nLab
quiver

Definition

The notion quiver is usually taken to be nothing but that of (finite) directed graph. However, saying quiver instead of graph indicates focus on a certain set of operation intended on that graph. Notably there is the notion of a quiver representation.

Now, one sees that a representation of a graph $G$ in the sense of quiver representation is nothing but a functor $\rho :Q:=F(G)\to \mathrm{Vect}$ from the free category $F(G)$ on the directed graph $Q$:

Given a directed graph $G$ with collection of vertices $G_0$ and collection of edges $G_1$, there is the free category $F(G)$ on the graph whose collection of objects coincides with the collection of vertices, and whose collection of morphisms consists of finite sequences of edges in $G$ that fit together head-to-tail. The composition operation in this free category is the concatenation of sequences of edges.

Here we taking advantage of the adjunction between Cat (the category of small categories) and DiGraph (the category of directed graphs). Namely, any category has an underlying directed graph:

$$U:\mathrm{Cat}\to \mathrm{DiGraph}$$U : Cat \to DiGraph

and the left adjoint of this functor gives the free category on a directed graph:

$$F:\mathrm{DiGraph}\to \mathrm{Cat}$$F : DiGraph \to Cat

Terminology Since this is the central operation on quivers that justifies there distinction from the plain concept of directed graph, we may adopt here the point of view that quiver is synonymous with free category .

So a representation of a quiver $Q=F(G)$ is a functor

$$R:Q\to \mathrm{Vect}$$R : Q \to Vect

Concretely, such a thing assigns a vector space to each vertex of the graph $G$, and a linear operator to each edge. Representations of quivers are fascinating things, with connections to ADE theory, quantum groups, string theory, and more.

Link to representation theory of algebras

For $Q$ a quiver, write $kQ$ for the path algebra of $Q$ over a ground field $k$. Now $kQ$ is an algebra with $k$-basis given by finite composable sequences of arrows in $Q$, including a “lazy path” of length zero at each vertex.

A module over $kQ$ is the same thing as a representation of $Q$, so the theory of representations of quivers can be viewed within the broader context of representation theory of (associative) algebras.

If $Q$ is acyclic, then $kQ$ is finite-dimensional as a vector space, so in studying representations of $Q$, we are really studying representations of a finite dimensional algebra, for which many interesting tools exist (Auslander-Reiten theory, tilting, etc.).

Remarks

In the literature, a quiver is usually defined as nothing but a directed graph. The point of this redundant use of terminology is that when people call their directed graphs “quivers”, they indicate that they want to perform certain operations on it, such as, notably, considering its quiver algebra. But this really means that they consider precisely the free category generated by the quiver. For that reason it makes sense to define, as above, that “quiver” really refers to the free category over a directed graph. This definition is arguably more sensible than the standard one, which is just redundant – but it is non-standard.

Literature

• Harm Derksen and Jerzy Weyman, Quiver representations AMS Notices, 2005.

• William Crawley-Boevy, Lectures on quiver representations.

• Alistair Savage, Finite-dimensional algebras and quivers, Encyclopedia of Mathematical Physics, eds. J.-P. Françoise, G.L. Naber and Tsou S.T., Oxford, Elsevier, 2006, volume 2, pp. 313-320.

These references need to be cleaned up. There are also many more to be added.

Discussion

Eric says: In the reference below

• Alistair Savage, Finite-dimensional algebras and quivers, Encyclopedia of Mathematical Physics, eds. J.-P. Françoise, G.L. Naber and Tsou S.T., Oxford, Elsevier, 2006, volume 2, pp. 313-320.

he defines a quiver as a directed graph. Then it seems like the path algebra is the category generated by the quiver. It seems the literature might be inconsistent in the naming conventions? Perhaps we need to settle the convention for nLab.

The wikipedia page also defines a quiver as a directed graph. It has a blurb about a categorical definition, but it doesn’t seem to be consistent with the previous definition (?)

Urs says: You are right, Eric, I think John “played Bourbaki” a bit when he wrote the above. I have added a comment in an attempt to clarify the situation.

John says: Yes, I played Bourbaki, trying to add some clarity to the usual treatment of quivers by recognizing that what people always do with quivers is treat them as the free categories on directed graphs! And after all, if you don’t want to take the free category on your directed graph, why bother calling it a ‘quiver’? You might as well call it a ‘directed graph’!

In short, my definition fits what people actually do with quivers, and saves ‘quiver’ from being a redundant synonym for ‘directed graph’.

By the way, quivers are called ‘quivers’ because they’re a bunch of arrows.

The Wikipedia entry on the mathematics of quivers says that a categorical definition of quiver can be given, but doesn’t actually give it: it says a quiver can be considered as a category, but it doesn’t say precisely which kind of category. But, people know which kind: a free category on a directed graph.

Also by the way, I’ve moved this down here because I imagine many pages on the $n$Lab will gradually acquire long and interesting discussions which we don’t want stuck right in the middle of the main stuff. Wikipedia has ‘talk pages’ for this purpose.

I removed the green ‘query’ box, which gets a bit inconvenient when the discussion gets long. But, it might be nice to have this sort of discussion be visibly different from the main article.

Eric says: What if we create a separate page called Discussion: quiver and link to it from here? Or something…

Hugh says: It is hard to defend Gabriel’s original choice to introduce the term “quiver” instead of talking about directed graphs. Nonetheless, the term having existed for more than thirty years as a synonym for “directed graph”, it seems to me that it would only cause confusion to try to change the definition now.

Let me emphasize that it really is used as a synonym – it doesn’t necessarily carry the implication that the representations of the graph are going to be considered. For example, representation theorists talk about the AR-quiver of a category all the time, and they hardly ever mean that they want to consider the representation theory of that quiver.

Edited to add: sorry, I wouldn’t have bothered saying this if I’d noticed that the definition had already been changed!

Toby: I'm inclined to rename this page to free category or free category on a quiver?.

Zoran: If we are going to talk free categories instead than one should be aware that some books (e.g. Gabriel-Zisman) say path category instead of free category. As far as quivers, what about “quivers with relations” ?

Toby: We have another meaning of path category, but that is a good term.