In [CATEGORY O: QUIVERS AND ENDOMORPHISM RINGS OF PROJECTIVES][1] by Stroppel,5.11 (page 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