The path algebra of this quiver is also the Yoneda algebra $$\mathrm{Ext}^\bullet(\mathcal O_{\mathbb P_1}\oplus \mathcal O_0,\mathcal O_{\mathbb P_1}\oplus \mathcal O_0)$$ of the direct sum of the structure sheaf and a skyscraper sheaf on the projective line, at least when ignoring the grading. This suggests that your quiver with relations is derived equivalent to the Kronecker quiver (two vertices and two parallel arrows), but with the degrees of the arrows differing by 1. A nice way to see this geometrically is via Fukaya categories of surfaces, see for example Section 3.4 of my paper arXiv:1409.8611 with Katzarkov and Kontsevich.

The path algebra of this quiver is also the Yoneda algebra $$\mathrm{Ext}^\bullet(\mathcal O_{\mathbb P_1}\oplus \mathcal O_0,\mathcal O_{\mathbb P_1}\oplus \mathcal O_0)$$ of the direct sum of the structure sheaf and a skyscraper sheaf on the projective line, at least when ignoring the grading. This suggests that your quiver with relations is derived equivalent to the Kronecker quiver (two vertices and two parallel arrows), but with the degrees of the arrows differing by 1. A nice way to see this geometrically is via Fukaya categories of surfaces, see for example Section 3.4 of my paper Flat surfaces and stability structures with Katzarkov and Kontsevich.

The path algebra of this quiver is also the Yoneda algebra $$\mathrm{Ext}^\bullet(\mathcal O_{\mathbb P_1}\oplus \mathcal O_0,\mathcal O_{\mathbb P_1}\oplus \mathcal O_0)$$ of the direct sum of the structure sheaf and a skyscraper sheaf on the projective line, at least when ignoring the grading. This suggests that your quiver with relations is derived equivalent to the Dynkin quiver (two vertices and two parallel arrows), but with the degrees of the arrows differing by 1. A nice way to see this geometrically is via Fukaya categories of surfaces, see for example Section 3.4 of my paper arXiv:1409.8611 with Katzarkov and Kontsevich.

The path algebra of this quiver is also the Yoneda algebra $$\mathrm{Ext}^\bullet(\mathcal O_{\mathbb P_1}\oplus \mathcal O_0,\mathcal O_{\mathbb P_1}\oplus \mathcal O_0)$$ of the direct sum of the structure sheaf and a skyscraper sheaf on the projective line, at least when ignoring the grading. This suggests that your quiver with relations is derived equivalent to the Kronecker quiver (two vertices and two parallel arrows), but with the degrees of the arrows differing by 1. A nice way to see this geometrically is via Fukaya categories of surfaces, see for example Section 3.4 of my paper arXiv:1409.8611 with Katzarkov and Kontsevich.