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Continuous cluster categories of type D

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Kiyoshi Igusa, Gordana Todorov

Representation Theory · Submitted 28 Sep 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We construct continuous Frobenius categories of type $D$.

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arXiv ID1309.7409 (version 1)
Pith IDCJBPUKUN, Pith's permanent record of this paper and its review
CitationGordana Todorov, Kiyoshi Igusa. "Continuous cluster categories of type D." arXiv:1309.7409 (2013). https://pith.science/paper/CJBPUKUN
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,131 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.73 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We construct continuous Frobenius categories of type $D$. The stable categories of these Frobenius categories are cluster categories which contain the standard cluster categories of type $D_n$. When $n=\infty$, maximal compatible sets of indecomposable objects are laminations of the punctured disk. Discrete laminations are clusters. This new construction is topological and it also gives an algebraic interpretation of the "tagged arcs" which occur in Schiffler's geometric description of clusters of type $D_n$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We construct continuous Frobenius categories of type $D$. contribution 1.00 0.97 0.47 asserted 1.00
s1 The stable categories of these Frobenius categories are cluster categories which contain the standard cluster categories of type $D_n$. result 0.93 0.80 0.34 asserted 1.00
s2 When $n=\infty$, maximal compatible sets of indecomposable objects are laminations of the punctured disk. result 0.97 0.76 0.18 asserted 1.00
s3 Discrete laminations are clusters. result 0.91 0.64 0.12 asserted 1.00
s4 This new construction is topological and it also gives an algebraic interpretation of the "tagged arcs" which occur in Schiffler's geometric description of clusters of type $D_n$. interpretation 0.79 0.90 0.21 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We construct continuous Frobenius categories of type $D$. The stable categories of these Frobenius categories are cluster categories which contain the standard cluster categories of type $D_n$. When $n=\infty$, maximal compatible sets of indecomposable objects are laminations of the punctured disk. Discrete laminations are clusters. This new construction is topological and it also gives an algebraic interpretation of the "tagged arcs" which occur in Schiffler's geometric description of clusters of type $D_n$.

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4 arXiv papers cite this in 8 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 5 sentences without a confident label

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Cites it as background 2 papers

What it cites · 7 sentences without a confident label

Uses the method or tool of 1 paper
  • We also show that, outside the exceptional mutation case $2(a)↔2(b)$ where $$ has exactly two singular objects of which $T$, the object being mutated, is one, the Octahedral Axiom is used, as in [ref] .

    Continuous cluster categories I · Cluster structure · 0.87

Cites as background 2 papers
  • Recall [ref] that the indecomposable objects are indexed by ordered pairs of points on the circle: $E(x,y)$ and that reversing the order gives an isomorphic but not equal object: $E(x,y)≅E(y,x+2π)$ We also use the notation $M(x,y)=E(x,y+π)$ so that $M(x,x)$ is a diameter of the circle.

    Continuous Frobenius categories · Clusters of continuous type $D$ · 0.86

  • (See [ref] for the general theory.

    Cluster categories coming from cyclic posets · Continuous Frobenius category · 0.97

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