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Decorated marked surfaces (Part B): Topological realizations

Yu Qiu Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

Representation Theory mathscidoc:2206.30008

Mathematische Zeitschrift, 288, 39-53, 2017.3
We study categories associated to a decorated marked surface S_△, which is obtained from an unpunctured marked surface S by adding a set of decorating points. For any triangulation T of S_△, let Γ_T be the associated Ginzburg dg algebra. We show that there is a bijection between reachable open arcs in S_△ and the reachable rigid indecomposables in the perfect derived category perΓ_T. This is the dual of the bijection, between simple closed arcs in S_△ and reachable spherical objects in the 3-Calabi-Yau category D_{fd}(ΓT), constructed in the prequel (Qiu in Math Ann 365:595–633, 2016). Moreover, we show that Amiot’s quotient perΓ_T/D_{fd}(ΓT) that defines the generalized cluster categories corresponds to the forgetful map S_△→S (forgetting the decorating points) in a suitable sense.
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@inproceedings{yu2017decorated,
  title={Decorated marked surfaces (Part B): Topological realizations},
  author={Yu Qiu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626164853417092466},
  booktitle={Mathematische Zeitschrift},
  volume={288},
  pages={39-53},
  year={2017},
}
Yu Qiu. Decorated marked surfaces (Part B): Topological realizations. 2017. Vol. 288. In Mathematische Zeitschrift. pp.39-53. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626164853417092466.
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