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.
14 questions from the last 365 days
9
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144
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Ringel's interpretation of quantum groups as Hall algebras at $q=1$
Let $Q$ be a finite-type quiver and let $\mathfrak{g}$ be the semisimple Lie algebra associated with the corresponding simply-laced Dynkin diagram. Let $U_v^+(\mathfrak{g})$ be the positive part of ...
- 3,446
2
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0
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45
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Hall algebra of constructible functions of affine quiver?
I have read in "Quiver Representations and Quiver Varieties" by Kirillov that Hall algebra of constructible functions are defined only for Dynkin quivers because they are of finite type. So ...
- 133
6
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1
answer
139
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Quiver variety, generically symplectic
Theorem 11.3.1 (iv) of "Noncommutative Geometry and Quiver algebras" by Crawley-Boevey, Etingof and Ginzburg claims that, for a dimension vector $\mathbf{d}\in\Sigma_0$, the quiver variety $...
- 985
2
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0
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72
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Does this finitely-generated algebra have a name?
I've been led to consider certain finitely-generated algebras that arise from some Coxeter groups (finite and affine Weyl groups at least). As a very concrete example, consider the infinite dihedral ...
- 13.3k
2
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70
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Does every SES of injective bounded cochain complexes split?
Question: Does every short exact sequence of injective bounded cochain complexes, $0\rightarrow I^\bullet\rightarrow J^\bullet\rightarrow K^\bullet\rightarrow 0$, split?
I am interested in a discrete ...
- 213
1
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0
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90
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Multiplicative bases, path algebras, and Ext algebras
I am interested in understanding when a multiplicative basis exists for finite dimensional algebras over an algebraically closed field, and, in particular, Ext-algebras that are finite dimensional.
It ...
- 71
0
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0
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103
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Matrix of the minimal projective presentation of a $\tau$-rigid module
Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$ of characteristics zero. Suppose $A$ is given by the bound quiver $(Q,I)$. We will use $P_l$ to denote the ...
- 839
3
votes
0
answers
102
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Projective resolution of a quiver with relations
How do we compute the projective resolution of a representation of a quiver with relations.
For example consider the Beilinson quiver $B_4$
$.
with the relations $\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^...
- 179
1
vote
0
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130
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A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$
I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)
I want to ...
- 839
3
votes
0
answers
107
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Auslander-Reiten sequences where irreducible morphisms are all epi/mono
Let's work in the setting of modules over an Artin algebra $A$, or a finite-dimensional $k$-algebra $A$, or if you like, modules over a connected quiver $Q$ without oriented cycles.
Let $M$ be such a ...
- 13.3k
4
votes
0
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184
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Proof of McKay correspondence without classifications
$\DeclareMathOperator\SU{SU}$I am wondering if there is any known proof of the McKay correspondence (I will give the precise statement that I mean by this) that doesn't use the classification of ...
- 141
9
votes
1
answer
236
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Formal smoothness of path algebras and connections
Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if
$$
\Omega^1_kA = \operatorname{Ker}(\...
- 985
0
votes
0
answers
70
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geometric objects in quiver variety corresponding to short exact sequences
I was studying quiver variety and known that representations of a quiver correspond to points in the corresponding quiver variety. So if give you a fixed triple representations $(M_1,M_2,M_3)$, I was ...
- 133
0
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0
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71
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"Approximating" ring of semi-invariants
I'm trying to calculate the semi-invariant ring for certain types of quivers. For a very brief introduction to semi-invariant rings of quiver please have a look at this wikipedia article at the ...
- 839