In CATEGORY OCategory $\mathcal O$: QUIVERS AND ENDOMORPHISM RINGS OF PROJECTIVESQuivers and endomorphism rings of projectives by Stroppel,5 5.11 (page 328-329pages 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 $\mathcal O$).