# When are path algebras of quivers hereditary.

Suppose we have a finite quiver with relations, possibly with oriented cycles. Is it known when the path algebra of this quiver (with relations) is hereditary?

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Let $A = kQ/I$ where $Q$ is a quiver and $I$ are the relations. Then $A$ is hereditary if and only if $Q$ is finite, connected, and acyclic, and $I = (0)$. This is essentially Theorem 1.7 in Chapter VII of Assem, Simson, Skowronski - Elements of the Representation Theory of Associative Algebras.
That's under the assumption that $A$ is finite dimensional. For the infinite-dimensional case a counterexample is the first Weyl algebra $A=\mathbb{C}\langle x,y\rangle/(xy-yx-1)$, which is hereditary, see e.g. [Lam: Lectures on Modules and Rings (2.32 (h))]. – Julian Kuelshammer Aug 20 '13 at 21:34