All Questions
Tagged with quivers cluster-algebras
16
questions
13
votes
2
answers
349
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 ...
- 131
8
votes
2
answers
878
views
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 ...
- 338
6
votes
1
answer
278
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 ...
- 6,041
4
votes
1
answer
115
views
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 ...
- 417
4
votes
1
answer
94
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 ...
- 143
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
3
votes
0
answers
95
views
Mutations in triangulated category and cluster algebra
Let $\mathcal{D}$ be an enhanced triangulated category (basically meaning that $\operatorname{Hom}$'s are complexes). There is the notion of mutation in an enhanced triangulated category: given a full ...
- 945
2
votes
1
answer
301
views
Cluster algebras of finite type
In the webpage, there is a result:
Theorem 1. Coefficient free cluster algebras without frozen variables are in bijection with Dynkin diagrams of type $A_n$, $B_n$, $C_n$, $D_n$, $E_6, E_7, E_8$, $...
- 6,041
2
votes
1
answer
168
views
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, ...
- 6,041
1
vote
1
answer
102
views
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 ...
- 6,041
1
vote
1
answer
187
views
Mutation of valued quivers
Mutations of valued quivers are defined in cluster algebras II, Proposition 8.1 on page 28. I have a question about the number $c'$. For example, let $a = 2, b=1, c=1$ and consider the quiver $Q$:
$1 ...
- 6,041
1
vote
1
answer
161
views
Why are exchange graphs of quivers with the same underlying graph but have different orientations isomorphic?
I know the fact that (undirected) exchange graphs of quivers with the same underlying undirected graph but have different orientations are isomorphic (i.e. quivers that are just finitely many arrow-...
- 417
1
vote
0
answers
117
views
Number of cluster variables associated to A type quivers
In a seminar/reading course about cluster algebras, we came across the fact that the number of cluster variables for the cluster algebra associated to mutating the quiver $A_n$ is $n(n+3)/2$ (rather, ...
- 315
1
vote
0
answers
53
views
When the dg cluster category of a quiver is saturated?
Let $Q$ be a finite quiver without oriented cycles. In https://arxiv.org/abs/0807.1960 , Keller defines the dg cluster category $C_Q$ of $Q$.
When is $C_Q$ smooth proper dg-category?
If $Q$ is a ...
- 357
0
votes
1
answer
179
views
Mutation equivalence of quivers
Given two orientations $Q, Q'$ of a Dyinkin diagram. Is it always true that after a sequence of mutations, $Q$ becomes $Q'$? Are the some references about this? Thank you very much.
- 6,041
0
votes
0
answers
175
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-...
- 775