Questions tagged [quiver]
A quiver is an oriented graph which might contain multiple edges and loops. The terminology is used in representation-theory of finite dimensional algebras, where one considers functors from this graph, viewed as a category, to the category of vector spaces.
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Minimally sufficient condition for composability of quiver morphisms?
Given three quivers $(X_0, X_1, \sigma, \tau)$, $(Y_0, Y_1, \phi, \psi)$, and $(Z_0, Z_1, \chi, \omega)$, and two morphisms $F := (F_0 : X_0 \to Y_0, F_1 : X_1 \to Y_1) : (X_0, X_1, \sigma, \tau) \to (...
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Dimension formula/algorithm for quiver varieties?
The title says the question. For a quiver and a dimension vector and a stability vector, we can construct a moduli space of semistable quiver representations with the given dimension vector. The ...
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Jordan and Dynkin quivers
I have a simple question. If I am not wrong, we can define $\tilde{A}_l$ affine Dynkin quiver of type A, for $l\geq1$. It has $l+1$ vertices that we can order from $0$ to $l$ such that $i$ is ...
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Adjoint functors induce in representations of quivers? (references)
Let $Q$ be a quiver and ${C}_1, C_2$ be two (small) categories. Let $F:C_1\rightarrow C_2$ and $G:C_2\rightarrow C_1$ be two (covariant) functors.
Is it true that if $F$ and $G$ induce functors
$$
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Endomorphism ring of a quiver representation
Consider the quiver with one loop, and consider the representation on $\mathbb{F}_{2}(t)$ described by the matrix $$W=\begin{pmatrix} 1 & t \\ 1 & 0 \end{pmatrix}.$$ I need to calculate the ...
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Correspondence between type A Nakajima Quiver varieties and weight spaces in the tensor product of fundamental representations of $\mathfrak{gl}(N)$
In the paper "New Quiver-Like varieties and Lie Superalgebras" by Rimanyi and Rozansky it is claimed that "There is a well-known correspondence between an $A_n$-type framed Nakajima ...
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Showing $J(A)=0$, where $A$ is the path algebra of an oriented cycle
Let $Q$ be a cyclic quiver with $n$ vertices and consider the path algebra $A=KQ$. I want to show that the Jacobson radical of $A$ is zero. Since the Jacobson radical is the intersection of all ...
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Path algebra on quivers of type ADE
Let $K$ be a field. For $n\in \mathbb{N}$ consider the equioriented $A_n$-quiver. It is easy to see that the path algebra $K[A_n]$ is isomorphic as a $K$-algebra to the algebra of upper triangular $n\...
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Connected components of $Q(\mathrm{s\tau\textrm{-}tilt} A)$
$\newcommand{\hy}{\textrm{-}}$
I'm reading about support $\tau$-tilting modules and their mutations. I'm trying to understand the mutation quiver.
Let $A$ be a finite dimensional algebra over an ...
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Fixed points of subtorus in Nakajima quiver variety (Lemma 3.2 in Nakajima's Quiver varieties and tensor products)
I am having trouble understanding the proof of lemma 3.2 in Nakajima's paper "Quiver varieties and tensor products," available on arxiv. This is where he shows that the fixed components of a ...
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$\text{soc}(V)$ and $\text{rad}(V)$ of a representation $V$ of an acyclic quiver $Q$
Let $Q$ be an acyclic quiver, and $V = (V_i, V_a)$ a representation of $Q$. I want to show that $$\text{soc}(V) = \bigoplus_{i \in Q_0} \left(\bigcap_{a \in Q_1, s(a) = i} \ker(V_a) \right), \hspace{...
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Some reference requests for path algebras
I can find some properties and characterizations of path algebras and its elements in representation theory books but I want some references that study and treat path algebras as invertible elements, ...
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Classify the finite-dimensional indecomposable representations of the quiver $A_n$
I have to classify the finite-dimensional indecomposable representations of the quiver $A_n$ up to isomorphism:
1 $\longrightarrow$ 2 $\longrightarrow \cdots
\longrightarrow$ n
Any idea of how to ...
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How does the center of a Path Algebra KQ for a quiver Q look like?
) I am attending a Foundation Course in Representation Theory and I am struggling with the following problem: determine the center of the path algebra KQ for a quiver Q.
First of all I've thought that ...
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Applications of representation theory in topology
I'm beginning to study representation theory, in particular, quiver representations. Since I have more familiarity with topology, I was wondering if there is any applications of these things to ...
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How to decompose representations of $1\to 2\leftarrow 3$?
For now let's consider the $n=2$-subspace quiver: $Q = \bullet \to \bullet \leftarrow \bullet$. I have shown (with the help of Kirillov) that to classify these representations, it suffices to classify ...
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Intuition for representation of a quiver inherited from the representation of the quiver's path algebra
Let $V$ be a representation of the path algebra $P_{Q}$. From this representation, we can construct a representation of $Q$ as follows: let $V_i =p_i V$, and for any edge $h$, let $x_h = a_h|_{p_h'} v ...
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Dimension of $(U+V+W)$
Let $U, V, W$ be subspaces of a vector space. I know that in general, the equation
$\dim(U+V+W)=\dim U+\dim V+\dim W-\dim(U \cap V)-\dim (U\cap W)-\dim (V \cap W)+\dim (U \cap V \cap W)$
doesn't hold. ...
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Is the Schofield semi-invariant defined at $V/IV$?
Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module ...
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How to determine if an invariant rational function is defined at the $\theta$-polystable point?
Background:
Let $A$ be a finite-dimensional (associative and unital) algebra over $\mathbb{C}$. Assume there is a quiver $Q=(Q_0,Q_1)$, where $Q_0$ is the set of vertices and $Q_1$ is the set of ...
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Example of a non-hereditary algebra whose $g$-vector fan is not complete
I'm reading the paper of Pierre-Guy Plamondon, Toshiya Yurikusa and Bernhard Keller "Tame Algebras Have Dense $g$-Vector Fans". I wanted to know some explicit examples of non-hereditary ...
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Why is a linear representation of a quiver a functor?
Let $Q:=(Q_0,Q_1,s,t)$ be a quiver and $k$ some field. A representation $(M,\rho)$ of $Q$ over $k$ is the following data:
A $k$-vector space $M_v$ for every $v\in Q_0$;
A $k$-linear map $\rho(a):M_{...
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References for Coefficient Quivers
I would like to study about Coefficient Quivers, but I cannot find a good reference.
I could find many papers working with coefficient quivers, but none of them give a
book or a "initial" ...
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Example of an irreducible component with an open set of infinitely many codimension 2 (codimension 3) orbits
Let $\mathbb{K}$ be an algebraically closed field of characteristics $0$. Let $A$ be a finite-dimensional (associative and unital) algebra over $\mathbb{K}$. Assume there is a quiver $Q=(Q_0,Q_1)$, ...
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Algebraic structure associated to non-ADE quivers
It is known that compact simple Lie algebras are classified by there Dynkin diagrams. Those Dynkin diagrams are the ADE quivers (plus the B,C,F,G-type quivers that I will not consider here).
Therefore,...
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Every root is both preprojective and preinjective in the root systems of Dynkin diagrams
I have a definition for preprojective and preinjective roots in a root system as:
If $\mathsf{R}$ is a root system and $C$ is Coxeter element adapted to an orientation of a quiver $Q$ without oriented ...
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Use the equivalence between representations and $kQ$-modules to find $\mathrm{rad}(P(i))$ and $\mathrm{top}(P(i))$
Question: I had the exercise below (not graded) and I think I have all the components to solve it, but somehow I can't seem to find a proper proof. To what right $kQ$-modules do I have to relate $P(i)$...
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Complete set of primitive orthogonal idempotents of the bound quiver algebra $KQ/I$ with $I$ ammissibile ideal.
I’m studying the book “elements of the representation theory of associative algebras, volume 1”. At the page 55 there is an interesting lemma which I don’t understand the prof.
($K$ is an ...
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Example of a brick-infinite, tame, triangular algebra of global dimension$\geq3$
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 ...
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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\...
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McKay correspondence for Irreps of $G \subset SU(2)$
We follow Alexander Kirilov's book "Quiver Representations and Quiver Varieties", Section 8.3: McKay correspondence: Let $G$ be a nontrivial finite subgroup in $SU(2)$. Let $Q(G)$
be the ...
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Notation for Modules Over Quiver Algebras
Disclaimer: I know that my question has probably answers in textbooks, but I have only found the answer for quiver representations, and not for modules over quiver algebras. I know that these two ...
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Is the connected assumption necessary in the following theorem?
I'm reading about the following facts from the book 'Elements of the Representation Theory of
Associative Algebras' by Assem, Simpson, and Skrowronski. Note that the algebras involved are not assumed ...
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Prove that R is an integral domain
I'm studying for my qualifying exam and I came across the following question in one of the old question bank.
Consider the affine space given by four $2\times 2$ matrices, i.e., $\mathbb{A}^{16}\cong ...
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When two slices give the same orientation and any orientation can be obtained from a slice.
In the book quiver representation and quiver varieties he lefts Lemma 6.13 to the reader, but I'm having trouble with it.
If $Q$ is a tree, any orientation of $Q$ can be obtained from a slice in $\...
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How to read off distinguished triangles and cluster-tilting objects in the cluster category of a Dynkin quiver?
I'm new to triangulated category and tilting theory. To illustrate, in $Q=A_4$ the module $M=kQ$ is cluster-tilting. While I know that $M$ satisfies $\mathrm{Ext}(M,M)=\mathrm{Hom}(M,M[1])=0$ by some ...
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A doubt in proof of existence of Auslander Reiten Sequences
I am providing the context. You can skip to the actual doubt below. My doubt doesnt need knowledge of quivers or even almost split sequences.
I am reading Theorem 5.4.10 from the book An introduction ...
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An application of Snake Lemma
I am stuck at undderstanding a claim made in Derksen's book An Introduction to Quiver Representations.
This is Lemma 2.4.3. But the part I am stuck at is purely homological algebra it seems.
Let $A$ ...
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Non-empty stable locus in 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 us ...
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Termination of Buchberger's algorithm for path algebras
I'm reading the paper Noncommutative Gröbner Bases, and Projective Resolutions by Edward L. Green, which presents a version of Buchberger's algorithm for path algebras.
I'm trying to show that the ...
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Dimensions of homomorphisms determines the module
I am reading this paper about degenerations for representations.
I am mentioning some context though all of it will not be needed I think for my question.
We basically have a finite dimensional ...
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Viewing Path Algebras as Matrix algebras
Given a Path Algebra $\mathbb{C}Q$ of an acyclic directed graph(also called a Quiver in this theory) $Q$. I am interested in finding a corresponding matrix algebra for it if possible.
I am able to do ...
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no. of minimum generating sets of $M_{n}(\mathbb{C})$
Problem: How many $n \times n$ pairs $(A, B)$ of (0 , 1) matrices are there which generate the whole algebra $M_{n \times n}(\mathbb{\mathbb{C}})$? I don't know if this question is open or hopelessly ...
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Cluster algebras from quivers vs. cluster algebras from skew-symmetrizable matrices
In a course on cluster algebras we first defined the cluster algebra arising from a quiver. Next we saw that each quiver gives rise to an exchange matrix that is skew-symmetric. So we generalized to ...
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Derived functor of a reflection (BGP) functor
I'm studying the book "quiver representations and quiver varieties" of Kirillov and I'm in Theorem 3.10:
The functor $\Phi_i^+$ is left exact. Moreover, $R^n\Phi_i^+(V)=0$ for all $n>1$, ...
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What if the semi-invariant ring is a polynomial ring or 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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Why do people study semi-invariant ring (in general)?
I'm interested in studying about semi-invariant ring in the context of Quiver representations. I started reading about it in some books and few places on the internet. But, nowhere do they mention why ...
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What is the quiver representation of an algebra, and how is it obtained from a decomposition into spaces of morphisms (the opposite algebra)?
I will make everything concrete so I have something to work with.
Let $A$ be an algebra, free, and 4-dimensional, and let $A = P_1 \oplus P_2$ be a decomposition into two 2-dimensional free modules.
I ...
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Gabriel's theorem
Need some help with Gabriel's theorem, doing the part "if a graph is of finite type, it must be a Dynkin graph" like this:
Let $\vec{Q}$ be of finite type. All representations is a direct ...
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Intuitive explanation of quiver $Q^{(n,s)}$.
On page 6 of this article, the author defined a new quiver.
Fix positive integers $n$ and $s$. We define the quiver $Q:=Q^{(n, s)}$ with the set $Q_{0}$ of vertices and the set $Q_{1}$ of arrows by
$$
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