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