Cluster categories for algebras of global dimension 2 and quivers with potential

Claire Amiot[1]

  • [1] Université Paris 7 Institut de Mathématiques de Jussieu Théorie des groupes et des représentations Case 7012 2 Place Jussieu 75251 Paris Cedex 05 (France)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 6, page 2525-2590
  • ISSN: 0373-0956

Abstract

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Let k be a field and A a finite-dimensional k -algebra of global dimension ≤ 2 . We construct a triangulated category 𝒞 A associated to A which, if  A is hereditary, is triangle equivalent to the cluster category of A . When 𝒞 A is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results also apply to quivers with potential. Namely, we introduce a cluster category 𝒞 ( Q , W ) associated to a quiver with potential ( Q , W ) . When it is Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra 𝒥 ( Q , W ) .

How to cite

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Amiot, Claire. "Cluster categories for algebras of global dimension 2 and quivers with potential." Annales de l’institut Fourier 59.6 (2009): 2525-2590. <http://eudml.org/doc/10463>.

@article{Amiot2009,
abstract = {Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\le 2$. We construct a triangulated category $\{\mathcal\{C\}\}_A $ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$. When $\{\mathcal\{C\}\}_A $ is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results also apply to quivers with potential. Namely, we introduce a cluster category $\{\mathcal\{C\}\}_\{(Q,W)\}$ associated to a quiver with potential $(Q,W)$. When it is Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra $\{\mathcal\{J\}\}(Q,W)$.},
affiliation = {Université Paris 7 Institut de Mathématiques de Jussieu Théorie des groupes et des représentations Case 7012 2 Place Jussieu 75251 Paris Cedex 05 (France)},
author = {Amiot, Claire},
journal = {Annales de l’institut Fourier},
keywords = {Cluster category; Calabi-Yau category; cluster-tilting; quiver with potential; preprojective algebra; cluster categories; Calabi-Yau categories; triangulated categories; quivers with potential; stable categories of modules; preprojective algebras},
language = {eng},
number = {6},
pages = {2525-2590},
publisher = {Association des Annales de l’institut Fourier},
title = {Cluster categories for algebras of global dimension 2 and quivers with potential},
url = {http://eudml.org/doc/10463},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Amiot, Claire
TI - Cluster categories for algebras of global dimension 2 and quivers with potential
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 6
SP - 2525
EP - 2590
AB - Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\le 2$. We construct a triangulated category ${\mathcal{C}}_A $ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$. When ${\mathcal{C}}_A $ is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results also apply to quivers with potential. Namely, we introduce a cluster category ${\mathcal{C}}_{(Q,W)}$ associated to a quiver with potential $(Q,W)$. When it is Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra ${\mathcal{J}}(Q,W)$.
LA - eng
KW - Cluster category; Calabi-Yau category; cluster-tilting; quiver with potential; preprojective algebra; cluster categories; Calabi-Yau categories; triangulated categories; quivers with potential; stable categories of modules; preprojective algebras
UR - http://eudml.org/doc/10463
ER -

References

top
  1. C. Amiot, On the structure of triangulated categories with finitely many indecomposables, Bull. Soc. Math. France 135 (2007), 435-474 Zbl1158.18005MR2430189
  2. C. Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, preprint (2008) Zbl1239.16011
  3. C. Amiot, Sur les petites catégories triangulées, (2008) 
  4. Handbook of Tilting Theory, 332 (2007), Angeleri-HügelL.L. Zbl1106.16300MR2385175
  5. H. Asashiba, The derived equivalence classification of representation-finite selfinjective algebras, J. Algebra 214 (1999), 182-221 Zbl0949.16013MR1684880
  6. I. Assem, T. Brüstle, R. Schiffler, Cluster-tilted algebras as trivial extensions Zbl1182.16009
  7. M. Auslander, Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976) (1978), 1-244. Lecture Notes in Pure Appl. Math., Vol. 37, Dekker, New York Zbl0383.16015MR480688
  8. M. Auslander, Isolated singularities and existence of almost split sequences, Representation theory, II (Ottawa, Ont., 1984) 1178 (1986), 194-242, Springer, Berlin Zbl0633.13007MR842486
  9. M. Auslander, I. Reiten, McKay quivers and extended Dynkin diagrams, Trans. Amer. Math. Soc. 293 (1986), 293-301 Zbl0588.20001MR814923
  10. M. Auslander, I. Reiten, Cohen-Macaulay and Gorenstein Artin algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) 95 (1991), 221-245, Birkhäuser, Basel Zbl0776.16003MR1112162
  11. R. Bautista, P. Gabriel, A. V. Roĭter, L. Salmerón, Representation-finite algebras and multiplicative bases, Invent. Math. 81 (1985), 217-285 Zbl0575.16012MR799266
  12. A. Beĭlinson, J. Bernstein, P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) 100 (1982), 5-171, Soc. Math. France, Paris Zbl0536.14011MR751966
  13. A. Berenstein, S. Fomin, A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 1-52 Zbl1135.16013MR2110627
  14. J. Białkowski, K. Erdmann, A. Skowroński, Deformed preprojective algebras of generalized Dynkin type, Trans. Amer. Math. Soc. 359 (2007), 2625-2650 (electronic) Zbl1117.16005MR2286048
  15. J. Białkowski, A. Skowroński, Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3 Zbl1164.16006
  16. J. Białkowski, A. Skowroński, Calabi-Yau stable module categories of finite type, (2006) Zbl1168.16003
  17. R. Bocklandt, Graded Calabi-Yau algebras of dimension 3, (2007) Zbl1132.16017MR2355031
  18. S. Brenner, M. C. R. Butler, A. D. King, Periodic algebras which are almost Koszul, Algebr. Represent. Theory 5 (2002), 331-367 Zbl1056.16003MR1930968
  19. A. B. Buan, O. Iyama, I. Reiten, J. Scott, Cluster structures for 2-Calabi-Yau categories and unipotent groups, (2007) Zbl1181.18006
  20. A. B. Buan, O. Iyama, I. Reiten, D. Smith, Mutation of cluster-tilting objects and potentials, (2008) Zbl1285.16012
  21. A. B. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572-618 Zbl1127.16011MR2249625
  22. A. B. Buan, R. Marsh, I. Reiten, Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), 323-332 (electronic) Zbl1123.16009MR2247893
  23. A. B. Buan, R. Marsh, I. Reiten, Cluster mutation via quiver representations, Comment. Math. Helv. 83 (2008), 143-177 Zbl1193.16016MR2365411
  24. A. B. Buan, R. Marsh, I. Reiten, G. Todorov, Clusters and seeds in acyclic cluster algebras, Proc. Amer. Math. Soc. 135 (2007), 3049-3060 (electronic) Zbl1190.16022MR2322734
  25. P. Caldero, F. Chapoton, Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv. 81 (2006), 595-616 Zbl1119.16013MR2250855
  26. P. Caldero, F. Chapoton, R. Schiffler, Quivers with relations arising from clusters ( A n case), Trans. Amer. Math. Soc. 358 (2006), 1347-1364 (electronic) Zbl1137.16020MR2187656
  27. P. Caldero, B. Keller, From triangulated categories to cluster algebras. II, Ann. Sci. École Norm. Sup. (4) 39 (2006), 983-1009 Zbl1115.18301MR2316979
  28. P. Caldero, B. Keller, From triangulated categories to cluster algebras, Invent. Math. 172 (2008), 169-211 Zbl1141.18012MR2385670
  29. J. Chuang, R. Rouquier, ??????? 
  30. H. Derksen, J. Weyman, A. Zelevinsky, Quivers with potentials and their representations I: Mutations, (2007) Zbl1204.16008MR2480710
  31. E. Dieterich, The Auslander-Reiten quiver of an isolated singularity, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) 1273 (1987), 244-264, Springer, Berlin Zbl0632.14004MR915179
  32. K. Erdmann, N. Snashall, On Hochschild cohomology of preprojective algebras. I, II, J. Algebra 205 (1998), 391-412, 413–434 Zbl0937.16013MR1632808
  33. K. Erdmann, N. Snashall, Preprojective algebras of Dynkin type, periodicity and the second Hochschild cohomology, Algebras and modules, II (Geiranger, 1996) 24 (1998), 183-193, Amer. Math. Soc., Providence, RI Zbl1034.16501MR1648626
  34. S. Fomin, A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529 (electronic) Zbl1021.16017MR1887642
  35. S. Fomin, A. Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), 63-121 Zbl1054.17024MR2004457
  36. S. Fomin, A. Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), 112-164 Zbl1127.16023MR2295199
  37. C. Fu, B. Keller, On cluster algebras with coefficients and 2-Calabi-Yau categories, (2007) Zbl1201.18007
  38. P. Gabriel, A. V. Roĭter, Representations of finite-dimensional algebras, Algebra, VIII 73 (1992), 1-177, Springer, Berlin Zbl0839.16001MR1239447
  39. Werner Geigle, Helmut Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) 1273 (1987), 265-297, Springer, Berlin Zbl0651.14006MR915180
  40. C. Geiß, B. Leclerc, J. Schröer, Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (to appear) (2006) Zbl1151.16009MR2427512
  41. C. Geiß, B. Leclerc, J. Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), 589-632 Zbl1167.16009MR2242628
  42. C. Geiß, B. Leclerc, J. Schröer, Auslander algebras and initial seeds for cluster algebras, J. London Math. Soc. (2) 75 (2007), 718-740 Zbl1135.17007MR2352732
  43. C. Geiß, B. Leclerc, J. Schröer, Cluster algebra structures and semi-canonical bases for unipotent groups, (2007) Zbl1135.17007
  44. V. Ginzburg, Calabi-Yau algebras, (2006) 
  45. D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), 339-389 Zbl0626.16008MR910167
  46. D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, 119 (1988), Cambridge University Press, Cambridge Zbl0635.16017MR935124
  47. D. Happel, A characterization of hereditary categories with tilting object, Invent. Math. 144 (2001), 381-398 Zbl1015.18006MR1827736
  48. D. Happel, U. Preiser, C. M. Ringel, Binary polyhedral groups and Euclidean diagrams, Manuscripta Math. 31 (1980), 317-329 Zbl0436.20005MR576503
  49. D. Happel, U. Preiser, C. M. Ringel, Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to D Tr -periodic modules, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) 832 (1980), 280-294, Springer, Berlin Zbl0446.16032MR607159
  50. D. Happel, I. Reiten, A characterization of the hereditary categories derived equivalent to some category of coherent sheaves on a weighted projective line, Proc. Amer. Math. Soc. 130 (2002), 643-651 (electronic) Zbl1043.16006MR1866014
  51. A. Heller, Stable homotopy categories, Bull. Amer. Math. Soc. 74 (1968), 28-63 Zbl0177.25605MR224090
  52. T. Holm, P. Jørgensen, Cluster categories and selfinjective algebras: type A, (2006) 
  53. T. Holm, P. Jørgensen, Cluster categories and selfinjective algebras: type D, (2006) 
  54. O. Iyama, Y. Yoshino, Mutations in triangulated categories and rigid Cohen-Macaulay modules, (2006) Zbl1140.18007
  55. B. Keller, Derived categories and universal problems, Comm. Algebra 19 (1991), 699-747 Zbl0722.18002MR1102982
  56. B. Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), 63-102 Zbl0799.18007MR1258406
  57. B. Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551-581 (electronic) Zbl1086.18006MR2184464
  58. B. Keller, On differential graded categories, International Congress of Mathematicians. Vol. II (2006), 151-190, Eur. Math. Soc., Zürich Zbl1140.18008MR2275593
  59. B. Keller, Calabi-Yau triangulated categories, (2008) Zbl1202.16014
  60. B. Keller, Deformed CY-completions and their duals, (2008) 
  61. B. Keller, On triangulated orbit categories, correction, (2008) 
  62. B. Keller, I. Reiten, Acyclic Calabi-Yau categories, (2006) Zbl1171.18008
  63. B. Keller, I. Reiten, Cluster-tilted algebras are Gorenstein and stably Calabi-Yau, Adv. Math. 211 (2007), 123-151 Zbl1128.18007MR2313531
  64. B. Keller, D. Vossieck, Sous les catégories dérivées, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 225-228 Zbl0628.18003MR907948
  65. B. Keller, D. Vossieck, Aisles in derived categories, Bull. Soc. Math. Belg. 40 (1988), 239-253 Zbl0671.18003MR976638
  66. B. Keller, D. Yang, Quiver mutation and derived equivalences, (2008) 
  67. M. Kontsevich, Y. Soibelman, Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I, (2006) Zbl1114.14027
  68. M. Künzer, On lifting diagrams in Frobenius categories, (2005) Zbl1108.18006
  69. R. Marsh, M. Reineke, A. Zelevinsky, Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), 4171-4186 (electronic) Zbl1042.52007MR1990581
  70. A. Neeman, Triangulated categories, 148 (2001), Princeton University Press, Princeton, NJ Zbl0974.18008MR1812507
  71. Y. Palu, On algebraic Calabi-Yau categories Zbl1154.16008
  72. Y. Palu, Grothendieck group and generalized mutation rule for 2-Calabi–Yau triangulated categories, (2008) Zbl1167.18003MR2497588
  73. I. Reiten, Calabi-Yau categories, (CIRM 2007) 
  74. I. Reiten, M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), 295-366 (electronic) Zbl0991.18009MR1887637
  75. C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), 199-224 Zbl0444.16018MR576602
  76. C. Riedtmann, Representation-finite self-injective algebras of class A n , Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) 832 (1980), 449-520, Springer, Berlin Zbl0455.16014MR607169
  77. C. Riedtmann, Many algebras with the same Auslander-Reiten quiver, Bull. London Math. Soc. 15 (1983), 43-47 Zbl0487.16021MR686347
  78. C. Riedtmann, Representation-finite self-injective algebras of class D n , Compos. Math. 49 (1983), 231-282 Zbl0514.16019MR704393
  79. C. M. Ringel, Tame algebras and integral quadratic forms, 1099 (1984), Springer-Verlag, Berlin Zbl0546.16013MR774589
  80. C. M. Ringel, The preprojective algebra of a quiver, Algebras and modules, II (Geiranger, 1996) 24 (1998), 467-480, Amer. Math. Soc., Providence, RI Zbl0928.16012MR1648647
  81. C. M. Ringel, Hereditary triangulated categories, Compos. Math. (2006) 
  82. G. Tabuada, On the structure of Calabi-Yau categories with a cluster tilting subcategory, Doc. Math. 12 (2007), 193-213 (electronic) Zbl1122.18007MR2302527
  83. M. Tepetla, Ph. D. 
  84. J.-L. Verdier, Catégories dérivées. Quelques résultats, Lect. Notes Math. 569 (1977), 262-311 Zbl0407.18008
  85. Jean-Louis Verdier, Des catégories dérivées des catégories abéliennes, Astérisque (1996) Zbl0882.18010MR1453167
  86. J. Xiao, B. Zhu, Relations for the Grothendieck groups of triangulated categories, J. Algebra 257 (2002), 37-50 Zbl1021.18004MR1942270
  87. J. Xiao, B. Zhu, Locally finite triangulated categories, J. Algebra 290 (2005), 473-490 Zbl1110.16013MR2153264
  88. Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, 146 (1990), Cambridge University Press, Cambridge Zbl0745.13003MR1079937

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