Algèbres amassées et applications [d'après Fomin-Zelevinsky, ...]

Keller, Bernhard

Séminaire Bourbaki, volume 2009/2010, exposés 1012-1026, Astérisque, no. 339 (2011), Exposé no. 1014, 28 p.

Zbl

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     author = {Keller, Bernhard},
     title = {Alg\`ebres amass\'ees et applications [d'apr\`es {Fomin-Zelevinsky,} ...]},
     booktitle = {S\'eminaire Bourbaki, volume 2009/2010, expos\'es 1012-1026},
     series = {Ast\'erisque},
     note = {talk:1014},
     pages = {63--90},
     year = {2011},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {339},
     zbl = {1375.13034},
     language = {fr},
     url = {https://www.numdam.org/item/AST_2011__339__63_0/}
}

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Keller, Bernhard. Algèbres amassées et applications [d'après Fomin-Zelevinsky, ...], dans Séminaire Bourbaki, volume 2009/2010, exposés 1012-1026, Astérisque, no. 339 (2011), Exposé no. 1014, 28 p.. https://www.numdam.org/item/AST_2011__339__63_0/

Bibliographie
Cité par

[1] C. Amiot - Cluster categories for algebras of global dimension $2$ and quivers with potential, Ann. Inst. Fourier (Grenoble) 59 (2009), p. 2525-2590. | MR | Zbl | EuDML | Numdam | DOI

[2] A. Berenstein, S. Fomin & A. Zelevinsky - Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122 (1996), p. 49-149. | MR | Zbl | DOI

[3] A. Berenstein, S. Fomin & A. Zelevinsky, Cluster algebras. III. Upper bounds and double bruhat cells, Duke Math. J. 126 (2005), p. 1-52. | MR | Zbl | DOI

[4] A. Berenstein & A. Zelevinsky - Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), p. 77-128. | MR | Zbl | DOI

[5] A. Berenstein & A. Zelevinsky, Quantum cluster algebras, Adv. Math. 195 (2005), p. 405-455. | MR | Zbl | DOI

[6] T. Bridgeland - Stability conditions on triangulated categories, Ann. of Math. 166 (2007), p. 317-345. | MR | Zbl | DOI

[7] A. B. Buan, O. Iyama, I. Reiten & D. Smith - Mutation of cluster-tilting objects and potentials, à paraître dans Am. J. Math. | MR | Zbl

[8] A. B. Buan & R. J. Marsh - Cluster-tilting theory, in Trends in representation theory of algebras and related topics, Contemp. Math., vol. 406, Amer. Math. Soc., 2006, p. 1-30. | MR | Zbl | DOI

[9] A. B. Buan, R. J. Marsh, M. Reineke, I. Reiten & G. Todorov - Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), p. 572-618. | MR | Zbl | DOI

[10] A. B. Buan, R. J. Marsh & I. Reiten - Cluster Mutation via quiver representations, Comment Math. Helv. 83 (2008), p. 143-177. | MR | Zbl | DOI

[11] A. B. Buan, R. J. Marsh, I. Reiten & G. Todorov - Clusters and seeds in acyclic cluster algebras, Proc. Amer. Math. Soc. 135 (2007), p. 3049-3060, with an appendix coauthored in addition by P. Caldero and B. Keller. | MR | Zbl | DOI

[12] P. Caldero & F. Chapoton - Cluster Algebras as Hall Algebras of Quiver Representations, Comment. Math. Helv. 81 (2006), p. 595-616. | MR | Zbl | DOI

[13] P. Caldero, F. Chapoton & R. Schiffler - Quivers with relations arising from clusters ( $A_{n}$ case), Trans. Amer. Math. Soc. 358 (2006), p. 1347-1364. | MR | Zbl | DOI

[14] P. Caldero & B. Keller - From triangulated categories to cluster algebras. II, Ann. Sci. École Norm. Sup. 39 (2006), p. 983-1009. | MR | Zbl | EuDML | Numdam | DOI

[15] P. Caldero & B. Keller, From Triangulated categories to cluster algebras, Invent. Math. 172 (2008), p. 169-211. | MR | Zbl | DOI

[16] P. Caldero & M. Reineke - On the quiver grassmannian in the acyclic case, J. Pure Appl. Algebra 212 (2008), p. 2369-2380. | MR | Zbl | DOI

[17] G. Cerulli Irelli - Canonically positive basis of cluster algebras of type $A_{2}^{(1)}$ , prépublication arXiv:0904.2543. | MR

[18] F. Chapoton - Enumerative properties of generalized associahedra, Sém. Lothar. Combin. 51 (2004/05) , Art. B51b. | MR | Zbl | EuDML

[19] F. Chapoton, S. Fomin & A. Zelevinsky - Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002), p. 537-566. | MR | Zbl | DOI

[20] V. Chari & A. Pressley - Quantum affine algebras, Comm. Math. Phys. 142 (1991), p. 261-283. | MR | Zbl | DOI

[21] L. Demonet - Categorification of skew-symmetrizable cluster algebras, prépublication arXiv:0909.1633. | MR | Zbl | DOI

[22] H. Derksen, J. Weyman & A. Zelevinsky - Quivers with potentials and their representations I: Mutations, Selecta Mathematica 14 (2008), p. 59-119. | MR | Zbl | DOI

[23] H. Derksen, J. Weyman & A. Zelevinsky, Quivers with potentials and their representations II: Applications to cluster algebras, prépublication arXiv:0904.0676. | MR | Zbl | DOI

[24] P. Di Francesco & R. Kedem - $Q$ -systems as cluster algebras. II. Cartan matrix of finite type and the polynomial property, Lett. Math. Phys. 89 (2009), p. 183-216. | MR | Zbl | DOI

[25] P. Di Francesco & R. Kedem, Positivity of the $T$ -system cluster algebra, prépublication arXiv:0908.3122. | MR | Zbl

[26] G. Dupont - Generic variables in acyclic cluster algebras, prépublication arXiv:0811.2909. | MR | Zbl | DOI

[27] M. Fekete - Über ein problem von Laguerre, Rend. Circ. Mat. Palermo 34 (1912), p. 89-100, 110-120. | JFM | DOI

[28] A. Felikson, M. Shapiro & P. Tumarkin - Skew-symmetric cluster algebras of finite mutation type, prépublication arXiv:0811.1703. | Zbl | DOI

[29] B. Feng, A. Hanany, Y.-H. He & A. M. Uranga - Toric duality as Seiberg duality and brane diamonds, J. High Energy Phys. 12 (2001), Paper 35, 29. | MR

[30] V. V. Fock & A. B. Goncharov - Cluster $ℋ$ -varieties, amalgamation, and Poisson-Lie groups, in Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser, 2006, p. 27-68. | MR | Zbl | DOI

[31] V. V. Fock & A. B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), p. 1-211. | MR | Zbl | EuDML | Numdam | DOI

[32] V. V. Fock & A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Super. 42 (2009), p. 865-930. | MR | Zbl | EuDML | Numdam | DOI

[33] V. V. Fock & A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm. II. The intertwiner, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Birkhäuser, 2009, p. 655-673. | MR | Zbl | DOI

[34] V. V. Fock & A. B. Goncharov, The quantum dilogarithm and representations of quantum cluster Varieties, Invent. Math. 175 (2009), p. 223-286. | MR | Zbl | DOI

[35] S. Fomin - Cluster algebras portal,http://www.math.lsa.umich.edu/~fomin/cluster.html.

[36] S. Fomin & N. Reading - Generalized cluster complexes and coxeter combinatorics, Int. Math. Res. Not. 2005 (2005), p. 2709-2757. | MR | Zbl | DOI

[37] S. Fomin, M. Shapiro & D. Thurston - Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (2008), p. 83-146. | MR | Zbl | DOI

[38] S. Fomin & A. Zelevinsky - Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), p. 497-529. | MR | Zbl | DOI

[39] S. Fomin & A. Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), p. 63-121. | MR | Zbl | DOI

[40] S. Fomin & A. Zelevinsky, Cluster algebras: notes for the CDM-03 conference, in Current developments in mathematics, 2003, Int. Press, Somerville, MA, 2003, p. 1-34. | MR | Zbl

[41] S. Fomin & A. Zelevinsky, $Y$ -systems and Generalized Associahedra, Ann. of Math. 158 (2003), p. 977-1018. | MR | Zbl | DOI

[42] S. Fomin & A. Zelevinsky, Cluster Algebras. IV. Coefficients, Compos. Math. 143 (2007), p. 112-164. | MR | Zbl | DOI

[43] E. Frenkel & A. Szenes - Thermodynamic Bethe ansatz and dilogarithm identities. I, Math. Res. Lett. 2 (1995), p. 677-693. | MR | Zbl | DOI

[44] C. J. Fu & B. Keller - On cluster algebras with coefficients and $2$ -Calabi-Yau categories, Trans. Amer. Math. Soc. 362 (2010), p. 859-895. | MR | Zbl

[45] P. Gabriel - Représentations indécomposables, Séminaire Bourbaki, vol. 1973/1974, exp. n° 444, Lecture Notes in Math., vol. 431, Springer, 1975, p. 143-169. | MR | Zbl | EuDML | Numdam | DOI

[46] D. Gaiotto, G. W. Moore & A. Neitzke - Wall-crossing, Hitchin systems and the WKB approximation, prépublication arXiv:0907.3987. | MR | Zbl | DOI

[47] C. Geiss, B. Leclerc & J. Schröer - Semicanonical bases and preprojective algebras, Ann. Sci. École Norm. Sup. 38 (2005), p. 193-253. | MR | Zbl | EuDML | Numdam | DOI

[48] C. Geiss, B. Leclerc & J. Schröer, Rigid Modules over preprojective algebras, Invent. Math. 165 (2006), p. 589-632. | MR | Zbl | DOI

[49] C. Geiss, B. Leclerc & J. Schröer, Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (Grenoble) 58 (2008), p. 825-876. | MR | Zbl | EuDML | Numdam | DOI

[50] C. Geiss, B. Leclerc & J. Schröer, Preprojective algebras and cluster algebras, in Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, p. 253-283. | MR | Zbl | DOI

[51] C. Geiss, B. Leclerc & J. Schröer, Cluster Algebra structures and semicanonical bases for unipotent groups, prépublication arXiv:math/0703039.

[52] M. Gekhtman, M. Shapiro & A. Vainshtein - Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (2003), p. 899-934, 1199. | MR | Zbl | DOI

[53] M. Gekhtman, M. Shapiro & A. Vainshtein, Cluster algebras and Weil-Petersson forms, Duke Math. J. 127 (2005), p. 291-311. | MR | Zbl | DOI

[54] M. Gekhtman, M. Shapiro & A. Vainshtein, On the properties of the exchange graph of a cluster algebra, Math. Res. Lett. 15 (2008), p. 321-330. | MR | Zbl | DOI

[55] V. Ginzburg - Calabi-Yau algebras, prépublication arXiv:math/0612139.

[56] F. Gliozzi & R. Tateo - Thermodynamic Bethe ansatz and three-fold triangulations, Internat. J. Modern Phys. A 11 (1996), p. 4051-4064. | MR | Zbl | DOI

[57] A. Henriques - A periodicity theorem for the octahedron recurrence, J. Algebraic Combin. 26 (2007), p. 1-26. | MR | Zbl | DOI

[58] D. Hernandez & B. Leclerc - Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2010), p. 265-341. | MR | Zbl | DOI

[59] A. Hubery - Acyclic cluster algebras via Ringel-Hall algebras, prépublication http://www.maths.leeds.ac.uk/~ahubery/Cluster.pdf.

[60] C. Ingalls & H. Thomas - Noncrossing partitions and representations of quivers, Compos. Math. 145 (2009), p. 1533-1562. | MR | Zbl | DOI

[61] R. Inoue, O. Iyama, B. Keller, A. Kuniba & T. Nakanishi - Periodicities of $T$ and $Y$ -systems, dilogarithm identities, and cluster algebras I: type $B_{r}$ , prépublication arXiv:1001.1880. | MR | Zbl

[62] R. Inoue, O. Iyama, B. Keller, A. Kuniba & T. Nakanishi, Periodicities of $T$ and $Y$ -systems, dilogarithm identities, and cluster algebras II: types $C_{r}, F_{4}$ , and $G_{2}$ , prépublication arXiv:1001.1881. | MR | Zbl

[63] R. Inoue, O. Iyama, A. Kuniba, T. Nakanishi & J. Suzuki - Periodicities of $T$ -systems and $Y$ -systems, Nagoya Math. J. 197 (2010), p. 59-174. | MR | Zbl | DOI

[64] O. Iyama & I. Reiten - Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras, Amer. J. Math. 130 (2008), p. 1087-1149. | MR | Zbl | DOI

[65] O. Iyama & Y. Yoshino - Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), p. 117-168. | MR | Zbl | DOI

[66] D. Joyce & Y. Song - A Theory of generalized Donaldson-Thomas invariants. II. Multiplicative identities for Behrend functions, prépublication arXiv:0901.2872. | MR | Zbl

[67] V. G. Kac - Infinite-dimensional Lie algebras, third ed., Cambridge Univ. Press, 1990. | MR | Zbl

[68] M. Kashiwara - Bases cristallines, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), p. 277-280. | MR | Zbl

[69] R. Kedem - $Q$ -systems as cluster algebras, J. Phys. A 41 (2008), 194011, 14. | MR | Zbl | DOI

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

[71] B. Keller, Cluster algebras, quiver representations and triangulated categories, in Triangulated categories, London Math. Soc. Lecture Note Ser., vol. 375, Cambridge Univ. Press, 2010, p. 76-160. | MR | Zbl | DOI

[72] B. Keller, The periodicity conjecture for Pairs of Dynkin diagrams, prépublication arXiv: 1001.1531. | MR | Zbl | DOI

[73] B. Keller, Quiver mutation in Java, applet Java http://www.math.jussieu.fr/~keller/quivermutation.

[74] B. Keller & D. Yang - Derived equivalences from mutations of quivers with potential, Advances in Math. 226 (2011), p. 2118-2168. | MR | Zbl | DOI

[75] M. Kontsevich & Y. Soibelman - Stability structures, Donaldson-Thomas invariants and cluster transformations, prépublication arXiv:0811.2435. | MR

[76] C. Krattenthaler - The $F$ -triangle of the generalised cluster complex, in Topics in discrete mathematics, Algorithms Combin., vol. 26, Springer, 2006, p. 93-126. | MR | Zbl | DOI

[77] A. Kuniba & T. Nakanishi - Spectra in conformal field theories from the Rogers dilogarithm, Modern Phys. Lett. A 7 (1992), p. 3487-3494. | MR | Zbl | DOI

[78] A. Kuniba, T. Nakanishi & J. Suzuki - Functional Relations in Solvable Lattice Models. I. Functional Relations and Representation Theory, Internat. J. Modern Phys. A 9 (1994), p. 5215-5266. | MR | Zbl | DOI

[79] D. Labardini-Fragoso - Quivers with potentials associated to triangulated surfaces, Proc. Lond. Math. Soc. 98 (2009), p. 797-839. | MR | Zbl | DOI

[80] B. Leclerc - Algèbres affines quantiques et algèbres amassées, notes d'un exposé au séminaire d'algèbre à l'institut Henri Poincaré le 14 janvier 2008.

[81] G. Lusztig - Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), p. 447-498. | MR | Zbl | DOI

[82] G. Lusztig, Total positivity in reductive groups, in Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser, 1994, p. 531-568. | MR | Zbl | DOI

[83] G. Lusztig, Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000), p. 129-139. | MR | Zbl | DOI

[84] R. J. Marsh, M. Reineke & A. Zelevinsky - Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), p. 4171-4186. | MR | Zbl | DOI

[85] G. Musiker - A Graph theoretic expansion formula for cluster algebras of type $B_{n}$ and $D_{n}$ , prepublication arXiv:0710.3574. | MR | Zbl

[86] G. Musiker, R. Schiffler & L. Williams - Positivity for cluster algebras from surfaces, prepublication arXiv:0906.0748. | MR | Zbl | DOI

[87] H. Nakajima - Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), p. 145-238. | MR | Zbl | DOI

[88] H. Nakajima, Quiver varieties and cluster algebras, prepublication arXiv:0905.0002. | MR | Zbl | DOI

[89] Y. Palu - Cluster characters for $2$ -Calabi-Yau triangulated categories, Ann. Inst. Fourier (Grenoble) 58 (2008), p. 2221-2248. | MR | Zbl | EuDML | Numdam | DOI

[90] F. Ravanini, A. Valleriani & R. Tateo - Dynkin TBAs, Internat. J. Modern Phys. A 8 (1993), p. 1707-1727. | MR | DOI

[91] M. Reineke - Cohomology of quiver moduli, functional equations & integrality of Donaldson-Thomas type invariants, prepublication arXiv:0903.0261. | MR | Zbl | DOI

[92] I. Reiten - Tilting theory and cluster algebras, prepublication http://www.institut.math.jussieu.fr/~keller/ictp2006/lecturenotes/reiten.pdf.

[93] C. M. Ringel - Some remarks concerning tilting modules and tilted algebras. Origin. Relevance. Future, in Handbook of tilting theory, London Math. Soc. Lecture Note Ser., vol. 332, Cambridge Univ. Press, 2007, p. 49-104. | MR

[94] G.-C. Rota, B. Sagan & P. R. Stein - A Cyclic derivative in noncommutative algebra, J. Algebra 64 (1980), p. 54-75. | MR | Zbl | DOI

[95] J. S. Scott - Grassmannians and cluster algebras, Proc. London Math. Soc. 92 (2006), p. 345-380. | MR | Zbl | DOI

[96] P. Sherman & A. Zelevinsky - Positivity and canonical bases in rank $2$ cluster algebras of finite and affine types, Mosc. Math. J. 4 (2004), p. 947-974, 982. | MR | Zbl | DOI

[97] J. D. Stasheff - Homotopy associativity of $H$ -spaces. I, Trans. Amer. Math. Soc. 108 (1963), p. 275-292 | MR | Zbl

J. D. Stasheff - Homotopy associativity of $H$ -spaces. II, Trans. Amer. Math. Soc. 108 (1963), p. 293-312. | MR | Zbl

[98] A. Szenes - Periodicity of $Y$ -systems and flat connections, Lett. Math. Phys. 89 (2009), p. 217-230. | MR | Zbl | DOI

[99] M. Varagnolo & E. Vasserot - Perverse sheaves and quantum Grothendieck rings, in Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), Progr. Math., vol. 210, Birkhäuser, 2003, p. 345-365. | MR | Zbl | DOI

[100] A. Y. Volkov - On the periodicity conjecture for $Y$ -systems, Comm. Math. Phys. 276 (2007), p. 509-517. | MR | Zbl | DOI

[101] J. Xiao & F. Xu - Green's formula with $𝐂^{*}$ -action and Caldero-Keller's formula for cluster algebras, prepublication arXiv:0707.1175. | MR | Zbl

[102] S. W. Yang & A. Zelevinsky - Cluster algebras of finite type via Coxeter elements and principal minors, Transform. Groups 13 (2008), p. 855-955. | MR | Zbl | DOI

[103] A. B. Zamolodchikov - On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991), p. 391-394. | MR | DOI

[104] A. Zelevinsky - From Littlewood-Richardson coefficients to cluster algebras in three lectures, in Symmetric functions 2001: surveys of developments and perspectives, NATO Sci. Ser. II Math. Phys. Chem., vol. 74, Kluwer Acad. Publ., 2002, p. 253-273. | MR | Zbl

[105] A. Zelevinsky, Cluster Algebras: origins, results and conjectures, in Advances in algebra towards millennium problems, SAS Int. Publ., Delhi, 2005, p. 85-105. | MR | Zbl

[106] A. Zelevinsky, What is . . . a Cluster Algebra?, Notices Amer. Math. Soc. 54 (2007), p. 1494-1495. | MR | Zbl

[107] A. Zelevinsky, Cluster Algebras, notes for 2004 IMCC, prepublication arXiv:math.RT/0407414. | Zbl | MR