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Dear All!

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 >40 years.

Are there any connections of this trend with other Maths? For, it seems like it is a self-contained topic and basically I wonder why people study quivers so much -- in a sense everything becomes clear after the initial results of Gabriel and the old result of Yuri Drozd about wild/tame dichotomy, and these things ought to become boring.

BTW, about the dichotomy theorem -- is it really necessary to study so hard whether a given problem is of tame or wild (representation) type? In particular, why some people try to lift tame/wild things to curves and surfaces -- would that really yield something interesting in geometry?

(I am currently attending lectures about these things and unfortunately we were not told a single word about motivation, and when I tried to learn from the lecturer if this is really "top" Maths as he claims, he basically replied "this is important because I am doing this")

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What is your background outside representation theory? In which areas would you like to see applications? –  Julian Kuelshammer Oct 12 '11 at 17:18
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That's quite a ridiculous rationale provided by your instructor for anyone else to think the topic is interesting. How did he determine that the topic was interesting himself -- because his instructor was doing it? –  KConrad Oct 12 '11 at 17:43
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Unfortunately, most lectures provide even less rationale. –  darij grinberg Oct 12 '11 at 18:40

8 Answers 8

up vote 17 down vote accepted

In addition to being a nice example for abelian, $A_{\infty}$ and Calabi-Yau categories, and being a prototypical example for Generalized Donaldson - Thomas Invariants and the Wall Crossing Phenomenon, the quivers have a lot of applications in variours different fields. Since the question is applications in addition to representation theory, I'm listing a few cases,

Most prominent is Algebraic Geometry, particularly Moduli problems and GIT (read motivations in Reinke's article)and Video lectures on quivers by Reineke at Newton Institute, Cambridge.

Recently, a correspondence has been proposed between Gromov - Witten invariants and Quivers. (Pandharipande - Gross).

Also in physics applications in String Theory, Supersymmetry, Black Holes and Particle physics.

Relation with quantum dilogrithm, number theory, and cluster algebras, read e.g. this review by Keller.

Also through the work of Nakajima, there is relation to Instantons of Yang-Mills theory, Hitchin Moduli spaces and the theory of Hyperkahler manifolds.

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This sounds more than exciting!!! I would never believe there is a connection to Differential Geometry! Thanks a lot for the links! –  Victor Oct 13 '11 at 8:54

Speaking of industrial applications, graph theory is quite popular among computer people. For example Facebook's EdgeRank determines who sees what in the news feed. And neo4j is a popular (or at least well-advertised) graph database, with gremlin a graph-traversal language. LinkedIn has a team devoted to "social network analysis" and graphs are hot as well in "complexity theory" including models of the brain. For example http://video.neo4j.org/RHqy/the-pathology-of-graph-databases-by-marko-a-rodriguez/ gave me a flavour of how the web-programming crowd sees graphs.

I don't know if any of the above interests you but at least outside of pure mathematics I believe quivers-as-directed-graphs have practical applications.

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I think the question really is about representations of quivers, rather than quivers-as-directed-graphs. –  darij grinberg Feb 27 '13 at 5:20

As Pace Nielsen already posted, the strength of quiver theory is to provide easy examples and counterexamples.

The first applications are of course inside representation theory and ring theory, because Gabriel's Theorem states, that if you have a property of a finite dimensional algebra over an algebraically closed field that can be detected in the module category, then it suffices to look at path algebras of quivers (with relations). For example for proving that an algebra is wild, it suffices to find a subquiver (with relations) that is known to be wild; and there are several lists of such quivers. This is useful in representation theory of Lie algebras and finite groups.

There is the connection with Lie theory (and other things that can be classified via Dynkin diagrams) via the Hall algebra.

Cluster theory, which is an advanced topic in representation theory of quivers, has applications in geometry.

If you are given an algebra, I think it is a natural question is, whether it is possible to classify all the indecomposable representations. If this is possible you can work on a parametrization and understand better the module category by working with the classification. The tame-wild dichotomy helps you there, it answers the question if it is possible. If an algebra is wild, it is not possible to classify all representations. You have to ask other questions.

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As was mentioned above, many moduli spaces have a quiver description; one of the most famous example is given by Nakajima quiver varieties, which are defined for any quiver (and they serve as the main example of symplectic complex varieties which are resolutions of an affine variety), but when the quiver is the affine quiver of ADE type, they describe moduli spaces of torsion free sheaves on the quotients ${\mathbb C}^2/\Gamma$ where $\Gamma$ is a finite subgroup of $SL(2)$ (these are also known as ALE spaces). These moduli spaces are very important in many places in mathematics and physics (gauge theory) and quiver description is very useful when you want to tackle some explicit problems related to them.

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If you are interested in representation theory of finite dimensional algebras (including group algebras and their blocks—and everyone is interested in representations of groups, even if they don't know it), then considering quivers (and bound quivers) is a natural thing to do: all algebras (up to the appropriate equivalence relation relevant in the context of representation theory) are quotients of paths algebras. An immense number of non-finite dimensional algebras are also quotients of path algebras, too.

So whatever motivation you have to be interested in representations of finite dimensional algebras immediately carries over to quivers and their algebras, plus the obvious plus that it becomes extraordinarily easy to build up examples.

Later. As to concrete motivations:

  • A classical example is the Kronecker-Wierstrass classification of the indecomposable representations of the quiver $\bullet\rightrightarrows\bullet$ , motivated by the conjugation classification of certain systems of ODEs. There is another example, from control theory, in Gabriel-Roiter's book. I am pretty sure these two examples are real, in that the non-representation-theoretic problem came before the representationists took over.

  • Similarly, the whole cluster explosion of the last 10 years should be a nice example of representation theory of quivers and friends and the methods involved in it helping understand (and solve, in many cases) problems exterior to the theory. Of course, here the source of the problems is also of representation-theoretic nature ---Lie theory--- but in a rather palpable sense this is a quite different part of the theory.

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In the framework of Algebraic Geometry there is the equivalence (introduced by Bondal, Kapranov, and Hille) between the category of homogeneous bundles $E$ on any Hermitian symmetric variety $X=G/P$ of ADE-type and the category of representations of a certain quiver $\mathcal{Q}_X$ with relations.

This allows in some cases the computation of the cohomology groups of such bundles.

See for instance the paper by Ottaviani and Rubei Quivers and the cohomology of homogeneous vector bundles, Duke Mathematical Journal Volume 132, Number 3 (2006), 459-508, and the references given there.

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Since I believe Victor did some work in semigroup theory, these examples may be interesting to him. If $S$ is a finite monoid such that each maximal subgroup is abelian and the structure matrix of each regular $\mathcal J$-class contains only $1$s (i.e. products of $\mathcal J$-equivalent idempotents are idempotents) then for any field $\Bbbk$ the monoid algebra $\Bbbk M$ is basic and so is the quotient of the path algebra of its quiver modulo an admissible ideal. See the paper http://arxiv.org/abs/1101.0416 for how to compute this quiver. Thus one can really use quiver theory to understand your semigroup's representation theory.

For instance, the quiver of a 2x2 rectangular band with adjoined identity consists of two vertices $1,2$ and an edge $x\colon 1\to 2$ and an edge $y\colon 2\to 1$. The admissible ideal is generated by the relation $xy=0$. Thus a representation of a $2\times 2$ rectangular band amounts to taking to vector spaces $V,W$ and linear maps $A\colon V\to W$ and $B\colon W\to V$ such that $AB=0$. One can also prove this directly.

If you like categories, viewed as an algebraic structure, then the path algebra is just the "category algebra" of the free category on the quiver. Incidence algebras are category algebras of posets and so one can view all of this as studying representation theory of category algebras modulo relations.

See the following survey for applications in theoretical physics: arXiv:math/0505082

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Thanks for the link to the paper! I'll read it and contact you back soon. –  Victor Oct 13 '11 at 9:05
    
Victor, Ken Brown used quivers to prove that every submodule of a projective modules is free for the algebra of a free left regular band on a finite set. –  Benjamin Steinberg Oct 13 '11 at 16:08

The formation of quiver algebras is useful in ring theory in attempts to construct examples/counter-examples. One can often classify when a quiver algebra has some property in terms of some intrinsic property of the quiver itself, and then finding counter-examples boils down to forming a quiver with a certain easy-to-see property.

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A surprising and nice example was an answer to a question in ring theory - construct rings which are not isomorphic to their opposites: mathoverflow.net/questions/64370/… –  Amritanshu Prasad Feb 27 '13 at 6:52

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