All Questions
Tagged with quivers rt.representation-theory
125
questions
34
votes
8
answers
7k
views
Are quivers useful outside of Representation Theory?
There is a trend, for some people, to study representations of quivers. The setting of the problem is undoubtedly natural, but representations of quivers are present in the literature for already >...
- 1,419
30
votes
3
answers
4k
views
Why did Gabriel invent the term "quiver"?
A quiver in representation theory is what is called in most other areas a directed graph. Does anybody know why Gabriel felt that a new name was needed for this object? I am more interested in why he ...
27
votes
3
answers
2k
views
How can classifying irreducible representations be a "wild" problem?
Let $q$ be a prime power and $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$-matrices. I've read and heard in several places (see e.g. this mathoverflow question) that classifying ...
- 2,390
21
votes
5
answers
3k
views
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: $${D}^b(\mathrm {Coh}(\Bbb P^1))\simeq {D}^b(\mathrm {Rep}(\bullet\rightrightarrows \bullet))$...
- 1,135
19
votes
2
answers
1k
views
Quiver representations and coherent sheaves
I've heard that under certain assumptions on an algebraic variety $X$ there exist a quiver $Q$ for which there is an equivalence $$D^b(\mathsf{Coh}(X))\simeq D^b(\mathsf{Rep}(Q))$$ between the ...
user95283
18
votes
7
answers
2k
views
ubiquity, importance of path algebras
I work in planar algebras and subfactors, where the idea of path algebras on a graph (alternately known as graph algebras, graph planar algebras, etc.) is quite useful. The particular result I'm ...
- 1,089
14
votes
2
answers
944
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....
- 6,567
14
votes
2
answers
2k
views
when are algebras quiver algebras ?
Good Morning from Belgium,
I'm no stranger to the mantra that quiver-algebras are an extremely powerful tool (see for example the representation theory of finite dimensional algebras). But what is a ...
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
13
votes
0
answers
604
views
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 $\...
- 3,114
13
votes
0
answers
555
views
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 ...
- 27.5k
12
votes
3
answers
1k
views
construct scheme from quivers?
I heard from some guys working in noncommutative geometry talking about the idea that one can construct the noncommutative space from quivers. I feel it is rather interesting. However, I can not image ...
- 1,265
11
votes
2
answers
1k
views
What do the local systems in Lusztig's perverse sheaves on quiver varieties look like?
In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined ...
- 43.4k
11
votes
1
answer
766
views
What's known about the stalks of Lusztig's perverse sheaves on quiver varieties?
Lusztig has defined a category of perverse sheaves on the moduli space of representations of a Dynkin quiver (see his paper) corresponding to canonical basis vectors.
I'm interested in the stalks ...
- 43.4k
11
votes
0
answers
184
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 $\mathcal{O}$ .
...
- 25.4k
10
votes
3
answers
1k
views
Are the underlying undirected graphs of two mutation-equivalent acylic quivers isomorphic?
Quiver mutation, defined by Fomin and Zelevinsky, is a combinatorial process. It is important in the representation theory of quivers, in the theory of cluster algebras, and in physics.
We consider ...
- 101
10
votes
1
answer
609
views
The Fukaya category of a simple singularity (reference request)
I have heard that for an ADE singularity $f$,
$ D^b\mathrm{Fuk}(f) \simeq D^b(\mathrm{Rep}\ Q)$
where $Q$ is the corresponding Dynkin quiver. (As one would hope, if $\mathrm{Fuk}$ is some kind of ...
- 843
9
votes
1
answer
455
views
Algorithm for finding quiver algebras
Im looking for an algorithm that does the following in a quick way:
Input: Natural number $r \geq 2$, natural number $s \geq 3$, prime power $q$.
Output:
Finds all two-sided ideals in $J^2/J^s \...
- 25.4k
8
votes
3
answers
1k
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 ...
- 1,149
8
votes
1
answer
959
views
Global dimenson of quivers with relations
Let Q be a finite quiver without loops. Then its global dimension is 1 if it contains at least one arrow.
I'm trying to get some intuition about how much the global dimension can grow when we ...
- 9,982
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
8
votes
1
answer
477
views
Does unique factorisation hold for quiver algebras?
Given a finite dimensional quiver algebra A=KQ/I. It can be (possibly) written as $A= B_1 \otimes_k B_2 ... \otimes_k B_r$ and the $B_i$ can not be decomposed into smaller algebras. Is this ...
- 25.4k
8
votes
1
answer
529
views
Construction of irreps of path algebra of cyclic quiver, classification of all finite-dimensional irreps
Originally posted here on Mathematics Stack Exchange.
Let $Q$ be a quiver with vertex set $\{1, 2, \ldots, n\}$ such that $Q$ has a single edge $i \to i + 1$, for every $i = 1, 2, \ldots, n - 1$, one ...
- 339
8
votes
1
answer
764
views
Quivers of selfinjective algebras.
Let's say a quiver $Q$ is covered by cycles if each of it’s arrows can be included in an oriented cycle.
It's easy to prove that if a path-algebra with relations $KQ/I$ (where $I$ is an admissible ...
- 83
7
votes
3
answers
783
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 ...
- 1,149
7
votes
3
answers
399
views
Smallest faithful representation of an upper-triangular matrix quotient
This is a curiosity question that came out of teaching abstract algebra.
Let $F$ be a field, and $n>1$ an integer.
Let $F^{n \leq n}$ be the $F$-algebra of all upper-triangular $n\times n$-matrices ...
- 32.7k
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
1
answer
333
views
Is this modified bound quiver algebra necessarily representation-finite?
Suppose that $A = kQ/I$ is a bound quiver algebra for $k$ an algebraically closed field, $Q=(Q_0, Q_1)$ a finite connected quiver with no oriented cycles with no multiple edges or self-loops, and $I$ ...
- 185
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
6
votes
1
answer
277
views
A question about saturation of quivers
Let $Q$ be an acylic quiver. Let $E$ and $F$ be finite dimensional representations, with $E$ indecomposable. Suppose that, for some positive integer $r$, the representation $F$ injects into $E^{\oplus ...
- 150k
6
votes
2
answers
712
views
Auslander-Reiten theory for Gorenstein algebras
In the paper "Cohen-Macauley and Gorenstein artin algebras", Auslander and Reiten have a short section about Auslander-Reiten theory in Gorenstein algebras (I always assume we have an artin algebra ...
- 25.4k
6
votes
1
answer
465
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 ...
- 25.4k
6
votes
1
answer
183
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)= \...
- 6,075
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
6
votes
2
answers
345
views
Is this algebra isomorphic to an incidence algebra?
This question is motivated by trying to establish a converse to Theorem 7.8 of our paper.
I have a finite poset $P$ with the following properties:
$P$ has binary meets (and hence a least element).
$...
- 36.8k
6
votes
2
answers
885
views
Do fixed point sets in equivariant crepant resolutions have the same cohomology? How about for the specific case of Nakajima quiver varieties?
A crepant resolution $f:Y\to X$ is a resolution of singularities with $f^*(K_X)=K_Y$. Crepant resolutions do not always exist, and when they exist they may not be unique. However, different crepant ...
- 2,362
6
votes
1
answer
300
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-...
- 757
6
votes
0
answers
205
views
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 ...
- 25.4k
5
votes
3
answers
765
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 \...
- 2,137
5
votes
3
answers
763
views
Acyclic quivers differing only in arrow directions: functorial isomorphism of representation categories?
Let $Q$ and $R$ be two acyclic quivers which differ only in the directions of their arrows (i. e., the underlying undirected graphs are the same).
1. Does there exist an isomorphism of additive ...
- 32.7k
5
votes
1
answer
992
views
Indecomposable modules over preprojective algebras
Would you please give some references concerning the number of indecomposable modules over preprojective algebras of type $A_n$?
More precisely, I need references about the following claim: The ...
- 3,038
5
votes
1
answer
356
views
When is a given quiver algebra a hopf algebra?
Given a finite dimensional selfinjective quiver algebra A over a finite field (or more generally an arbitrary field). Whats the best way to check if the algebra A has a Hopf algebra structure or not? ...
- 25.4k
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
504
views
what is the injective hull of indecomposable module of preprojective algebra
Let $Q$ be a ADE type quiver and $s_i$ ($i$ runs through the vertices of $Q$) be the simple $\Lambda$-module with 1-dimensional vector space at vertex $i$ and zero-dim at other vertices. Here $\Lambda$...
- 849
5
votes
1
answer
498
views
analog of Lusztig nilpotent scheme
Fix a quiver $Q$ without loop. Denote the set of vectices of $Q$ by $I$.
Let $\Lambda_V$ be the Lusztig nilpotent scheme with associated vector space $V$ over $I$. Briefly speaking, when $Q$ is a $ADE$...
- 849
5
votes
1
answer
495
views
Morita equivalence of acyclic categories
(Crossposted from math.SE.)
Call a category acyclic if only the identity morphisms are invertible and the endomorphism monoid of every object is trivial. Let $C, D$ be two finite acyclic categories. ...
- 114k
5
votes
0
answers
93
views
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 ...
- 36.8k
5
votes
0
answers
344
views
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-...
- 757
5
votes
0
answers
213
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?
- 373
5
votes
0
answers
307
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 ...
- 6,075