Abstract
In this article, we give a definition and a classification of ‘higher’ simple-minded systems in triangulated categories generated by spherical objects with negative Calabi–Yau dimension. We also study mutations of this class of objects and that of ‘higher’ Hom-configurations and Riedtmann configurations. This gives an explicit analogue of the ‘nice’ mutation theory exhibited in cluster-tilting theory.
Funding source: Fundação para a Ciência e a Tecnologia
Award Identifier / Grant number: SFRH/BPD/90538/2012
Funding statement: The author would like to thank Fundação para a Ciência e Tecnologia, for their financial support through Grant SFRH/BPD/90538/2012.
References
[1] T. Aihara and O. Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. (2) 85 (2012), no. 2, 633–668. 10.1112/jlms/jdr055Search in Google Scholar
[2] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), 317–345. 10.4007/annals.2007.166.317Search in Google Scholar
[3] A. B. Buan, R. J. Marsh, M. Reineke, I. Reiten and G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572–618. 10.1016/j.aim.2005.06.003Search in Google Scholar
[4] A. B. Buan, I. Reiten and H. Thomas, Three kinds of mutations, J. Algebra 339 (2011), no. 1, 97–113. 10.1016/j.jalgebra.2011.04.030Search in Google Scholar
[5] R. Coelho Simões, Hom-configurations and noncrossing partitions, J. Algebraic Combin. 35 (2012), no. 2, 311–343. 10.1007/s10801-011-0305-5Search in Google Scholar
[6] R. Coelho Simões, Maximal rigid objects as noncrossing bipartite graphs, Algebr. Represent. Theory 16 (2013), 1243–1272. 10.1007/s10468-012-9355-1Search in Google Scholar
[7] R. Coelho Simões, Hom-configurations in triangulated categories generated by spherical objects, J. Pure Appl. Algebra 219 (2015), 3322–3336. 10.1016/j.jpaa.2014.10.017Search in Google Scholar
[8] R. Coelho Simões and D. Pauksztello, Torsion pairs in a triangulated category generated by a spherical object, J. Algebra 448 (2016), 1–47. 10.1016/j.jalgebra.2015.09.011Search in Google Scholar
[9] A. Dugas, Torsion pairs and simple-minded systems in triangulated categories, Appl. Categ. Structures 23 (2015), no. 3, 507–526. 10.1007/s10485-014-9365-8Search in Google Scholar
[10] C. Fu and D. Yang, The Ringel–Hall Lie algebra of a spherical object, J. Lond. Math. Soc. (2) 85 (2012), no. 12, 511–533. 10.1112/jlms/jdr064Search in Google Scholar
[11] T. Holm and P. Jørgensen, On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon, Math. Z. 270 (2012), no. 1–2, 277–295. 10.1007/s00209-010-0797-zSearch in Google Scholar
[12] T. Holm and P. Jørgensen, Cluster-tilting vs. weak cluster-tilting in Dykin type A infinity, Forum Math. 27 (2015), 1117–1137. 10.1515/forum-2012-0093Search in Google Scholar
[13] T. Holm, P. Jørgensen and D. Yang, Sparseness of t-structures and negative Calabi–Yau dimension in triangulated categories generated by a spherical object, Bull. Lond. Math. Soc. 45 (2013), 120–130. 10.1112/blms/bds072Search in Google Scholar
[14] P. Jørgensen, Auslander–Reiten theory over topological spaces, Comment. Math. Helv. 79 (2004), 160–182. 10.1007/s00014-001-0795-4Search in Google Scholar
[15] B. Keller, D. Yang and G. Zhou, The Hall algebra of a spherical object, J. Lond. Math. Soc. (2) 80 (2009), 771–784. 10.1112/jlms/jdp054Search in Google Scholar
[16] A. King and Y. Qiu, Exchange graphs of acyclic Calabi–Yau categories, Adv. Math. 285 (2015), 1106–1154. 10.1016/j.aim.2015.08.017Search in Google Scholar
[17] S. Koenig and Y. Liu, Simple-minded systems in stable module categories, Q. J. Math. 63 (2012), no. 3, 653–674. 10.1093/qmath/har009Search in Google Scholar
[18] S. Koenig and D. Yang, Silting objects, simple-minded collections, t-structures, co-t-structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403–438. 10.4171/dm/451Search in Google Scholar
[19]
P. Ng,
A characterization of torsion theories in the cluster category of type
[20] Z. Pogorzały, Algebras stably equivalent to self-injective special biserial algebras, Comm. Algebra 22 (1994), no. 4, 1127–1160. 10.1080/00927879408824898Search in Google Scholar
[21]
C. Riedtmann,
Representation-finite selfinjective algebras of class
[22] Y. Zhou and B. Zhu, Mutation of torsion pairs in triangulated categories and its geometric realization, preprint (2011), https://arxiv.org/abs/1105.3521v1. 10.1007/s10468-017-9740-xSearch in Google Scholar
[23] B. Zhu, Generalized cluster complexes via quiver representations, J. Algebraic Combin. 27 (2008), no. 1, 35–54. 10.1007/s10801-007-0074-3Search in Google Scholar
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