Skip to main content
×
Home
    • Aa
    • Aa
  • Access
  • Cited by 47
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Buan, Aslak Bakke Palu, Yann and Reiten, Idun 2016. Algebras of finite representation type arising from maximal rigid objects. Journal of Algebra, Vol. 446, p. 426.


    Chang, Wen and Zhu, Bin 2016. On rooted cluster morphisms and cluster structures in 2-Calabi–Yau triangulated categories. Journal of Algebra, Vol. 458, p. 387.


    Gratz, Sira 2016. Mutation of Torsion Pairs in Cluster Categories of Dynkin Type D. Applied Categorical Structures, Vol. 24, Issue. 1, p. 79.


    Guo, Jin Yun 2016. On n-translation algebras. Journal of Algebra, Vol. 453, p. 400.


    Guo, Jin Yun and Luo, Deren 2016. On n-cubic Pyramid Algebras. Algebras and Representation Theory, Vol. 19, Issue. 4, p. 991.


    Holm, Thorsten and Jørgensen, Peter 2016. Generalised friezes and a modified Caldero–Chapoton map depending on a rigid object, II. Bulletin des Sciences Mathématiques, Vol. 140, Issue. 4, p. 112.


    Jensen, Bernt Tore King, Alastair D. and Su, Xiuping 2016. A categorification of Grassmannian cluster algebras. Proceedings of the London Mathematical Society, Vol. 113, Issue. 2, p. 185.


    Kalck, Martin and Yang, Dong 2016. Relative singularity categories I: Auslander resolutions. Advances in Mathematics, Vol. 301, p. 973.


    Leclerc, B. 2016. Cluster structures on strata of flag varieties. Advances in Mathematics,


    Lu, Ming 2016. Singularity Categories of some 2-CY-tilted Algebras. Algebras and Representation Theory,


    Mizuno, Yuya and Yamaura, Kota 2016. Higher APR tilting preserves n-representation infiniteness. Journal of Algebra, Vol. 447, p. 56.


    Chang, Wen Zhang, Jie and Zhu, Bin 2015. On support τ-tilting modules over endomorphism algebras of rigid objects. Acta Mathematica Sinica, English Series, Vol. 31, Issue. 9, p. 1508.


    Igusa, Kiyoshi and Todorov, Gordana 2015. Cluster Categories Coming from Cyclic Posets. Communications in Algebra, Vol. 43, Issue. 10, p. 4367.


    Igusa, Kiyoshi and Todorov, Gordana 2015. Continuous Cluster Categories I. Algebras and Representation Theory, Vol. 18, Issue. 1, p. 65.


    Kalck, Martin Iyama, Osamu Wemyss, Michael and Yang, Dong 2015. Frobenius categories, Gorenstein algebras and rational surface singularities. Compositio Mathematica, Vol. 151, Issue. 03, p. 502.


    Oppermann, Steffen Reiten, Idun and Thomas, Hugh 2015. Quotient closed subcategories of quiver representations. Compositio Mathematica, Vol. 151, Issue. 03, p. 568.


    Qiu, Yu 2015. C-sortable words as green mutation sequences. Proceedings of the London Mathematical Society, p. pdv046.


    Xu, Jinde and Ouyang, Baiyu 2015. Maximal rigid objects without loops in connected 2-CY categories are cluster-tilting objects. Journal of Algebra and Its Applications, Vol. 14, Issue. 05, p. 1550071.


    Baumann, Pierre Kamnitzer, Joel and Tingley, Peter 2014. Affine Mirković-Vilonen polytopes. Publications mathématiques de l'IHÉS, Vol. 120, Issue. 1, p. 113.


    Iyama, Osamu and Reiten, Idun 2014. Introduction to τ-tilting theory. Proceedings of the National Academy of Sciences, Vol. 111, Issue. 27, p. 9704.


    ×

Cluster structures for 2-Calabi–Yau categories and unipotent groups

  • A. B. Buan (a1), O. Iyama (a2), I. Reiten (a3) and J. Scott (a4)
  • DOI: http://dx.doi.org/10.1112/S0010437X09003960
  • Published online: 01 July 2009
Abstract
Abstract

We investigate cluster-tilting objects (and subcategories) in triangulated 2-Calabi–Yau and related categories. In particular, we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi–Yau categories contains, as special cases, the cluster categories and the stable categories of preprojective algebras of Dynkin graphs. For these 2-Calabi–Yau categories, we construct cluster-tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We discuss connections with cluster algebras and subcluster algebras related to unipotent groups, in both the Dynkin and non-Dynkin cases.

    • Send article to Kindle

      To send this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Cluster structures for 2-Calabi–Yau categories and unipotent groups
      Your Kindle email address
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      Cluster structures for 2-Calabi–Yau categories and unipotent groups
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      Cluster structures for 2-Calabi–Yau categories and unipotent groups
      Available formats
      ×
Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2]I. Assem , T. Brüstle and R. Schiffler , Cluster-tilted algebras as trivial extensions, Bull. London Math. Soc. 40 (2008), 151162.

[4]M. Auslander , Coherent functors, in Proceedings of the conference on categorical algebra (La Jolla, CA, 1965) (Springer, Berlin, 1966), 189231.

[6]M. Auslander and I. Reiten , Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), 111152.

[9]M. Auslander and S. O. Smalø , Preprojective modules over Artin algebras, J. Algebra 66 (1980), 61122.

[10]D. Baer , W. Geigle and H. Lenzing , The preprojective algebra of a tame hereditary Artin algebra, Comm. Algebra 15 (1987), 425457.

[11]A. Berenstein , S. Fomin and A. Zelevinsky , Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 152.

[12]A. Berenstein and A. Zelevinsky , Total positivity in Schubert varieties, Comment. Math. Helv. 72 (1997), 128166.

[13]S. Billey and V. Lakshmibai , Singular loci of Schubert varieties, Progress in Mathematics, vol. 182 (Birkhäuser, Boston, MA, 2000).

[15]R. Bocklandt , Graded Calabi Yau algebras of dimension 3, J. Pure Appl. Algebra 212 (2008), 1432.

[16]S. Brenner , M. Butler and A. King , Periodic algebras which are almost Koszul, Algebr. Represent. Theory 5 (2002), 331367.

[17]A. Buan and R. Marsh , Cluster-tilting theory, in Trends in representation theory of algebras and related topics (Queretaro, Mexico, 11–14 August 2004), Contemporary Mathematics, vol. 406 (American Mathematical Society, Providence, RI, 2006), 130.

[18]A. Buan , R. Marsh , M. Reineke , I. Reiten and G. Todorov , Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572618.

[19]A. Buan , R. Marsh and I. Reiten , Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), 323332.

[23]I. Burban , O. Iyama , B. Keller and I. Reiten , Cluster tilting for one-dimensional hypersurface singularities, Adv. Math. 217 (2008), 24432484.

[25]P. Caldero , F. Chapoton and R. Schiffler , Quivers with relations arising from clusters (An case), Trans. Amer. Math. Soc. 358 (2006), 13471364.

[27]P. Caldero and B. Keller , From triangulated categories to cluster algebras, Invent. Math. 172 (2008), 169211.

[29]W. Crawley-Boevey , On the exceptional fibres of Kleinian singularities, Amer. J. Math. 122 (2000), 10271037.

[30]K. Erdmann and T. Holm , Maximal n-orthogonal modules for selfinjective algebras, Proc. Amer. Math. Soc. 136 (2008), 30693078.

[31]S. Fomin and A. Zelevinsky , Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), 335380.

[32]S. Fomin and A. Zelevinsky , Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), 497529.

[33]S. Fomin and A. Zelevinsky , Cluster algebras II: Finite type classification, Invent. Math. 154 (2003), 63121.

[37]C. Geiss , B. Leclerc and J. Schröer , Rigid modules over preprojective algebras, Invent. Math. 165 (2006), 589632.

[38]C. Geiss , B. Leclerc and J. Schröer , Auslander algebras and initial seeds for cluster algebras, J. London Math. Soc. (2) 75 (2007), 718740.

[41]C. Geiss , B. Leclerc and J. Schröer , Partial flag varieties and preprojective algebras, Ann. Inst. Fourier 58 (2008), 825876.

[43]D. Happel , On Gorenstein algebras, in Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), Progress in Mathematics, vol. 95 (Birkhäuser, Basel, 1991), 389404.

[44]D. Happel and L. Unger , On a partial order of tilting modules, Algebr. Represent. Theory 8 (2005), 147156.

[46]K. Igusa , Notes on the no loops conjecture, J. Pure Appl. Algebra 69 (1990), 161176.

[48]O. Iyama , Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), 2250.

[49]O. Iyama , Auslander correspondence, Adv. Math. 210 (2007), 5182.

[52]O. Iyama and Y. Yoshino , Mutation in triangulated categories and rigid Cohen–Macaulay modules, Invent. Math. 172 (2008), 117168.

[53]V. G. Kac and D. H. Peterson , Regular functions on certain infinite-dimensional groups, in Arithmetic and geometry, Progress in Mathematics, vol. 36 (Birkhäuser, Boston, 1983), 141166.

[55]B. Keller , Chain complexes and stable categories, Manuscripta Math. 67 (1990), 379417.

[56]B. Keller , Derived categories and their uses, in Handbook of algebra, vol. 1 (North-Holland, Amsterdam, 1996), 671701.

[58]B. Keller , Calabi–Yau triangulated categories, in Trends in representation theory of algebras and related topics, EMS series of Congress Reports (European Mathematical Society, 2008), 467490.

[60]B. Keller and I. Reiten , Cluster-tilted algebras are Gorenstein and stably Calabi–Yau, Adv. Math. 211 (2007), 123151.

[61]S. König and B. Zhu , From triangulated categories to abelian categories—cluster tilting in a general framework, Math. Z 258 (2008), 143160.

[62]H. Lenzing , Nilpotente Elemente in Ringen von endlicher globaler Dimension, Math. Z. 108 (1969), 313324.

[63]G. Lusztig , Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365421.

[64]G. Lusztig , Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000), 129139.

[65]R. Marsh , M. Reineke and A. Zelevinsky , Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), 41714186.

[66]Y. Palu , Grothendieck group and generalized mutation rule for 2-Calabi–Yau triangulated categories, J. Pure Appl. Algebra 213 (2008), 14381449.

[68]I. Reiten and M. van den Bergh , Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), 295366.

[73]S. O. Smalö , Torsion theories and tilting modules, Bull. London Math. Soc. 16 (1984), 518522.

[75]A. Yekutieli , Dualizing complexes, Morita equivalence and the derived Picard group of a ring, J. London Math. Soc. (2) 60 (1999), 723746.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords: