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?



Quiver with two objects and two arrows composing to zero
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](https://www.math.uni-bielefeld.de/%7Esek/select/Nakayama-alg1.pdf)).
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](https://arxiv.org/abs/1607.05965) for this algebra and its "bigger brothers" and where they appear.


