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.
77 questions with no upvoted or accepted answers
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The derived category of integral representations of a Dynkin quiver
Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write $\...
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Are the extra vertices in Nakajima's doubling of a quiver related to Langlands duality?
To define a Nakajima quiver variety associated to a quiver $Q = (Q_0,Q_1)$
(vertices and arrows), one first doubles it to $Q^\heartsuit$ by attaching
an extra vertex to every old vertex in $Q_0$. Then ...
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12
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Quivers as noncommutative curves
I've heard that an idea behind noncommutative geometry (in dim 1) is to study "noncommutative" analogues of $\text{Coh}(\text{curve})$, rather than the curve directly. Apparently the ...
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Quiver and relations for blocks of category $\mathcal{O}$
In Vybornov - Perverse sheaves, Koszul IC-modules, and the quiver for the category $\mathscr O$ an algorithm is presented to calculate quiver and relations for blocks of category $\mathcal{O}$ .
...
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Ringel's interpretation of quantum groups as Hall algebras at $q=1$
Let $Q$ be a finite-type quiver and let $\mathfrak{g}$ be the semisimple Lie algebra associated with the corresponding simply-laced Dynkin diagram. Let $U_v^+(\mathfrak{g})$ be the positive part of ...
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Modern proof of a theorem of Dickson on finite representation type
In Theorem 3.1 the paper S. Dickson, On algebras of finite representation type
Trans. Amer. Math. Soc. 135 (1969), 127-141, Dickson gives a sufficient condition for an algebra to have infinite ...
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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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Tensor product of quivers
As a special case of a general construction I have constructed "accidentally" a tensor product of quivers aka directed multigraphs (aka directed graphs for category theorists). Probably this ...
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What representation theoretic properties does the semi-invariant ring tell us?
I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)?
I have been studying about semi-...
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Finite-dimensional irreducible representations of 2-loop quiver
What are the finite-dimensional irreducible representations of the quiver with one vertex and two loops?
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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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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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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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Homological dimension of completed path algebras.
Let A = c[Q]/I be a finite dimensional quotient of a path algebra over a quiver Q, with I being the ideal of relations.
Is it true that the I-adic completion of A has finite homological dimension?
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Hirzebruch-Riemann-Roch for quiver varieties?
These days, I attended a workshop at North Carolina State University. The key lecturer is Professor Nakajima. He introduced two types of quiver variety. One of them is affine, another one is quasi-...
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Proof of McKay correspondence without classifications
$\DeclareMathOperator\SU{SU}$I am wondering if there is any known proof of the McKay correspondence (I will give the precise statement that I mean by this) that doesn't use the classification of ...
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Road map for learning cluster algebras
I'm a PhD student and I would like learn about cluster algebras. I'm wondering what is a good reference (i.e., has detailed explanations, examples, etc) to learn from the basic and what are some of ...
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Field elements in quiver and relations
Let $A=KQ/I$ be a quiver algebra such that the coefficients of the relations in the admissible ideal $I$ consist only of the field elements $0,1$ and $-1$.
Question 1: Is it true for every basic ...
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128
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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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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-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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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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Projective resolution of a quiver with relations
How do we compute the projective resolution of a representation of a quiver with relations.
For example consider the Beilinson quiver $B_4$
$.
with the relations $\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^...
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107
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Auslander-Reiten sequences where irreducible morphisms are all epi/mono
Let's work in the setting of modules over an Artin algebra $A$, or a finite-dimensional $k$-algebra $A$, or if you like, modules over a connected quiver $Q$ without oriented cycles.
Let $M$ be such a ...
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111
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Example of an irreducible component with an open set of infinitely many codimension 2 (codimension 3) orbits
Let $\mathbb{K}$ be an algebraically closed field of characteristics $0$. Let $A$ be a finite dimensional (associative and unital) algebra over $\mathbb{K}$. Assume there is a quiver $Q=(Q_0,Q_1)$, ...
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A conceptual explanation for the Kirchoff matrix theorem in terms of the quiver algebra
On the wikipedia page for the Kirchoff matrix theorem, they state a souped up version of the theorem:
Let $G$ be a finite undirected loopless graph and let us form the square matrix $L$ indexed by the ...
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Bound quiver algebras with relations of the form $x_ix_j=$sum of paths of length $\geqslant 3$
While working with homotopes and isotopes of finite dimensional algebras I often encounter algebras isomorphic to a path algebra of a bound quiver, i.e. $k[\Gamma]/I$, where the relations $I$ have the ...
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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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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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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!
- 163
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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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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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Invariant Subvarieties of Variety of Quiver Representations
I'd like to understand a special case of the following rather general algebraic geometry question:
Given an algebraic group $G$ acting on a variety $V$, can we describe the $G$-invariant subvarieties ...
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Hall algebra of constructible functions of affine quiver?
I have read in "Quiver Representations and Quiver Varieties" by Kirillov that Hall algebra of constructible functions are defined only for Dynkin quivers because they are of finite type. So ...
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Does this finitely-generated algebra have a name?
I've been led to consider certain finitely-generated algebras that arise from some Coxeter groups (finite and affine Weyl groups at least). As a very concrete example, consider the infinite dihedral ...
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Does every SES of injective bounded cochain complexes split?
Question: Does every short exact sequence of injective bounded cochain complexes, $0\rightarrow I^\bullet\rightarrow J^\bullet\rightarrow K^\bullet\rightarrow 0$, split?
I am interested in a discrete ...
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Example of a triangular string algebra that is rep infinite, but $\tau$-tilting finite
Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$. Hence, $A$ can be realized as the path algebra of a bound quiver $(Q,I)$, where $I\subseteq\mathbb{K}Q$ is an ...
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Number of admissible quotient algebras
Let $Q$ be a finite connected quiver. An admissible quotient algebra is an algebra of the form $KQ/I$ with an admissible ideal $I$.
Question 1: Is there a nice closed formula for the number of ...
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Two notions of stability
Let $Q$ be a finite quiver (i.e. an oriented graph). A representation of $Q$ is by definition a module over the path algebra of $Q$. More concretely, a representation associates to every vertex $v \in ...
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Getting an equivariant morphism
Let $X\subset\mathbb{A}^n$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristic zero. Suppose we have two linearly reductive algebraic groups $G$, $G'$ ...
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Quiver and relations for Hopf algebras associated to quiver algebras
Let $A=KQ/I$ be a finite dimensional quiver algebra with admissible relations $I$.
$A$ can be made into a restricted Lie algebra over a field of characteristic $p$ via
$[x,y]=xy-yx$ and $x^{p}=x^p$. ...
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102
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When do two path algebras share an underlying graph?
Suppose $Q$ and $Q'$ are two quivers. I am curious as to what relation $\mathbb{C}Q$ bears to $\mathbb{C}Q'$ when $Q$ and $Q'$ share the same underlying graph and only differ by direction.
Since ...
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91
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Quiver and relations for group algebras of p-groups
Let $G$ be a finite $p$-group and $K$ a field of characteristic $p$.
$KG$ is isomorphic to a quiver algebra $KQ/I$ with admissible ideal $I$.
Question 1: Does there always exist such an $I$ where the ...
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268
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Understanding a proof of a result of Schofield
I'm reading a paper of Aidan Schofield- "General Representations of Quivers" and I'm trying to understand the proof of Theorem 3.3. I'm having trouble understanding the argument that's ...
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Potential on a quiver
I found two definitions of potential on a quiver.
Selfinjective quivers with potential and 2-representation-finite algebras, Martin Herschend and Osamu Iyama 2.1 Quivers with potential. Let $Q$ be a ...
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135
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How to compute the derived functor of bounded derived categories of hereditary algebra?
Let $\Lambda$ be
a finite dimensional algebra given by the quiver
$$\cdot\leftarrow\cdot\leftarrow\cdot\rightarrow\cdot.$$
It can be view as an triangulated matrix algebra.
$$\Lambda={A\ \ M\choose0\ ...
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$G$-module representations of a profinite quiver
I have a profinite directed graph $\Gamma$, i.e., I can think of $\Gamma$ as the inverse limit of a directed system of finite directed graphs under inclusion. To each vertex of the graph a $G$-module ...
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48
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The isomorphism class of the 1-representation of a complete quiver
Let $Q$ be a quiver with vertex set $Q_0$ and the arrows $Q_1$. A quiver self $Q$ is said to be complete if it has no loops and for every arrow in $Q_1$ the opposite arrow is also in $Q_1$.
A ...
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Injective morphisms between preprojective representations
Let $Q$ be an acyclic quiver. Is it true that if $P$ is a preprojective representation of $Q$ and $r\geq 0$, there exists $s\geq r$ and a preprojective $P'$ with an injective morphism
$$
P\rightarrow \...
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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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