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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
92 views

Prove that $B$ is a directing module

Let $A\cong\mathbb{K}Q/I$ be a finite dimensional, associative, basic $\mathbb{K}$ algebra, where $\mathbb{K}$ is algebraically closed and $Q$ is a finite Gabriel quiver on $n$ vertices and $I\...
  • 391
2 votes
0 answers
65 views

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 ...
  • 22.5k
2 votes
0 answers
177 views

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 ...
  • 391
0 votes
1 answer
150 views

Quiver representations over any commutative ring

I'm reading a paper of Aidan Schofield "General Representations of Quivers" and he defines quiver representation over any commutative ring. See the below image. Towards the end, he has this ...
  • 391
1 vote
1 answer
167 views

A result of Schofield in the case of quivers with relations

Let $Q$ be a quiver without oriented cycles. A result of Schofield says that, for dimension vectors $\alpha$ and $\beta$ of $Q$, $\beta\hookrightarrow\alpha$ iff $\operatorname{ext}(\beta, \alpha-\...
  • 391
5 votes
3 answers
688 views

Quiver with two objects and two arrows composing to zero

In the description of the integral Adams spectral sequence, representations of the following quiver (with relations) arise naturally: We have two objects $A, B$, we have two arrows $\pi: A \...
6 votes
1 answer
274 views

Prove that $\overline{a}_{11}$ is a prime element in $R$

Consider the affine space given by four $2\times 2$ matrices, i.e., $\mathbb{A}^{16}\cong M(\mathbb{C})_{2\times 2}^4$. Now, consider the algebraic set $V$ given by the vanishing of the relation $AB-...
  • 391
4 votes
1 answer
79 views

How much is known about the non-degeneracy of Quiver-with-potential associated to closed punctured surfaces?

The potential of the quiver associated to surfaces is the canonical one given by Labardini-Fragoso's 2009 paper, who proved that the the QP associated to surfaces whose boundary is nonempty is rigid ...
1 vote
0 answers
58 views

A sufficient condition for automorphism of an exact sequence

I asked A sufficient condition for Automorphism of an exact sequence earlier on Math.StackExchange but did not get any response so am posting it here. I am given the following commutative diagram with ...
1 vote
0 answers
123 views

Non-empty stable locus of an irreducible component

I have a vague question: Let $X$ be an algebraic pre-scheme and $G$ be a linear reductive group. Consider the G.I.T. quotient $X{/\!/}G$. Is there any result (maybe in some special case) which tells ...
  • 391
2 votes
2 answers
116 views

Finding exceptional regular representations of $\tilde{D}_4$ efficiently

Let $A$ be the path algebra of the quiver $\tilde{D}_4$. I would like to find its exceptional regular representations with as little computation as possible. Of course, we can compute the whole ...
3 votes
1 answer
85 views

Auslander-Reiten quiver of quiver algebra kQ where Q is of extended dynkin type D4~

I am looking for literature about the Auslander-Reiten quiver of the quiver algebra $kQ$, where $Q$ is of extended dynkin type $\tilde{D_4}$ and $k$ is an algebraically closed field. Does somebody ...
2 votes
0 answers
82 views

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 ...
  • 553
0 votes
1 answer
221 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 $\...
14 votes
2 answers
719 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....
4 votes
2 answers
726 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 ...
5 votes
0 answers
269 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-...
  • 391
1 vote
0 answers
165 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: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$. ...
3 votes
0 answers
157 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 ...
  • 7,062
5 votes
0 answers
150 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?
  • 343
5 votes
2 answers
344 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 ...
  • 553
12 votes
0 answers
311 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 ...
  • 4,234
3 votes
0 answers
59 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 ...
2 votes
0 answers
126 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\ ...
  • 81
2 votes
0 answers
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 ...
  • 21
13 votes
2 answers
265 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 ...
5 votes
2 answers
301 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: ...
12 votes
1 answer
458 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 $1$($\...
  • 1,558
2 votes
0 answers
41 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 ...
  • 1,159
4 votes
0 answers
77 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 ...
  • 22.5k
10 votes
0 answers
142 views

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 $\mathcal{O}$ . ...
  • 22.5k
8 votes
1 answer
471 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 ...
6 votes
1 answer
306 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 ...
  • 513
3 votes
0 answers
120 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 ...
  • 1,119
7 votes
3 answers
674 views

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 ...
  • 1,119
5 votes
0 answers
222 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 ...
  • 2,199
3 votes
1 answer
211 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 ...
  • 31
2 votes
0 answers
42 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 \...
  • 323
6 votes
1 answer
160 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)= \...
  • 5,928
2 votes
0 answers
98 views

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 ...
  • 22.5k
3 votes
1 answer
145 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 $\theta$-...
3 votes
1 answer
200 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, ...
  • 1,119
3 votes
1 answer
165 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 ...
6 votes
2 answers
289 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 ...
0 votes
1 answer
136 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....
  • 849
8 votes
3 answers
924 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 ...
  • 1,119
3 votes
0 answers
129 views

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 ...
  • 6,261
3 votes
1 answer
108 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 ...
4 votes
0 answers
254 views

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 ...
  • 5,928
1 vote
0 answers
73 views

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 "...
  • 22.5k