Quiver with two objects and two arrows composing to zero In the description of the integral Adams spectral sequence, representations of the following quiver (with relations) arise naturally:

*

*We have two objects $A, B$,

*we have two arrows $\pi: A \rightarrow B$ and $\delta: B \rightarrow A$ and

*we have a single relation $\delta \circ \pi = 0$.

Does this quiver arise naturally in other contexts? What is known about its representation theory?
 A: 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.
A: This is a Nakayama algebra with Kupisch series [2,3] and the representation theory of Nakayama algebras (a Nakayama algebra is a quiver algebra with admissible relations whose quiver is a linear oriented line or a cycle) is completely understood in nearly all details. (see for example Farnsteiner - Nakayama algebras: Kupisch series and Morita type).
It has 5 indecomposable modules and global dimension 2 and appears very often as a block of algebras in geometric representation theory for example for blocks of category $\mathcal{O}$ and it is also the simplest non-trivial representaiton-finite block of a Schur algebra. See also section 6.1 of Chan and Marczinzik - On representation-finite gendo-symmetric biserial algebras for this algebra and its "bigger brothers" and where they appear.
A: 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 $\mathcal O$).
