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I work in planar algebras and subfactors, where the idea of path algebras on a graph (alternately known as graph algebras, graph planar algebras, etc.) is quite useful. The particular result I'm thinking of is a forthcoming result of Jones and some others; it says that any subfactor planar algebra can be found inside the planar algebra of its principal graph. If you're not into subfactors/planar algebras, the importance of this result is that it says you know a concrete place to begin looking for a particular abstract object.

At Birge Huisgen-Zimmermann's talk on quivers at the AMS meeting at Riverside last weekend, I encountered what seemed to be a similar result: Gabriel's theorem, which says that any finite-dimensional algebra is equivalent (Morita equivalent I think?) to a path algebra modulo some relations. (As far as I can tell, "quiver" is a fancy word for a directed finite graph). I also know, though I don't know why, that path algebras are used in particular constructions in C*-algebras. This got me thinking:

  1. What are some other places that path algebras appear, and what are they used for?

  2. Why is this idea so useful in these different fields? Is it simply that path algebras are a convenient place to do calculations? Or is there some philosophical reason path algebras are important?

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As far as I can tell, "quiver" is a fancy word for a directed finite graph

Yes. It doesn't even have to be finite.

Or is there some philosophical reason path algebras are important?

A huge application of path algebras lately is the path algebra of a quiver of Dynkin type. Following the ideas of Lusztig and Ringel, the representation varieties of these quivers are a main method to categorify quantum groups. A big share of the interest in quivers is either this specific purpose, or generalizations of features of this application. Lusztig's papers on this and his book are a big revelation.

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You also have the rather new field of Leavitt Path Algebras (in which I happen to be working right now), where you take a field $K$ and a directed graph $E$, generate its extended graph $E'$ (add to $E$ its own edges reversed, denoted as $e^*$ for every edge $e$), and compute the Leavitt path algebra of $E$, $L(E)$, as the path algebra $KE'$ modulo some relations called the Cuntz-Krieger relations, inherited from the $C^*$-algebras setting, concretely:

(CK1) $e^* f=\delta_{ef}$ for any two edges $e,f$ of $E'$.

(CK2) $\sum_{e\in s^{-1}(v)}ee^* = v$, for $v$ a vertex which emits a nonzero finite number of edges, and $s^{-1}(v)$ the set of those edges.

(One can look at (CK1) and (CK2) as an abstract generalization of the product of matrix units).

These associative algebras provide us simultaneously with a purely algebraic analog of $C^*$-algebras of graph and a generalization of the Leavitt algebras (some associative algebras which do not satisfy the IBN property).

The full matrix rings over $K$ of order $n$ then arise as the Leavitt path algebras of the graphs with $n$ (consecutive) vertices and $n-1$ arrows, one between every pair of consecutive vertices.

Another simple example of Leavitt path algebra is the ring of Laurent polynomials over $K$, $K[x,x^{-1}]$, which appears associated to the graph with one vertex and a single loop.

The theory of LPAs is useful, and even beautiful, because:

  • They provide simple, visually attractive representations of well-known algebras.

  • They allow us to look at their algebraic properties by means of the combinatorial properties of their associated graphs. This happens to equip us with some rather powerful tools.

  • Conversely, they also enable "algebraic engineering", since they give us a straightforward, visual way to construct new algebras, customized with any algebraic or ring-theoretic properties we may desire. For example, we can show an algebra generated by five elements such that it is exchange but not purely innitely simple, by constructing a particular (small) graph with some (easy) graph-theoretic features.

Some references:

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I once mentioned in a talk (to a group of algebraic combinatorialists) that "A quiver is just a directed graph". An audience-member stuck up his hand to say "A quiver is a directed graph with pretensions."

Representation theory of path algebras provides a very useful way to approach cluster algebras, acyclic cluster algebras in particular. Roughly speaking, for acyclic cluster algebras, the cluster variables correspond to the indecomposable modules E which satisfy Ext1(E,E)=0. This answer is connected to Greg's answer (via the connection of cluster algebras to Lusztig's canonical bases).

For me, path algebras of quivers have provided a lovely setting in which to learn about homological algebra, because the objects involved are simple enough that you can understand them quite concretely. (For example: an indecomposable object in the bounded derived category of a path algebra of a quiver without oriented cycles, is a chain which is non-zero in only one degree. Nonetheless, the category still has a lot of non-trivial (triangulated) structure.)

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Can't we just agree that a quiver is the free category on a directed graph? –  Qiaochu Yuan Nov 15 '09 at 16:09
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I don't think the existence of synonyms is a big problem. Words with multiple conflicting definitions, on the other hand, are annoying. Your proposal fixes a synonym by introducing a conflict among definitions of "quiver". Even if everyone in the world made the change today, there would still be over thirty years of representation theory literature written using the old meaning of quiver. So I don't think your proposed change is an improvement. –  Hugh Thomas Nov 15 '09 at 17:10
    
I guess this is an issue for the nLab: ncatlab.org/nlab/show/quiver –  Qiaochu Yuan Nov 15 '09 at 18:34
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Actually, the nLab seems to have given up on redefining "quiver". –  Hugh Thomas Nov 16 '09 at 3:20
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They are also of interest in non-commutative geometry. There is this result "If formally smooth algebras are the non-commutative analogon of manifolds then path algebras of quivers are non-commutative version of affine spaces!" See here:

http://www.neverendingbooks.org/NEBPDFS/34.pdf

and here

http://arxiv.org/abs/math/0406618

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I want to say something like, "quivers (with relations) are finitely-generated unital algebras over $k^{\oplus n}$ ", but it's fairly vacuous, and I'm about to head off to the airport, so I don't have time to really think it through.

In physics, that is pretty much the situation once you throw in the word graded (it's easy to prove a version of Gabriel's theorem in this case). Quivers arise when you have a finite set of objects in a (pretriangulated dg-/A${}_\infty$/stable infinity/triangulated/whatever) category of D-branes, and the endomorphism algebra of the sum of these objects is presented as the path algebra of a quiver. If those objects form a nice generator, you get the usual equivalence between the original category and the derived category of quiver reps. The simple reps corresponding to the nodes of the quiver are called "fractional branes" in the physics literature, and the arrows in the quiver correspond to massless string states in the physics (as they are given by Ext^1s b/w the simple reps.)

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One idea comes to mind for number two: Bases! If we have a unital inclusion of finite dimensional semi-simple complex algebras, the isomorphisms with the path algebras fix bases and make our lives easier.

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I'm not sure direct sums of matrix algebras (which, by Wedderburn, is what you're talking about in your answer) are particularly interesting examples of path algebras... –  Yemon Choi Nov 14 '09 at 20:55
    
But I'm generally suspicious of bases ... –  Emily Peters Nov 14 '09 at 21:51
    
@Yemon - they are very interesting indeed. If you are interested in inclusions of finite von Neumann algebras, they are instrumental in calculating the basic construction, conditional expectations, Pimsner-Popa bases, etc. See Jones and Sunder's book Introduction to Subfactors or Goodman, de la Hapre, and Jones' book Coxeter Graphs and Towers of Algebras for all sorts of subfactor applications, including commuting squares, Ocneanu compactness, and more. –  Dave Penneys Nov 14 '09 at 21:59
    
@Dave: I think I am misunderstanding what "path algebra" means, so I need to go and look it up. But you are right (and I was overly glib) in that tracking the inclusions of subalgebras is interesting (cf. Bratteli diagrams) –  Yemon Choi Nov 14 '09 at 22:54
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I really cannot see how this is an answer to the question. It is rather rare that a path algebra be semisimple: as soon as there is an arrow in the quiver, $Ext^1$ is non-zero... Maybe there are two different meanings of "path algebra" being mixed, but in the context of Gabriel's theorem, a path algebra is isomorphic to a semisimple algebra iff there are no arrows in the quiver. –  Mariano Suárez-Alvarez Nov 14 '09 at 23:03
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There are applications to physics among these are the quiver gauge theories. Here is one paper on quiver gauge theories:

http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.4259v2.pdf

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