Cluster categories for algebras of global dimension 2 and quivers with potential
Annales de l'Institut Fourier, Volume 59 (2009) no. 6, pp. 2525-2590.

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).

Soient k un corps et A une k-algèbre de dimension finie et de dimension globale 2. On construit une catégorie triangulée 𝒞 A associée à A, qui est triangle-équivalente à la catégorie amassée 𝒞 A si A est héréditaire. Lorsque 𝒞 A est Hom-finie, on prouve qu’elle est 2-Calabi-Yau et munie d’un objet amas-basculant canonique. Cette nouvelle classe de catégories contient certaines sous-catégories stables de modules sur une algèbre préprojective introduite par Geiss-Leclerc-Schröer et par Buan-Iyama-Reiten-Scott. Ces résultats s’appliquent aussi aux carquois à potentiel. Plus précisément, on introduit une catégorie amassée 𝒞(Q,W) associée à un carquois à potentiel (Q,W). Quand il est Jacobi-fini, on prouve que cette catégorie est munie d’un objet amas-basculant dont l’algèbre d’endomorphismes est isomorphe à l’algèbre jacobienne.

DOI: 10.5802/aif.2499
Classification: 16G20,  16E45
Keywords: Cluster category, Calabi-Yau category, cluster-tilting, quiver with potential, preprojective algebra
Amiot, Claire 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)
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Amiot, Claire. Cluster categories for algebras of global dimension 2 and quivers with potential. Annales de l'Institut Fourier, Volume 59 (2009) no. 6, pp. 2525-2590. doi : 10.5802/aif.2499. http://archive.numdam.org/articles/10.5802/aif.2499/

[1] Amiot, C. On the structure of triangulated categories with finitely many indecomposables, Bull. Soc. Math. France, Volume 135 (2007), pp. 435-474 | Numdam | MR | Zbl

[2] Amiot, C. Cluster categories for algebras of global dimension 2 and quivers with potential, preprint (2008), pp. arXiv:math. RT/0805.1035

[3] Amiot, C. Sur les petites catégories triangulées (2008) (http://www.institut.math.jussieu.fr/~amiot/these.pdf)

[4] Angeleri-Hügel, L.; Happel, D.; Krause, H. Handbook of Tilting Theory, London Mathematical society, 332, Cambridge University press, 2007 | MR | Zbl

[5] Asashiba, H. The derived equivalence classification of representation-finite selfinjective algebras, J. Algebra, Volume 214 (1999) no. 1, pp. 182-221 | DOI | MR | Zbl

[6] Assem, I.; Brüstle, T.; Schiffler, R. Cluster-tilted algebras as trivial extensions (preprint, arXiv:math. RT/0601537)

[7] Auslander, M. Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), Dekker, New York, 1978, p. 1-244. Lecture Notes in Pure Appl. Math., Vol. 37 | MR | Zbl

[8] Auslander, M. Isolated singularities and existence of almost split sequences, Representation theory, II (Ottawa, Ont., 1984) (Lecture Notes in Math.), Volume 1178, Springer, Berlin, 1986, pp. 194-242 | MR | Zbl

[9] Auslander, M.; Reiten, I. McKay quivers and extended Dynkin diagrams, Trans. Amer. Math. Soc., Volume 293 (1986) no. 1, pp. 293-301 | DOI | MR | Zbl

[10] Auslander, M.; Reiten, I. Cohen-Macaulay and Gorenstein Artin algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) (Progr. Math.), Volume 95, Birkhäuser, Basel, 1991, pp. 221-245 | MR | Zbl

[11] Bautista, R.; Gabriel, P.; Roĭter, A. V.; Salmerón, L. Representation-finite algebras and multiplicative bases, Invent. Math., Volume 81 (1985) no. 2, pp. 217-285 | DOI | MR | Zbl

[12] Beĭlinson, A.; Bernstein, J.; Deligne, P. Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) (Astérisque), Volume 100, Soc. Math. France, Paris, 1982, pp. 5-171 | MR | Zbl

[13] Berenstein, A.; Fomin, S.; Zelevinsky, A. Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J., Volume 126 (2005) no. 1, pp. 1-52 | DOI | MR | Zbl

[14] Białkowski, J.; Erdmann, K.; Skowroński, A. Deformed preprojective algebras of generalized Dynkin type, Trans. Amer. Math. Soc., Volume 359 (2007) no. 6, p. 2625-2650 (electronic) | DOI | MR | Zbl

[15] Białkowski, J.; Skowroński, A. Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3 (to appear in Algebra and Discrete Mathematics) | Zbl

[16] Białkowski, J.; Skowroński, A. Calabi-Yau stable module categories of finite type (2006) (preprint, http://www.mat.uni.torun.pl/preprints/) | Zbl

[17] Bocklandt, R. Graded Calabi-Yau algebras of dimension 3 (2007) (preprint, arXiv:math. RA/0603558) | MR | Zbl

[18] Brenner, S.; Butler, M. C. R.; King, A. D. Periodic algebras which are almost Koszul, Algebr. Represent. Theory, Volume 5 (2002) no. 4, pp. 331-367 | DOI | MR | Zbl

[19] Buan, A. B.; Iyama, O.; Reiten, I.; Scott, J. Cluster structures for 2-Calabi-Yau categories and unipotent groups (2007) (preprint, arXiv:math. RT/0701557)

[20] Buan, A. B.; Iyama, O.; Reiten, I.; Smith, D. Mutation of cluster-tilting objects and potentials (2008) (preprint, arXiv:math. RT/08043813)

[21] Buan, A. B.; Marsh, R.; Reineke, M.; Reiten, I.; Todorov, G. Tilting theory and cluster combinatorics, Adv. Math., Volume 204 (2006) no. 2, pp. 572-618 | DOI | MR | Zbl

[22] Buan, A. B.; Marsh, R.; Reiten, I. Cluster-tilted algebras, Trans. Amer. Math. Soc., Volume 359 (2007) no. 1, p. 323-332 (electronic) | DOI | MR | Zbl

[23] Buan, A. B.; Marsh, R.; Reiten, I. Cluster mutation via quiver representations, Comment. Math. Helv., Volume 83 (2008) no. 1, pp. 143-177 | DOI | MR

[24] Buan, A. B.; Marsh, R.; Reiten, I.; Todorov, G. Clusters and seeds in acyclic cluster algebras, Proc. Amer. Math. Soc., Volume 135 (2007) no. 10, p. 3049-3060 (electronic) (With an appendix coauthored in addition by P. Caldero and B. Keller) | DOI | MR

[25] Caldero, P.; Chapoton, F. Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv., Volume 81 (2006) no. 3, pp. 595-616 | DOI | MR | Zbl

[26] Caldero, P.; Chapoton, F.; Schiffler, R. Quivers with relations arising from clusters (A n case), Trans. Amer. Math. Soc., Volume 358 (2006) no. 3, p. 1347-1364 (electronic) | DOI | MR | Zbl

[27] Caldero, P.; Keller, B. From triangulated categories to cluster algebras. II, Ann. Sci. École Norm. Sup. (4), Volume 39 (2006) no. 6, pp. 983-1009 | Numdam | MR | Zbl

[28] Caldero, P.; Keller, B. From triangulated categories to cluster algebras, Invent. Math., Volume 172 (2008), pp. 169-211 | DOI | MR | Zbl

[29] Chuang, J.; Rouquier, R. ??????? (preprint in preparation)

[30] Derksen, H.; Weyman, J.; Zelevinsky, A. Quivers with potentials and their representations I: Mutations (2007) (preprint, arXiv:math. RA/07040649) | MR

[31] Dieterich, E. The Auslander-Reiten quiver of an isolated singularity, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) (Lecture Notes in Math.), Volume 1273, Springer, Berlin, 1987, pp. 244-264 | MR | Zbl

[32] Erdmann, K.; Snashall, N. On Hochschild cohomology of preprojective algebras. I, II, J. Algebra, Volume 205 (1998) no. 2, p. 391-412, 413–434 | DOI | MR | Zbl

[33] Erdmann, K.; Snashall, N. Preprojective algebras of Dynkin type, periodicity and the second Hochschild cohomology, Algebras and modules, II (Geiranger, 1996) (CMS Conf. Proc.), Volume 24, Amer. Math. Soc., Providence, RI, 1998, pp. 183-193 | MR | Zbl

[34] Fomin, S.; Zelevinsky, A. Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, p. 497-529 (electronic) | DOI | MR | Zbl

[35] Fomin, S.; Zelevinsky, A. Cluster algebras. II. Finite type classification, Invent. Math., Volume 154 (2003) no. 1, pp. 63-121 | DOI | MR | Zbl

[36] Fomin, S.; Zelevinsky, A. Cluster algebras. IV. Coefficients, Compos. Math., Volume 143 (2007) no. 1, pp. 112-164 | DOI | MR | Zbl

[37] Fu, C.; Keller, B. On cluster algebras with coefficients and 2-Calabi-Yau categories (2007) (preprint, arXiv:math. RT/07103152)

[38] Gabriel, P.; Roĭter, A. V. Representations of finite-dimensional algebras, Algebra, VIII (Encyclopaedia Math. Sci.), Volume 73, Springer, Berlin, 1992, pp. 1-177 (With a chapter by B. Keller) | MR

[39] Geigle, Werner; Lenzing, Helmut A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) (Lecture Notes in Math.), Volume 1273, Springer, Berlin, 1987, pp. 265-297 | MR | Zbl

[40] Geiß, C.; Leclerc, B.; Schröer, J. Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (to appear) (2006), pp. arXiv:math. RT/0609138 | Numdam | MR | Zbl

[41] Geiß, C.; Leclerc, B.; Schröer, J. Rigid modules over preprojective algebras, Invent. Math., Volume 165 (2006) no. 3, pp. 589-632 | DOI | MR | Zbl

[42] Geiß, C.; Leclerc, B.; Schröer, J. Auslander algebras and initial seeds for cluster algebras, J. London Math. Soc. (2), Volume 75 (2007) no. 3, pp. 718-740 | DOI | MR | Zbl

[43] Geiß, C.; Leclerc, B.; Schröer, J. Cluster algebra structures and semi-canonical bases for unipotent groups (2007) (preprint, arXiv:math. RT/0703039)

[44] Ginzburg, V. Calabi-Yau algebras (2006) (arXiv:math. AG/0612139)

[45] Happel, D. On the derived category of a finite-dimensional algebra, Comment. Math. Helv., Volume 62 (1987) no. 3, pp. 339-389 | DOI | MR | Zbl

[46] Happel, D. Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, 119, Cambridge University Press, Cambridge, 1988 | MR | Zbl

[47] Happel, D. A characterization of hereditary categories with tilting object, Invent. Math., Volume 144 (2001) no. 2, pp. 381-398 | DOI | MR | Zbl

[48] Happel, D.; Preiser, U.; Ringel, C. M. Binary polyhedral groups and Euclidean diagrams, Manuscripta Math., Volume 31 (1980) no. 1-3, pp. 317-329 | DOI | MR | Zbl

[49] Happel, D.; Preiser, U.; Ringel, C. M. 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) (Lecture Notes in Math.), Volume 832, Springer, Berlin, 1980, pp. 280-294 | MR | Zbl

[50] Happel, D.; Reiten, I. A characterization of the hereditary categories derived equivalent to some category of coherent sheaves on a weighted projective line, Proc. Amer. Math. Soc., Volume 130 (2002) no. 3, p. 643-651 (electronic) | DOI | MR | Zbl

[51] Heller, A. Stable homotopy categories, Bull. Amer. Math. Soc., Volume 74 (1968), pp. 28-63 | DOI | MR | Zbl

[52] Holm, T.; Jørgensen, P. Cluster categories and selfinjective algebras: type A (2006) (preprint, arXiv:math. RT/0610728)

[53] Holm, T.; Jørgensen, P. Cluster categories and selfinjective algebras: type D (2006) (preprint, arXiv:math. RT/0612451)

[54] Iyama, O.; Yoshino, Y. Mutations in triangulated categories and rigid Cohen-Macaulay modules (2006) (preprint, arXiv:math. RT/0607736) | Zbl

[55] Keller, B. Derived categories and universal problems, Comm. Algebra, Volume 19 (1991) no. 3, pp. 699-747 | DOI | MR | Zbl

[56] Keller, B. Deriving DG categories, Ann. Sci. École Norm. Sup. (4), Volume 27 (1994) no. 1, pp. 63-102 | Numdam | MR | Zbl

[57] Keller, B. On triangulated orbit categories, Doc. Math., Volume 10 (2005), p. 551-581 (electronic) | MR | Zbl

[58] Keller, B. On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 151-190 | MR | Zbl

[59] Keller, B. Calabi-Yau triangulated categories (2008) (preprint, http://www.math.jussieu.fr/~keller/publ/KellerCYtriangCat.pdf)

[60] Keller, B. Deformed CY-completions and their duals (2008) (in preparation)

[61] Keller, B. On triangulated orbit categories, correction (2008) (http://people.math.jussieu.fr/~keller/publ/corrTriaOrbit.pdf)

[62] Keller, B.; Reiten, I. Acyclic Calabi-Yau categories (2006) (preprint, arXiv:math. RT/0610594)

[63] Keller, B.; Reiten, I. Cluster-tilted algebras are Gorenstein and stably Calabi-Yau, Adv. Math., Volume 211 (2007) no. 1, pp. 123-151 | DOI | MR | Zbl

[64] Keller, B.; Vossieck, D. Sous les catégories dérivées, C. R. Acad. Sci. Paris Sér. I Math., Volume 305 (1987) no. 6, pp. 225-228 | MR | Zbl

[65] Keller, B.; Vossieck, D. Aisles in derived categories, Bull. Soc. Math. Belg., Volume 40 (1988), pp. 239-253 | MR | Zbl

[66] Keller, B.; Yang, D. Quiver mutation and derived equivalences (2008) (in preparation)

[67] Kontsevich, M.; Soibelman, Y. Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I (2006) (preprint, arXiv:math. RA/0606241)

[68] Künzer, M. On lifting diagrams in Frobenius categories (2005) (Homotopy, Homology and Applications)

[69] Marsh, R.; Reineke, M.; Zelevinsky, A. Generalized associahedra via quiver representations, Trans. Amer. Math. Soc., Volume 355 (2003) no. 10, p. 4171-4186 (electronic) | DOI | MR | Zbl

[70] Neeman, A. Triangulated categories, Annals of Mathematics Studies, 148, Princeton University Press, Princeton, NJ, 2001 | MR | Zbl

[71] Palu, Y. On algebraic Calabi-Yau categories (Ph.D. in preparation)

[72] Palu, Y. Grothendieck group and generalized mutation rule for 2-Calabi–Yau triangulated categories (2008) (preprint, arXiv:math. RT/08033907) | MR | Zbl

[73] Reiten, I. Calabi-Yau categories (CIRM 2007) (Talk at the meeting: Calabi-Yau algebras and N-Koszul algebras)

[74] Reiten, I.; Van den Bergh, M. Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc., Volume 15 (2002) no. 2, p. 295-366 (electronic) | DOI | MR | Zbl

[75] Riedtmann, C. Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv., Volume 55 (1980) no. 2, pp. 199-224 | DOI | MR | Zbl

[76] Riedtmann, C. Representation-finite self-injective algebras of class A n , Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) (Lecture Notes in Math.), Volume 832, Springer, Berlin, 1980, pp. 449-520 | MR | Zbl

[77] Riedtmann, C. Many algebras with the same Auslander-Reiten quiver, Bull. London Math. Soc., Volume 15 (1983) no. 1, pp. 43-47 | DOI | MR | Zbl

[78] Riedtmann, C. Representation-finite self-injective algebras of class D n , Compos. Math., Volume 49 (1983) no. 2, pp. 231-282 | Numdam | MR | Zbl

[79] Ringel, C. M. Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, 1099, Springer-Verlag, Berlin, 1984 | MR | Zbl

[80] Ringel, C. M. The preprojective algebra of a quiver, Algebras and modules, II (Geiranger, 1996) (CMS Conf. Proc.), Volume 24, Amer. Math. Soc., Providence, RI, 1998, pp. 467-480 | MR | Zbl

[81] Ringel, C. M. Hereditary triangulated categories, Compos. Math. (2006), pp. (to appear)

[82] Tabuada, G. On the structure of Calabi-Yau categories with a cluster tilting subcategory, Doc. Math., Volume 12 (2007), p. 193-213 (electronic) | MR | Zbl

[83] Tepetla, M. Ph. D. (in preparation)

[84] Verdier, J.-L. Catégories dérivées. Quelques résultats, Lect. Notes Math., Volume 569 (1977), pp. 262-311 | DOI | Zbl

[85] Verdier, Jean-Louis Des catégories dérivées des catégories abéliennes, Astérisque (1996) no. 239, pp. xii+253 pp. (1997) (With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis) | MR | Zbl

[86] Xiao, J.; Zhu, B. Relations for the Grothendieck groups of triangulated categories, J. Algebra, Volume 257 (2002) no. 1, pp. 37-50 | DOI | MR | Zbl

[87] Xiao, J.; Zhu, B. Locally finite triangulated categories, J. Algebra, Volume 290 (2005) no. 2, pp. 473-490 | DOI | MR | Zbl

[88] Yoshino, Y. Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, 146, Cambridge University Press, Cambridge, 1990 | MR | Zbl

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