Let be a field and a finite-dimensional -algebra of global dimension . We construct a triangulated category associated to which, if is hereditary, is triangle equivalent to the cluster category of . When 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 associated to a quiver with potential . 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 .
Soient un corps et une -algèbre de dimension finie et de dimension globale . On construit une catégorie triangulée associée à , qui est triangle-équivalente à la catégorie amassée si est héréditaire. Lorsque 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 associée à un carquois à potentiel . 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.
Keywords: Cluster category, Calabi-Yau category, cluster-tilting, quiver with potential, preprojective algebra
Mot clés : catégorie amassée, catégorie de Calabi-Yau, amas-basculant, carquois à potentiel, algèbre préprojective
@article{AIF_2009__59_6_2525_0, author = {Amiot, Claire}, title = {Cluster categories for algebras of global dimension 2 and quivers with potential}, journal = {Annales de l'Institut Fourier}, pages = {2525--2590}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {6}, year = {2009}, doi = {10.5802/aif.2499}, mrnumber = {2640929}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2499/} }
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 SP - 2525 EP - 2590 VL - 59 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2499/ DO - 10.5802/aif.2499 LA - en ID - AIF_2009__59_6_2525_0 ER -
%0 Journal Article %A Amiot, Claire %T Cluster categories for algebras of global dimension 2 and quivers with potential %J Annales de l'Institut Fourier %D 2009 %P 2525-2590 %V 59 %N 6 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2499/ %R 10.5802/aif.2499 %G en %F AIF_2009__59_6_2525_0
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] On the structure of triangulated categories with finitely many indecomposables, Bull. Soc. Math. France, Volume 135 (2007), pp. 435-474 | Numdam | MR | Zbl
[2] Cluster categories for algebras of global dimension 2 and quivers with potential, preprint (2008), pp. arXiv:math. RT/0805.1035
[3] Sur les petites catégories triangulées (2008) (http://www.institut.math.jussieu.fr/~amiot/these.pdf)
[4] Handbook of Tilting Theory, London Mathematical society, 332, Cambridge University press, 2007 | MR | Zbl
[5] The derived equivalence classification of representation-finite selfinjective algebras, J. Algebra, Volume 214 (1999) no. 1, pp. 182-221 | DOI | MR | Zbl
[6] Cluster-tilted algebras as trivial extensions (preprint, arXiv:math. RT/0601537)
[7] 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] 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] McKay quivers and extended Dynkin diagrams, Trans. Amer. Math. Soc., Volume 293 (1986) no. 1, pp. 293-301 | DOI | MR | Zbl
[10] 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] Representation-finite algebras and multiplicative bases, Invent. Math., Volume 81 (1985) no. 2, pp. 217-285 | DOI | MR | Zbl
[12] 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] Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J., Volume 126 (2005) no. 1, pp. 1-52 | DOI | MR | Zbl
[14] Deformed preprojective algebras of generalized Dynkin type, Trans. Amer. Math. Soc., Volume 359 (2007) no. 6, p. 2625-2650 (electronic) | DOI | MR | Zbl
[15] Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3 (to appear in Algebra and Discrete Mathematics) | Zbl
[16] Calabi-Yau stable module categories of finite type (2006) (preprint, http://www.mat.uni.torun.pl/preprints/) | Zbl
[17] Graded Calabi-Yau algebras of dimension 3 (2007) (preprint, arXiv:math. RA/0603558) | MR | Zbl
[18] Periodic algebras which are almost Koszul, Algebr. Represent. Theory, Volume 5 (2002) no. 4, pp. 331-367 | DOI | MR | Zbl
[19] Cluster structures for 2-Calabi-Yau categories and unipotent groups (2007) (preprint, arXiv:math. RT/0701557)
[20] Mutation of cluster-tilting objects and potentials (2008) (preprint, arXiv:math. RT/08043813)
[21] Tilting theory and cluster combinatorics, Adv. Math., Volume 204 (2006) no. 2, pp. 572-618 | DOI | MR | Zbl
[22] Cluster-tilted algebras, Trans. Amer. Math. Soc., Volume 359 (2007) no. 1, p. 323-332 (electronic) | DOI | MR | Zbl
[23] Cluster mutation via quiver representations, Comment. Math. Helv., Volume 83 (2008) no. 1, pp. 143-177 | DOI | MR
[24] 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] Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv., Volume 81 (2006) no. 3, pp. 595-616 | DOI | MR | Zbl
[26] Quivers with relations arising from clusters ( case), Trans. Amer. Math. Soc., Volume 358 (2006) no. 3, p. 1347-1364 (electronic) | DOI | MR | Zbl
[27] 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] From triangulated categories to cluster algebras, Invent. Math., Volume 172 (2008), pp. 169-211 | DOI | MR | Zbl
[29] ??????? (preprint in preparation)
[30] Quivers with potentials and their representations I: Mutations (2007) (preprint, arXiv:math. RA/07040649) | MR
[31] 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] On Hochschild cohomology of preprojective algebras. I, II, J. Algebra, Volume 205 (1998) no. 2, p. 391-412, 413–434 | DOI | MR | Zbl
[33] 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] Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, p. 497-529 (electronic) | DOI | MR | Zbl
[35] Cluster algebras. II. Finite type classification, Invent. Math., Volume 154 (2003) no. 1, pp. 63-121 | DOI | MR | Zbl
[36] Cluster algebras. IV. Coefficients, Compos. Math., Volume 143 (2007) no. 1, pp. 112-164 | DOI | MR | Zbl
[37] On cluster algebras with coefficients and 2-Calabi-Yau categories (2007) (preprint, arXiv:math. RT/07103152)
[38] 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] 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] Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (to appear) (2006), pp. arXiv:math. RT/0609138 | Numdam | MR | Zbl
[41] Rigid modules over preprojective algebras, Invent. Math., Volume 165 (2006) no. 3, pp. 589-632 | DOI | MR | Zbl
[42] 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] Cluster algebra structures and semi-canonical bases for unipotent groups (2007) (preprint, arXiv:math. RT/0703039)
[44] Calabi-Yau algebras (2006) (arXiv:math. AG/0612139)
[45] On the derived category of a finite-dimensional algebra, Comment. Math. Helv., Volume 62 (1987) no. 3, pp. 339-389 | DOI | MR | Zbl
[46] 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] A characterization of hereditary categories with tilting object, Invent. Math., Volume 144 (2001) no. 2, pp. 381-398 | DOI | MR | Zbl
[48] Binary polyhedral groups and Euclidean diagrams, Manuscripta Math., Volume 31 (1980) no. 1-3, pp. 317-329 | DOI | MR | Zbl
[49] Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to -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] 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] Stable homotopy categories, Bull. Amer. Math. Soc., Volume 74 (1968), pp. 28-63 | DOI | MR | Zbl
[52] Cluster categories and selfinjective algebras: type A (2006) (preprint, arXiv:math. RT/0610728)
[53] Cluster categories and selfinjective algebras: type D (2006) (preprint, arXiv:math. RT/0612451)
[54] Mutations in triangulated categories and rigid Cohen-Macaulay modules (2006) (preprint, arXiv:math. RT/0607736) | Zbl
[55] Derived categories and universal problems, Comm. Algebra, Volume 19 (1991) no. 3, pp. 699-747 | DOI | MR | Zbl
[56] Deriving DG categories, Ann. Sci. École Norm. Sup. (4), Volume 27 (1994) no. 1, pp. 63-102 | Numdam | MR | Zbl
[57] On triangulated orbit categories, Doc. Math., Volume 10 (2005), p. 551-581 (electronic) | MR | Zbl
[58] On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 151-190 | MR | Zbl
[59] Calabi-Yau triangulated categories (2008) (preprint, http://www.math.jussieu.fr/~keller/publ/KellerCYtriangCat.pdf)
[60] Deformed CY-completions and their duals (2008) (in preparation)
[61] On triangulated orbit categories, correction (2008) (http://people.math.jussieu.fr/~keller/publ/corrTriaOrbit.pdf)
[62] Acyclic Calabi-Yau categories (2006) (preprint, arXiv:math. RT/0610594)
[63] Cluster-tilted algebras are Gorenstein and stably Calabi-Yau, Adv. Math., Volume 211 (2007) no. 1, pp. 123-151 | DOI | MR | Zbl
[64] Sous les catégories dérivées, C. R. Acad. Sci. Paris Sér. I Math., Volume 305 (1987) no. 6, pp. 225-228 | MR | Zbl
[65] Aisles in derived categories, Bull. Soc. Math. Belg., Volume 40 (1988), pp. 239-253 | MR | Zbl
[66] Quiver mutation and derived equivalences (2008) (in preparation)
[67] Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I (2006) (preprint, arXiv:math. RA/0606241)
[68] On lifting diagrams in Frobenius categories (2005) (Homotopy, Homology and Applications)
[69] Generalized associahedra via quiver representations, Trans. Amer. Math. Soc., Volume 355 (2003) no. 10, p. 4171-4186 (electronic) | DOI | MR | Zbl
[70] Triangulated categories, Annals of Mathematics Studies, 148, Princeton University Press, Princeton, NJ, 2001 | MR | Zbl
[71] On algebraic Calabi-Yau categories (Ph.D. in preparation)
[72] Grothendieck group and generalized mutation rule for 2-Calabi–Yau triangulated categories (2008) (preprint, arXiv:math. RT/08033907) | MR | Zbl
[73] Calabi-Yau categories (CIRM 2007) (Talk at the meeting: Calabi-Yau algebras and -Koszul algebras)
[74] Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc., Volume 15 (2002) no. 2, p. 295-366 (electronic) | DOI | MR | Zbl
[75] Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv., Volume 55 (1980) no. 2, pp. 199-224 | DOI | MR | Zbl
[76] Representation-finite self-injective algebras of class , 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] Many algebras with the same Auslander-Reiten quiver, Bull. London Math. Soc., Volume 15 (1983) no. 1, pp. 43-47 | DOI | MR | Zbl
[78] Representation-finite self-injective algebras of class , Compos. Math., Volume 49 (1983) no. 2, pp. 231-282 | Numdam | MR | Zbl
[79] Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, 1099, Springer-Verlag, Berlin, 1984 | MR | Zbl
[80] 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] Hereditary triangulated categories, Compos. Math. (2006), pp. (to appear)
[82] On the structure of Calabi-Yau categories with a cluster tilting subcategory, Doc. Math., Volume 12 (2007), p. 193-213 (electronic) | MR | Zbl
[83] Ph. D. (in preparation)
[84] Catégories dérivées. Quelques résultats, Lect. Notes Math., Volume 569 (1977), pp. 262-311 | DOI | Zbl
[85] 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) | Numdam | MR | Zbl
[86] Relations for the Grothendieck groups of triangulated categories, J. Algebra, Volume 257 (2002) no. 1, pp. 37-50 | DOI | MR | Zbl
[87] Locally finite triangulated categories, J. Algebra, Volume 290 (2005) no. 2, pp. 473-490 | DOI | MR | Zbl
[88] Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, 146, Cambridge University Press, Cambridge, 1990 | MR | Zbl
Cited by Sources: