@incollection{AST_2011__339__63_0, 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}, publisher = {Soci\'et\'e math\'ematique de France}, number = {339}, year = {2011}, zbl = {1375.13034}, language = {fr}, url = {http://www.numdam.org/item/AST_2011__339__63_0/} }
TY - CHAP AU - Keller, Bernhard TI - Algèbres amassées et applications [d'après Fomin-Zelevinsky, ...] BT - Séminaire Bourbaki, volume 2009/2010, exposés 1012-1026 AU - Collectif T3 - Astérisque N1 - talk:1014 PY - 2011 SP - 63 EP - 90 IS - 339 PB - Société mathématique de France UR - http://www.numdam.org/item/AST_2011__339__63_0/ LA - fr ID - AST_2011__339__63_0 ER -
%0 Book Section %A Keller, Bernhard %T Algèbres amassées et applications [d'après Fomin-Zelevinsky, ...] %B Séminaire Bourbaki, volume 2009/2010, exposés 1012-1026 %A Collectif %S Astérisque %Z talk:1014 %D 2011 %P 63-90 %N 339 %I Société mathématique de France %U http://www.numdam.org/item/AST_2011__339__63_0/ %G fr %F AST_2011__339__63_0
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. http://www.numdam.org/item/AST_2011__339__63_0/
[1] Cluster categories for algebras of global dimension and quivers with potential, Ann. Inst. Fourier (Grenoble) 59 (2009), p. 2525-2590. | DOI | EuDML | Numdam | MR | Zbl
-[2] Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122 (1996), p. 49-149. | DOI | MR | Zbl
, & -[3] Cluster algebras. III. Upper bounds and double bruhat cells, Duke Math. J. 126 (2005), p. 1-52. | DOI | MR | Zbl
, & ,[4] Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), p. 77-128. | DOI | MR | Zbl
& -[5] Quantum cluster algebras, Adv. Math. 195 (2005), p. 405-455. | DOI | MR | Zbl
& ,[6] Stability conditions on triangulated categories, Ann. of Math. 166 (2007), p. 317-345. | DOI | MR | Zbl
-[7] Mutation of cluster-tilting objects and potentials, à paraître dans Am. J. Math. | MR | Zbl
, , & -[8] Cluster-tilting theory, in Trends in representation theory of algebras and related topics, Contemp. Math., vol. 406, Amer. Math. Soc., 2006, p. 1-30. | DOI | MR | Zbl
& -[9] Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), p. 572-618. | DOI | MR | Zbl
, , , & -[10] Cluster Mutation via quiver representations, Comment Math. Helv. 83 (2008), p. 143-177. | DOI | MR | Zbl
, & -[11] 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. | DOI | MR | Zbl
, , & -[12] Cluster Algebras as Hall Algebras of Quiver Representations, Comment. Math. Helv. 81 (2006), p. 595-616. | DOI | MR | Zbl
& -[13] Quivers with relations arising from clusters ( case), Trans. Amer. Math. Soc. 358 (2006), p. 1347-1364. | DOI | MR | Zbl
, & -[14] From triangulated categories to cluster algebras. II, Ann. Sci. École Norm. Sup. 39 (2006), p. 983-1009. | DOI | EuDML | Numdam | MR | Zbl
& -[15] From Triangulated categories to cluster algebras, Invent. Math. 172 (2008), p. 169-211. | DOI | MR | Zbl
& ,[16] On the quiver grassmannian in the acyclic case, J. Pure Appl. Algebra 212 (2008), p. 2369-2380. | DOI | MR | Zbl
& -[17] Canonically positive basis of cluster algebras of type , prépublication arXiv:0904.2543. | MR
-[18] Enumerative properties of generalized associahedra, Sém. Lothar. Combin. 51 (2004/05) , Art. B51b. | EuDML | MR | Zbl
-[19] Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002), p. 537-566. | DOI | MR | Zbl
, & -[20] Quantum affine algebras, Comm. Math. Phys. 142 (1991), p. 261-283. | DOI | MR | Zbl
& -[21] Categorification of skew-symmetrizable cluster algebras, prépublication arXiv:0909.1633. | DOI | MR | Zbl
-[22] Quivers with potentials and their representations I: Mutations, Selecta Mathematica 14 (2008), p. 59-119. | DOI | MR | Zbl
, & -[23] Quivers with potentials and their representations II: Applications to cluster algebras, prépublication arXiv:0904.0676. | DOI | MR | Zbl
, & ,[24] -systems as cluster algebras. II. Cartan matrix of finite type and the polynomial property, Lett. Math. Phys. 89 (2009), p. 183-216. | DOI | MR | Zbl
& -[25] Positivity of the -system cluster algebra, prépublication arXiv:0908.3122. | MR | Zbl
& ,[26] Generic variables in acyclic cluster algebras, prépublication arXiv:0811.2909. | DOI | MR | Zbl
-[27] Über ein problem von Laguerre, Rend. Circ. Mat. Palermo 34 (1912), p. 89-100, 110-120. | DOI | JFM
-[28] Skew-symmetric cluster algebras of finite mutation type, prépublication arXiv:0811.1703. | DOI | Zbl
, & -[29] Toric duality as Seiberg duality and brane diamonds, J. High Energy Phys. 12 (2001), Paper 35, 29. | MR
, , & -[30] Cluster -varieties, amalgamation, and Poisson-Lie groups, in Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser, 2006, p. 27-68. | DOI | MR | Zbl
& -[31] Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), p. 1-211. | DOI | EuDML | Numdam | MR | Zbl
& ,[32] Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Super. 42 (2009), p. 865-930. | DOI | EuDML | Numdam | MR | Zbl
& ,[33] 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. | DOI | MR | Zbl
& ,[34] The quantum dilogarithm and representations of quantum cluster Varieties, Invent. Math. 175 (2009), p. 223-286. | DOI | MR | Zbl
& ,[35] Cluster algebras portal,http://www.math.lsa.umich.edu/~fomin/cluster.html.
-[36] Generalized cluster complexes and coxeter combinatorics, Int. Math. Res. Not. 2005 (2005), p. 2709-2757. | DOI | MR | Zbl
& -[37] Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (2008), p. 83-146. | DOI | MR | Zbl
, & -[38] Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), p. 497-529. | DOI | MR | Zbl
& -[39] Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), p. 63-121. | DOI | MR | Zbl
& ,[40] 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] -systems and Generalized Associahedra, Ann. of Math. 158 (2003), p. 977-1018. | DOI | MR | Zbl
& ,[42] Cluster Algebras. IV. Coefficients, Compos. Math. 143 (2007), p. 112-164. | DOI | MR | Zbl
& ,[43] Thermodynamic Bethe ansatz and dilogarithm identities. I, Math. Res. Lett. 2 (1995), p. 677-693. | DOI | MR | Zbl
& -[44] On cluster algebras with coefficients and -Calabi-Yau categories, Trans. Amer. Math. Soc. 362 (2010), p. 859-895. | MR | Zbl
& -[45] Représentations indécomposables, Séminaire Bourbaki, vol. 1973/1974, exp. n° 444, Lecture Notes in Math., vol. 431, Springer, 1975, p. 143-169. | DOI | EuDML | Numdam | MR | Zbl
-[46] Wall-crossing, Hitchin systems and the WKB approximation, prépublication arXiv:0907.3987. | DOI | MR | Zbl
, & -[47] Semicanonical bases and preprojective algebras, Ann. Sci. École Norm. Sup. 38 (2005), p. 193-253. | DOI | EuDML | Numdam | MR | Zbl
, & -[48] Rigid Modules over preprojective algebras, Invent. Math. 165 (2006), p. 589-632. | DOI | MR | Zbl
, & ,[49] Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (Grenoble) 58 (2008), p. 825-876. | DOI | EuDML | Numdam | MR | Zbl
, & ,[50] 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. | DOI | MR | Zbl
, & ,[51] Cluster Algebra structures and semicanonical bases for unipotent groups, prépublication arXiv:math/0703039.
, & ,[52] Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (2003), p. 899-934, 1199. | DOI | MR | Zbl
, & -[53] Cluster algebras and Weil-Petersson forms, Duke Math. J. 127 (2005), p. 291-311. | DOI | MR | Zbl
, & ,[54] On the properties of the exchange graph of a cluster algebra, Math. Res. Lett. 15 (2008), p. 321-330. | DOI | MR | Zbl
, & ,[55] Calabi-Yau algebras, prépublication arXiv:math/0612139.
-[56] Thermodynamic Bethe ansatz and three-fold triangulations, Internat. J. Modern Phys. A 11 (1996), p. 4051-4064. | DOI | MR | Zbl
& -[57] A periodicity theorem for the octahedron recurrence, J. Algebraic Combin. 26 (2007), p. 1-26. | DOI | MR | Zbl
-[58] Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2010), p. 265-341. | DOI | MR | Zbl
& -[59] Acyclic cluster algebras via Ringel-Hall algebras, prépublication http://www.maths.leeds.ac.uk/~ahubery/Cluster.pdf.
-[60] Noncrossing partitions and representations of quivers, Compos. Math. 145 (2009), p. 1533-1562. | DOI | MR | Zbl
& -[61] Periodicities of and -systems, dilogarithm identities, and cluster algebras I: type , prépublication arXiv:1001.1880. | MR | Zbl
, , , & -[62] Periodicities of and -systems, dilogarithm identities, and cluster algebras II: types , and , prépublication arXiv:1001.1881. | MR | Zbl
, , , & ,[63] Periodicities of -systems and -systems, Nagoya Math. J. 197 (2010), p. 59-174. | DOI | MR | Zbl
, , , & -[64] Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras, Amer. J. Math. 130 (2008), p. 1087-1149. | DOI | MR | Zbl
& -[65] Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), p. 117-168. | DOI | MR | Zbl
& -[66] A Theory of generalized Donaldson-Thomas invariants. II. Multiplicative identities for Behrend functions, prépublication arXiv:0901.2872. | MR | Zbl
& -[67] Infinite-dimensional Lie algebras, third ed., Cambridge Univ. Press, 1990. | MR | Zbl
-[68] Bases cristallines, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), p. 277-280. | MR | Zbl
-[69] -systems as cluster algebras, J. Phys. A 41 (2008), 194011, 14. | DOI | MR | Zbl
-[70] On triangulated orbit categories, Doc. Math. 10 (2005), p. 551-581. | EuDML | MR | Zbl
-[71] 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. | DOI | MR | Zbl
,[72] The periodicity conjecture for Pairs of Dynkin diagrams, prépublication arXiv: 1001.1531. | DOI | MR | Zbl
,[73] Quiver mutation in Java, applet Java http://www.math.jussieu.fr/~keller/quivermutation.
,[74] Derived equivalences from mutations of quivers with potential, Advances in Math. 226 (2011), p. 2118-2168. | DOI | MR | Zbl
& -[75] Stability structures, Donaldson-Thomas invariants and cluster transformations, prépublication arXiv:0811.2435. | MR
& -[76] The -triangle of the generalised cluster complex, in Topics in discrete mathematics, Algorithms Combin., vol. 26, Springer, 2006, p. 93-126. | DOI | MR | Zbl
-[77] Spectra in conformal field theories from the Rogers dilogarithm, Modern Phys. Lett. A 7 (1992), p. 3487-3494. | DOI | MR | Zbl
& -[78] Functional Relations in Solvable Lattice Models. I. Functional Relations and Representation Theory, Internat. J. Modern Phys. A 9 (1994), p. 5215-5266. | DOI | MR | Zbl
, & -[79] Quivers with potentials associated to triangulated surfaces, Proc. Lond. Math. Soc. 98 (2009), p. 797-839. | DOI | MR | Zbl
-[80] Algèbres affines quantiques et algèbres amassées, notes d'un exposé au séminaire d'algèbre à l'institut Henri Poincaré le 14 janvier 2008.
-[81] Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), p. 447-498. | DOI | MR | Zbl
-[82] Total positivity in reductive groups, in Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser, 1994, p. 531-568. | DOI | MR | Zbl
,[83] Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000), p. 129-139. | DOI | MR | Zbl
,[84] Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), p. 4171-4186. | DOI | MR | Zbl
, & -[85] A Graph theoretic expansion formula for cluster algebras of type and , prepublication arXiv:0710.3574. | MR | Zbl
-[86] Positivity for cluster algebras from surfaces, prepublication arXiv:0906.0748. | DOI | MR | Zbl
, & -[87] Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), p. 145-238. | DOI | MR | Zbl
-[88] Quiver varieties and cluster algebras, prepublication arXiv:0905.0002. | DOI | MR | Zbl
,[89] Cluster characters for -Calabi-Yau triangulated categories, Ann. Inst. Fourier (Grenoble) 58 (2008), p. 2221-2248. | DOI | EuDML | Numdam | MR | Zbl
-[90] Dynkin TBAs, Internat. J. Modern Phys. A 8 (1993), p. 1707-1727. | DOI | MR
, & -[91] Cohomology of quiver moduli, functional equations & integrality of Donaldson-Thomas type invariants, prepublication arXiv:0903.0261. | DOI | MR | Zbl
-[92] Tilting theory and cluster algebras, prepublication http://www.institut.math.jussieu.fr/~keller/ictp2006/lecturenotes/reiten.pdf.
-[93] 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] A Cyclic derivative in noncommutative algebra, J. Algebra 64 (1980), p. 54-75. | DOI | MR | Zbl
, & -[95] Grassmannians and cluster algebras, Proc. London Math. Soc. 92 (2006), p. 345-380. | DOI | MR | Zbl
-[96] Positivity and canonical bases in rank cluster algebras of finite and affine types, Mosc. Math. J. 4 (2004), p. 947-974, 982. | DOI | MR | Zbl
& -[97] Homotopy associativity of -spaces. I, Trans. Amer. Math. Soc. 108 (1963), p. 275-292 | MR | Zbl
-Homotopy associativity of -spaces. II, Trans. Amer. Math. Soc. 108 (1963), p. 293-312. | MR | Zbl
-[98] Periodicity of -systems and flat connections, Lett. Math. Phys. 89 (2009), p. 217-230. | DOI | MR | Zbl
-[99] 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. | DOI | MR | Zbl
& -[100] On the periodicity conjecture for -systems, Comm. Math. Phys. 276 (2007), p. 509-517. | DOI | MR | Zbl
-[101] Green's formula with -action and Caldero-Keller's formula for cluster algebras, prepublication arXiv:0707.1175. | MR | Zbl
& -[102] Cluster algebras of finite type via Coxeter elements and principal minors, Transform. Groups 13 (2008), p. 855-955. | DOI | MR | Zbl
& -[103] On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991), p. 391-394. | DOI | MR
-[104] 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] Cluster Algebras: origins, results and conjectures, in Advances in algebra towards millennium problems, SAS Int. Publ., Delhi, 2005, p. 85-105. | MR | Zbl
,[106] What is . . . a Cluster Algebra?, Notices Amer. Math. Soc. 54 (2007), p. 1494-1495. | MR | Zbl
,[107] Cluster Algebras, notes for 2004 IMCC, prepublication arXiv:math.RT/0407414. | MR | Zbl
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