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

  • A. B. Buan (a1), O. Iyama (a2), I. Reiten (a3) and J. Scott (a4)

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.

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[1]Angeleri Hügel, L., Happel, D. and Krause, H., Handbook of tilting theory, London Mathematical Society Lecture Note Series, vol. 332 (Cambridge University Press, Cambridge, 2007).
[2]Assem, I., Brüstle, T. and Schiffler, R., Cluster-tilted algebras as trivial extensions, Bull. London Math. Soc. 40 (2008), 151162.
[3]Assem, I., Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras. 1: Techniques of representation theory, London Mathematical Society Student Texts, vol. 65 (Cambridge University Press, Cambridge, 2006).
[4]Auslander, M., Coherent functors, in Proceedings of the conference on categorical algebra (La Jolla, CA, 1965) (Springer, Berlin, 1966), 189231.
[5]Auslander, M., Applications of morphisms determined by modules, in Proceedings of the conference on representation theory of algebras (Temple Univ., Philadelphia, 1976), Lecture Notes in Pure and Applied Mathematics, vol. 37 (Dekker, New York, 1978), 245327.
[6]Auslander, M. and Reiten, I., Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), 111152.
[7]Auslander, M. and Reiten, I., DTr-periodic modules and functors, in Representation theory of algebras (Cocoyoc, 1994), CMS Conference Proceedings, vol. 18 (American Mathematical Society, Providence, RI, 1996), 3950.
[8]Auslander, M., Reiten, I. and Smalø, S. O., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36 (Cambridge University Press, Cambridge, 1997).
[9]Auslander, M. and Smalø, S. O., Preprojective modules over Artin algebras, J. Algebra 66 (1980), 61122.
[10]Baer, D., Geigle, W. and Lenzing, H., The preprojective algebra of a tame hereditary Artin algebra, Comm. Algebra 15 (1987), 425457.
[11]Berenstein, A., Fomin, S. and Zelevinsky, A., Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 152.
[12]Berenstein, A. and Zelevinsky, A., Total positivity in Schubert varieties, Comment. Math. Helv. 72 (1997), 128166.
[13]Billey, S. and Lakshmibai, V., Singular loci of Schubert varieties, Progress in Mathematics, vol. 182 (Birkhäuser, Boston, MA, 2000).
[14]Björner, A. and Brenti, F., Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231 (Springer, New York, 2005).
[15]Bocklandt, R., Graded Calabi Yau algebras of dimension 3, J. Pure Appl. Algebra 212 (2008), 1432.
[16]Brenner, S., Butler, M. and King, A., Periodic algebras which are almost Koszul, Algebr. Represent. Theory 5 (2002), 331367.
[17]Buan, A. and Marsh, R., 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]Buan, A., Marsh, R., Reineke, M., Reiten, I. and Todorov, G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572618.
[19]Buan, A., Marsh, R. and Reiten, I., Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), 323332.
[20]Buan, A., Marsh, R. and Reiten, I., Cluster mutation via quiver representations, Comm. Math. Helv 83 (2008), 143177.
[21]Buan, A., Marsh, R., Reiten, I. and Todorov, G., Clusters and seeds for acyclic cluster algebras; with an appendix by Buan A., Caldero P., Keller B., Marsh R., Reiten I. and Todorov G., Proc. Amer. Math. Soc. 135 (2007), 30493060.
[22]Buan, A. and Reiten, I., Acyclic quivers of finite mutation type, Int. Math. Res. Not. (2006), 110.
[23]Burban, I., Iyama, O., Keller, B. and Reiten, I., Cluster tilting for one-dimensional hypersurface singularities, Adv. Math. 217 (2008), 24432484.
[24]Caldero, P. and Chapoton, F., Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv. 81 (2006), 595616.
[25]Caldero, P., Chapoton, F. and Schiffler, R., Quivers with relations arising from clusters (A n case), Trans. Amer. Math. Soc. 358 (2006), 13471364.
[26]Caldero, P. and Keller, B., From triangulated categories to cluster algebras II, Ann. Sci. Ecole Norm. Sup. (4) 39 (2006), 9831009.
[27]Caldero, P. and Keller, B., From triangulated categories to cluster algebras, Invent. Math. 172 (2008), 169211.
[28]Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, Princeton, NJ, 1956).
[29]Crawley-Boevey, W., On the exceptional fibres of Kleinian singularities, Amer. J. Math. 122 (2000), 10271037.
[30]Erdmann, K. and Holm, T., Maximal n-orthogonal modules for selfinjective algebras, Proc. Amer. Math. Soc. 136 (2008), 30693078.
[31]Fomin, S. and Zelevinsky, A., Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), 335380.
[32]Fomin, S. and Zelevinsky, A., Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), 497529.
[33]Fomin, S. and Zelevinsky, A., Cluster algebras II: Finite type classification, Invent. Math. 154 (2003), 63121.
[34]Fomin, S. and Zelevinsky, A., Cluster algebras IV: Coefficients, Compositio Math. 143 (2007), 112164.
[35]Fu, C. and Keller, B., On cluster algebras with coefficients and 2-Calabi-Yau categories, Preprint (2007), arXiv:0710.3152, Trans. Amer. Math. Soc., to appear.
[36]Geiss, C., Leclerc, B. and Schröer, J., Semicanonical bases and preprojective algebras, Ann. Sci. Ecole Norm. Sup. (4) 38 (2005), 193253.
[37]Geiss, C., Leclerc, B. and Schröer, J., Rigid modules over preprojective algebras, Invent. Math. 165 (2006), 589632.
[38]Geiss, C., Leclerc, B. and Schröer, J., Auslander algebras and initial seeds for cluster algebras, J. London Math. Soc. (2) 75 (2007), 718740.
[39]Geiss, C., Leclerc, B. and Schröer, J., Semicanonical bases and preprojective algebras II: A multiplication formula, Compositio Math. 143 (2007), 13131334.
[40]Geiss, C., Leclerc, B. and Schröer, J., Cluster algebra structures and semicanonical bases for unipotent groups, Preprint (2007), arxiv:math.RT/0703039.
[41]Geiss, C., Leclerc, B. and Schröer, J., Partial flag varieties and preprojective algebras, Ann. Inst. Fourier 58 (2008), 825876.
[42]Happel, D., Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119 (Cambridge University Press, Cambridge, 1988).
[43]Happel, D., 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]Happel, D. and Unger, L., On a partial order of tilting modules, Algebr. Represent. Theory 8 (2005), 147156.
[45]Hubery, A., Acyclic cluster algebras via Ringel-Hall algebras, Preprint. Available from www.maths.leeds.ac.uk/∼ahubery.
[46]Igusa, K., Notes on the no loops conjecture, J. Pure Appl. Algebra 69 (1990), 161176.
[47]Ingalls, C. and Thomas, H., Noncrossing partitions and representations of quivers, Preprint (2006), arxiv:math.RT/0612219, Compositio Math., to appear.
[48]Iyama, O., Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), 2250.
[49]Iyama, O., Auslander correspondence, Adv. Math. 210 (2007), 5182.
[50]Iyama, O., d-CalabiYau algebras and d-cluster-tilting subcategories, Preprint. Available from www.math.nagoya-u.ac.jp/∼iyama/papers.html.
[51]Iyama, O. and Reiten, I., Fomin–Zelevinsky mutation and tilting modules over Calabi–Yau algebras, Amer. J. Math. 130 (2008), 10891149.
[52]Iyama, O. and Yoshino, Y., Mutation in triangulated categories and rigid Cohen–Macaulay modules, Invent. Math. 172 (2008), 117168.
[53]Kac, V. G. and Peterson, D. H., Regular functions on certain infinite-dimensional groups, in Arithmetic and geometry, Progress in Mathematics, vol. 36 (Birkhäuser, Boston, 1983), 141166.
[54]Kac, V. G. and Peterson, D. H., Defining relations of certain infinite-dimensional groups, in Proceedings of the Cartan conference (Lyon, 1984), Astérisque, numero hors serie (Société Mathématique de France, Paris, 1985), 165208.
[55]Keller, B., Chain complexes and stable categories, Manuscripta Math. 67 (1990), 379417.
[56]Keller, B., Derived categories and their uses, in Handbook of algebra, vol. 1 (North-Holland, Amsterdam, 1996), 671701.
[57]Keller, B., On triangulated orbit categories, Documenta Math. 10 (2005), 551581.
[58]Keller, B., Calabi–Yau triangulated categories, in Trends in representation theory of algebras and related topics, EMS series of Congress Reports (European Mathematical Society, 2008), 467490.
[59]Keller, B. and Reiten, I., Acyclic Calabi–Yau categories, Compositio Math. 144 (2008), 13321348.
[60]Keller, B. and Reiten, I., Cluster-tilted algebras are Gorenstein and stably Calabi–Yau, Adv. Math. 211 (2007), 123151.
[61]König, S. and Zhu, B., From triangulated categories to abelian categories—cluster tilting in a general framework, Math. Z 258 (2008), 143160.
[62]Lenzing, H., Nilpotente Elemente in Ringen von endlicher globaler Dimension, Math. Z. 108 (1969), 313324.
[63]Lusztig, G., Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365421.
[64]Lusztig, G., Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000), 129139.
[65]Marsh, R., Reineke, M. and Zelevinsky, A., Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), 41714186.
[66]Palu, Y., Grothendieck group and generalized mutation rule for 2-Calabi–Yau triangulated categories, J. Pure Appl. Algebra 213 (2008), 14381449.
[67]Pressley, A. and Segal, G., Loop groups, Oxford Mathematical Monographs (Oxford University Press, New York, 1986).
[68]Reiten, I. and van den Bergh, M., Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), 295366.
[69]Rickard, J., Morita theory for derived categories, J. London Math. Soc. (2) 29 (1989), 436456.
[70]Ringel, C. M., Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099 (Springer, Berlin, 1984).
[71]Ringel, C. M., Some remarks concerning tilting modules and tilted algebras. Origin. Relevance. Future, in Handbook of tilting theory, London Mathematical Society Lecture Note Series, vol. 332 (Cambridge University Press, Cambridge, 2007), 413469.
[72]Scott, J., Block-Toeplitz determinants, chess tableaux, and the type   GeissLeclercSchröer φ-map, Preprint (2007), arXiv:0707.3046.
[73]Smalö, S. O., Torsion theories and tilting modules, Bull. London Math. Soc. 16 (1984), 518522.
[74]Tabuada, G., On the structure of Calabi–Yau categories with a cluster tilting subcategory, Documenta Math. 12 (2007), 193213.
[75]Yekutieli, A., Dualizing complexes, Morita equivalence and the derived Picard group of a ring, J. London Math. Soc. (2) 60 (1999), 723746.
[76]Yoshino, Y., Cohen–Macaulay modules over Cohen–Macaulay rings, London Mathematical Society Lecture Note Series, vol. 146 (Cambridge University Press, Cambridge, 1990).
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