Quiver representation theory MOC

Acyclic quiver

Let Γ be a quiver. An oriented cycle is a nontrivial path 𝑝 with dom⁡𝑝 =cod⁡𝑝, i.e. a non-identity endomorphism in the free category Γ――. The quiver Γ is called acyclic iff it has no oriented cycles.

Properties

• Γ is acyclic iff Mor⁡Γ―― is finite, or equivalently the path algebra is finite-dimensional.

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