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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.

A quiver is an oriented graph which might contain multiple edges and loops. Sometimes it is assumed to be finite.

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. To a quiver one can associate its path algebra, which is the $k$-algebra with basis given by all paths in the quiver and multiplication given by concatenation (or zero). These path algebras are basic hereditary algebras. And Gabriel's theorem says that up to Morita equivalence all finite dimensional algebras over an algebraically closed field arise as quotients of these path algebras.

Important books to learn more about the subject include:

  • Assem, Simson, Skowronski: Elements of the representation theory of associative algebras
  • Auslander, Reiten, Smalø: Representation theory of Artin algebras
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Why are (representations of ) quivers such a big deal?

Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, Gabriel gives a complete classification of…
Hui Yu
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Path Algebra for Categories

For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", and the like), where $P$ consists of paths in…
Tunococ
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Is there a natural way to regard a valued quiver as a category?

You can think of a quiver, a collection of vertices $Q_0$ and arrows $Q_1$, as a free category with objects $Q_0$ and morphisms generated by the morphisms of $Q_1$. More precisely this is a functor from $\mathsf{Quiv}$ to $\mathsf{Cat}$ that sends a…
Mike Pierce
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When are infinite dimensional path algebras hereditary

The title says mostly everything. Suppose we have a quiver, maybe with relations and cycles. Is it known when the path algebra modulo relations is hereditary? Especially in the case that the path algebra is infinite dimensional. I would appreciate…
O. Straser
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When are path algebras of quivers hereditary.

Suppose we have a finite quiver with relations, possibly with oriented cycles. Is it known when the path algebra of this quiver (with relations) is hereditary?
O. Straser
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Specific projective dimension of a module over bound quiver

Suppose $K$ is an algebraically closed field, and $A$ is the algebra presented by the quiver $$\require{AMScd} \begin{CD} 1 @>>> 2\\ @V{}VV @V{}VV \\ 3 @>>> 4 @>>> 5 \end{CD} $$ bound by $1\to 2\to 4 = 1\to3\to 4$ and $3\to4\to5 = 0$. What is the…
Jack Schmidt
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Reflection Functor of a Quiver

Fix a Quiver $Q=(Q_{0},Q_{1})$, where $Q_{0}$ is the set of vertices and $Q_{1}$ is the set of edges. For $j\in Q_{0}$ define $\sigma_{j}Q$ to be the quiver in which all that include the vertex $j$ are flipped. Suppose $i\in Q_{0}$ is a sink (all…
user551642
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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., $\mathbb{A}^{16}\cong M(\mathbb{C})_{2\times 2}^4$. Now, consider the…
It'sMe
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Computing Hochschild cohomology of an algebra in GAP

I would like to calculate Hochschild cohomology of a path algebra of a quiver (modulo some relations) using GAP. Creating the quiver and its path algebra modulo relations is explained in detail in the QPA manual. After having defined an algebra $A$…
Earthliŋ
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Significance of adjoint relationship with Ext instead of Hom

For categories $\mathcal{C}$ and $\mathcal{D}$, a pair of functors $\mathcal{C} \xrightarrow{\;R\;} \mathcal{D}$ and $\mathcal{C} \xleftarrow{\;L\;} \mathcal{D}\,$ are an adjoint pair if for any objects $X$ of $\mathcal{C}$ and $Y$ of $\mathcal{D}$…
Mike Pierce
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how to get the injective envelope and projective cover of a given module

Given a bound quiver $(Q, I)$ and a representation $M$ of $Q$, how to get the injective envelope and projective cover of $M$? how to give the corresponding essential monomorphism and superfluous epimorphism? Is there a general or specific method?
Aimin Xu
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Representations of a quiver and sheaves on P^1

We know from Beilinson that there's an equivalence of derived categories $D^b Rep(Q) \simeq D^b Coh(\mathbb{P}^1)$ where the lefthandside is the derived category of bounded complexes of representations of the Kronecker quiver $* => *$ and the…
AlgGeomQ
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Can this puzzle be solved using the representation theory of quivers?

This riddle originates in the youtube video here. It's mathematical content was summarised here as follows: There's a $5\times 5$ grid of nodes, all nodes are (bidirectionally) connected to their vertical and horizontal neighbour(s) (so no…
Oscar Cunningham
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From a lower triangular matrix to its quiver representation

I have a question about quivers: Suppose we have an arbitrary lower triangular matrix algebra $A= \{\begin{pmatrix} a & 0 & 0\\ c & b & 0\\ e&d&a \end{pmatrix}: a , b, c, d \in \mathbb C \}$. How can we find the related quiver to this algebra? I…
Nikita
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Question for recommending a good textbook in representation of quivers

I am taking representation of quivers, and the lecture notes seems not enough. So could you recommend a good textbook for this course. There is a new book "Quiver Representations, by Ralf Schiffler" costing more than $60, and I don't know whether…
Dan
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