Abstract
Triangulated categories coming from cyclic posets were originally introduced by the authors in a previous paper as a generalization of the constructions of various triangulated categories with cluster structures. We give an overview, and then analyze “triangulation clusters” which are those corresponding to topological triangulations of the 2-disk. Locally finite nontriangulation clusters give topological triangulations of the “cactus space” associated to the “cactus cyclic poset”.
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References
Amiot C. On generalized cluster categories. In: Representations of Algebras and Related Topics. EMS Series of Congress Reports. Zürich: Eur Math Soc, 2011, 1–53
Buan A B, Iyama O, Reiten I, et al. Cluster structures for 2-Calabi-Yau categories and unipotent groups. Compos Math, 2009, 145: 1035–1079
Buan A B, Iyama O, Reiten I, et al. Mutation of cluster-tilting objects and potentials. Amer J Math, 2011, 133: 835–887
Buan A B, Marsh R J, Reiten I, et al. Tilting theory and cluster combinatorics. Adv Math, 2006, 204: 572–618
Caldero P, Chapoton F, Schiffler R. Quivers with relations arising from clusters (A n case). Trans Amer Math Soc, 2006, 358: 1347–1364
Drinfeld V. On the notion of geometric realization. Mosc Math J, 2004, 4: 619–626
Gratz S, Holm T, Jørgensen P. Cluster tilting subcategories and torsion pairs in Igusa-Todorov cluster categories of Dynkin type A ∞. Math Z, 2019, in press
Happel D. Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Mathematical Society Lecture Note Seres, vol. 119. Cambridge: Cambridge University Press, 1988
Holm T, Jørgensen P. On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon. Math Z, 2012, 270: 277–295
Holm T, Jørgensen P. Cluster tilting vs. weak cluster tilting in Dynkin type A infinity. Forum Math, 2015, 27: 1117–1137
Igusa K, Todorov G. Continuous Frobenius categories. In: Algebras, Quivers and Representations. Berlin-Heidelberg: Springer-Verlag, 2013, 115–143
Igusa K, Todorov G. Cluster categories coming from cyclic posets. Comm Algebra, 2015, 43: 4367–4402
Igusa K, Todorov G. Continuous cluster categories I. Algebr Represent Theory, 2015, 18: 65–101
Jørgensen P, Yakimov M. c-vectors of 2-Calabi-Yau categories and Borel subalgebras of sl ∞. ArXiv:1710.06492, 2017
Liu S, Paquette C. Cluster categories of type A ∞∞ and triangulations of the infinite strip. Math Z, 2017, 286: 197–222
Megiddo N. Partial and complete cyclic orders. Bull Amer Math Soc, 1976, 82: 274–276
Shi X P. On flatness properties of cyclic S-posets. Semigroup Forum, 2008, 77: 248–266
van Roosmalen A-C. Hereditary uniserial categories with Serre duality. Algebr Represent Theory, 2012, 15: 1291–1322
Zhang X, Zhang W L, Knauer U. A characterization of a pomonoid S-posets are regular injective. Categor Gen Algebr Structures Appl, 2013, 1: 103–117
Acknowledgements
The authors thank Professors Bin Zhu, Fang Li and Zongzhu Lin for their hospitality at Tsinghua University and at the Chern Institute of Mathematics during the Workshop on Cluster Algebras and Related Topics, July 10–13, 2017. A series of lectures on this topic were given by the authors, which motivated the beginning of this paper and subsequent work for the rest of the paper.
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Igusa, K., Todorov, G. Cyclic posets and triangulation clusters. Sci. China Math. 62, 1289–1316 (2019). https://doi.org/10.1007/s11425-018-9507-3
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DOI: https://doi.org/10.1007/s11425-018-9507-3