In Category $\mathcal O$: Quivers and endomorphism rings of projectives by Stroppel, 5.11 (pages 328–329), this quiver is described as $\rm{Rep}(\mathcal O_0(\mathfrak{sl}_2))$. The representation theory is well-understood (for example see Humphrey's book on category $\mathcal O$).
Nicolas Hemelsoet
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