

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### $M^{ss}_{(2,2)}(K_3,(-1,1))$ is isomorphic to $M_{\mathbb{P}^2}(0,2)$

Suppose $K_3$ is the Kronecker quiver with 3 arrows, and $M^{ss}_{(2,2)}(K_3,(-1,1))$ is the moduli space of semi stable representation of dimension $(2,2)$ wrt the weight $(-1,1)$. It is claim in the ...

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### Path algebras are formally smooth

In Ginzburg's notes Lectures on Noncommutative Geometry, he claim that the path algebra of a quiver is formally smooth (See Section 19.2).
I have two questions. First, how to show this claim and ...

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### dimension vector of indecomposable module over preprojective algebra

It is well-known that there are finitely many indecomposable module over the preprojective algebra associated to a quiver $Q$ if and only if $Q=A_2,A_3,A_4$ and tame type for $A_5$ and wild for others....

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### Intuition behind the canonical projective resolution of a quiver representation

Let $Q$ be a finite acyclic quiver, and $X$ some representation of $Q$. For $i \in Q_0$ define the $kQ$-modules $P_i = kQe_i$, and $X(i) = e_i X$. The representation $X$ has a canonical projective ...

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### Hochschild homology and Chern character quiver with potential

I am a beginner in quiver theory so this question might not be suitable for mathoverflow.
Let $(Q,W)$ be a quiver with potential and let $\Gamma$ be the Ginzburg DG-algebra associated to $(Q,W)$. Is ...

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### an inverse semigroup (and perhaps a $C^*\!$-algebra) associated with a directed graph

The following inverse semigroup associated to a directed graph came up in my research. I've read that from an inverse semigroup one may derive a $C^*\!$-algebra whose generators are partial isometries ...

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### Kac's theorem for quiver representations over an arbitrary ground field

Let $Q$ be a quiver without loops (cycles of length 1). Kac proved that if $K$ is algebraically closed, the dimension vectors of indecomposable representations of $Q$ over $K$ are exactly the ...

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### Endomorphism ring of a cotilting module

Given a basic tilting or cotilting right module $T$ over an algebra $A$ given by quiver and relations, is there a "linear-algebra method" to decide whether $\operatorname{End}_A(T) \cong A?$
Here "...

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### (Non-)formality for ADE preprojective algebras

Given a quiver $Q$, I can associate to $Q$ a certain 2-Calabi-Yau (dg-)algebra $\Gamma_Q$ by a 2-dimensional version of the "Ginzburg dga" construction: i.e., start by doubling $Q$, and then impose ...

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### Indecomposable representations of quivers of finite fields

Given a path algebra $A=KQ$ with a wild quiver $Q$ over a finite field.
There should be only a finite number of indecomposable modules of a given dimension for the algebra $A$. Are there example of ...

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### Endomorphism algebras of indecomposable quiver representations

Let $Q$ be a wild quiver without oriented cycles and let $V$ be an indecomposable representation of $Q$. Assume that $V_i\neq 0$ for each vertex $i$ of $Q$. The base field $k$ is algebraically closed. ...

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### Why are coherent sheaves on $\Bbb P^1$ derived equivalent to representations of the Kronecker quiver?

I'm looking for an explanation or a reference to why there is this equivelence of triangulated categories: $${D}^b(\mathrm {Coh}(\Bbb P^1))\simeq {D}^b(\mathrm {Rep}(\bullet\rightrightarrows \bullet))$...

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### Indecomposable representations of euclidean quivers

The classification of indecomposable representations of a Euclidean quiver is well-known over an algebraically closed field. I am interested in an analogous classification, but over an arbitrary field....

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### Isomorphic quiver representations “after adding some zeros”

Let $Q$ be a quiver, with dimension vector $d$ and let $e$ be another dimension vector, such that $d_v\leq e_v$ for every vertex $v$ of $Q$. If $M$ is a $K$-representation of $Q$ of dimension vector $...

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### radical and socle of the path algebra

Let $Q$ be an infinite quiver without oriented cycle.
Is it true that the radical of $KQ$ is generated by all the arrows?
What can we say about its socle?
Thank you!

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### Automorphisms of weighted quiver

I am reading this paper strongly primitiv species with potentials I : mutations.
In page 6, they give the definition of weighted quiver: a weighted quiver is a pair $(Q,d)$, where $Q$ is a loop-...

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### What is the status of a problem about cluster categories?

Let $H$ be a hereditary algebra of Dynkin type. There is a cluster category $\mathcal{C}_H$ defined by Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov in Tilting ...

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### Graded quivers vs “ordinary” quivers and derived categories

I have heard the "slogan" that graded quivers are (derived) equivalent to ordinary quivers (with this "result" being attributed to Keller) and am looking for a precise statement and a reference.
By a ...

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### Global dimension of quiver algebra

Given a representation-finite (finite dimensional over a field) quiver algebra of finite global dimension. Is $eAe$ isomorphic to the field for at least one primitive idempotent $e$?
This is true for ...

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### Predecessors and Successors of regular silting objects in bounded derived categories of wild hereditary algebras

Let $\Lambda$ be a wild hereditary algebra and let $T$ be one of its regular silting objects (i.e. all indecomposable direct summands of $T$ are shifts of indecomposable regular modules). What do we ...

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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 ...