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The title says mostly everything.

Suppose we have a quiver, maybe with relations and cycles. Is it known when the path algebra modulo relations is hereditary. Especially in the case that the path algebra is infinite dimensional.

I would appreciate any reference.

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If an associative algebra $A$ is finite dimensional, basic, connected and defined over an algebraically closed field $K$, then $A$ is hereditary if and only if it is isomorphic to the path algebra of a finite, connected and acyclic quiver (see Thm. VII.1.7 of Assem-Simson-Skowroński). However, this still leaves the more interesting cases of your question unanswered... – Matthew Pressland Aug 29 '13 at 15:27

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