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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?

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  • $\begingroup$ As you can see from the variety of answers, this algebra arises naturally in many contexts — it is a Nakayama algebra, a gentle algebra, an extended Khovanov arc algebra, etc. Are there also natural higher-dimensional analogues in your setting? $\endgroup$
    – Severin Barmeier
    Commented Jul 5, 2022 at 6:10
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    $\begingroup$ The short answer is yes (although these are not representations over a field). Informally, you get the quiver I described by considering integral homology and mod p homology at the same time, the two maps between them being the natural projection and the Bockstein. There are natural "higher height" analogues of integral homology, namely truncated Brown-Peterson spectra, whose coefficient ring is now n-dimensional, with a distinguished regular sequence of generators. Then one gets the n-length analogue of this quiver by considering homology with coefficients in the corresponding quotient. $\endgroup$
    – Piotr Pstrągowski
    Commented Jul 10, 2022 at 12:22

3 Answers 3

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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.

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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 $\mathcal O$).

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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.

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