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?



ubiquity, importance of path algebras
*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.


