All Questions
Tagged with quivers ra.rings-and-algebras
31
questions
7
votes
2
answers
826
views
Does anyone recognize this quiver-with-relations?
Below I describe an infinite (but locally finite) quiver with relations. My question is whether anyone recognizes it and can provide appropriate pointers to the literature. I'm mainly interested in ...
- 12.2k
7
votes
1
answer
246
views
Description of modules over self-injective algebras of finite representation type
Is there any description of indecomposable modules and irreducible morphisms over self-injective algebras of finite representation type? I am interested mainly in such a description for nonstandard ...
- 71
6
votes
2
answers
453
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 ...
- 593
6
votes
1
answer
387
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 ...
- 523
6
votes
1
answer
243
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 ...
- 403
5
votes
2
answers
344
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:
...
- 961
5
votes
2
answers
475
views
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 ...
- 217
5
votes
1
answer
850
views
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 ...
- 483
5
votes
1
answer
387
views
When are infinite dimensional path algebras hereditary?
I allready asked this on MO, but did not get any answer.
Given a finite quiver with relations. When is the path algebra modulo relations hereditary?
If the path algebra is finite dimensional or ...
- 2,504
5
votes
0
answers
408
views
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?
- 51
4
votes
0
answers
224
views
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 ...
- 757
4
votes
0
answers
423
views
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 ...
- 1,685
3
votes
2
answers
353
views
Do morphisms of finitely-decomposable Quiver representations map indecomposables nicely?
Consider two quivers $Q$ and $Q'$ of type $A_n$, laid out horizontally like so:
Given representations of $Q$ and $Q'$, Gabriel's theorem guarantees the existence of finitely many indecomposables for ...
- 15.3k
3
votes
1
answer
279
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
3
votes
1
answer
139
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 ...
- 951
3
votes
0
answers
76
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 ...
- 961
3
votes
0
answers
131
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 ...
- 1,149
3
votes
0
answers
151
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 ...
- 7,100
3
votes
0
answers
355
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!
- 163
2
votes
1
answer
130
views
Rep infinite, but $\tau$-tilting finite
Let $A$ be a finite dimensional algebra over an algebraically closed field. I'm trying to better understand the difference between $A$ being representation infinite and $A$ being $\tau$-tilting ...
- 757
2
votes
1
answer
279
views
Indecomposable extensions of regular simple modules by preprojectives
Given four points in general position on $\mathbb{P}^2$ there exists a projection to $\mathbb{P}^1$ collapsing these four pairwise to two points. Its kernel is some fifth point on $\mathbb{P}^2$.
In ...
- 843
2
votes
0
answers
80
views
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 ...
- 757
2
votes
0
answers
98
views
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 ...
- 420
2
votes
0
answers
233
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 ...
- 757
2
votes
0
answers
103
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
1
vote
1
answer
205
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-\...
- 757
1
vote
0
answers
115
views
Quiver representations and the standard matrix decompositions
Many matrix decompositions - like the Jordan Normal Form, the SVD, the spectral theorem, the Takagi decomposition - have the property that they express a matrix $M$ as the form:
$$M = A D B$$
where $D$...
- 4,752
1
vote
0
answers
56
views
Structure of tame concealed algebra of Euclidean type
I wanted to know some references where people have studied the representation theory of tame concealed algebra of Euclidean type. What do we know about the structure of their module category? What ...
- 757
1
vote
0
answers
123
views
Example of a brick-infinite, tame, triangular algebra of global dimension$\geq 3$
I'm trying to compute some examples and I'm unable to come up with a following example:
What is(are) the example(s) of an acyclic quiver $Q$ with relations such that the 2-Kronecker quiver is NOT a ...
- 757
1
vote
0
answers
108
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\...
- 757
0
votes
0
answers
86
views
Connected components of $Q(\mathrm{s\tau-tilt}A)$
I'm reading about support $\tau$-tilting modules and their mutations. I'm trying to understand the mutation quiver.
Let $A$ be a finite dimensional algebra over an algebraically closed field, which is ...
- 757