In [Category $\mathcal O$: Quivers and endomorphism rings of projectives][1] 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 $\mathcal O$). 

  [1]: http://www.math.uni-bonn.de/ag/stroppel/Quivers.pdf