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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92 views

Quiver varieties associated to D_4

Let $Q=(I,\Omega)$ be the $D_4$ affine quiver. We choose as dimension vector $(2,1,1,1,1)$ (where $2$ is on the central vertex). As this dimension vector is indivisible, we can choose a generic $\...
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571 views

Invariants of matrices (by simultaneous $\mathrm{GL}_n$ conjugation) over arbitrary rings

$\DeclareMathOperator\GL{GL}$Let $R$ be a commutative ring, let $R[n] := R[M_d^{\oplus n}]$ be the polynomial ring on $nd^2$ variables corresponding to the coordinates of $n$-many $d\times d$ matrices....
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634 views

Research topics in representation theory of algebras [closed]

I was wondering what are some of the hot topics in quiver representation or representation theory of algebras that can lead to good mathematics and is important to many mathematicians and top ...
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244 views

When is the semi-invariant ring is a polynomial ring or a hypersurface?

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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162 views

What is this algebraic object (special case of a semigroup)?

Let $(M,*)$ be a finite semigroup. Further we demand the following: Zero element: $\exists0\in M \forall m\in M:0*m=0=m*0$. Left cancelation: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$. ...
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147 views

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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124 views

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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262 views

Tensor of finite-dimensional algebra over perfect field is semisimple

Let $K$ be a field and let $\Lambda_{1}$ and $\Lambda_{2}$ be two finite-dimensional $K$-algebras with Jacobson radicals $J_{1}$ and $J_{2}$ respectively. How to show or where can I find the proof of ...
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277 views

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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39 views

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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119 views

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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84 views

$G$-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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213 views

Quiver representations of type $D_n$ mutation class

I was wondering if there is a classification of the indecomposable quiver representations of (not necessarily acyclic) quivers that are mutation equivalent to the $D_n$ Dynkin diagram. Such quivers ...
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2answers
260 views

Checking if Hochschild cohomology $\mathit{HH}^2(A)=0$

I am trying to compute the Hochschild cohomology of a particular bound quiver path algebra. The quiver $Q$ consists of one vertex and four loops $x,y, h_1,h_2$, and the relations $I$ are generated by: ...
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430 views

Embedding of a derived category into another derived category

I am considering the following two cases: Assume that there is an embedding: $D^b(\mathcal{A})\xrightarrow{\Phi} D^b(\mathbb{P}^2)$and the homological dimension of $\mathcal{A}$ is equal to $1$($\...
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38 views

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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73 views

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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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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1answer
394 views

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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1answer
244 views

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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103 views

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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202 views

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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170 views

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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40 views

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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147 views

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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1answer
140 views

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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1answer
164 views

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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1answer
162 views

$M^{ss}_{(2,2)}(K_3,(-1,1))$ 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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278 views

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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1answer
123 views

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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793 views

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 ...
3
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1answer
100 views

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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1answer
142 views

(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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116 views

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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1answer
162 views

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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5answers
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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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2answers
291 views

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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60 views

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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165 views

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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127 views

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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1answer
254 views

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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1answer
461 views

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 ...
6
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1answer
332 views

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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36 views

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 ...
3
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1answer
205 views

$\mathrm{Ext}$ 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 ...