Variétés carquois de Nakajima [d'après Nakajima, Lusztig, Varagnolo, Vasserot, Crawley-Boevey, ...]
Séminaire Bourbaki - Volume 2006/2007 - Exposés 967-981, Astérisque, no. 317 (2008), Talk no. 976, 50 p.
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Schiffmann, Olivier. Variétés carquois de Nakajima [d'après Nakajima, Lusztig, Varagnolo, Vasserot, Crawley-Boevey, ...], in Séminaire Bourbaki - Volume 2006/2007  - Exposés 967-981, Astérisque, no. 317 (2008), Talk no. 976, 50 p. http://archive.numdam.org/item/AST_2008__317__295_0/

[1] M. F. Atiyah, N. J. Hitchin, V. G. Drinfel'D & Y. I. Manin - Construction of instantons, Phys. Lett. A 65 (1978), p. 185-187. | DOI | MR | Zbl

[2] V. Baranovsky, V. Ginzburg & A. Kuznetsov - Quiver varieties and a noncommutative 2 , Compositio Math. 134 (2002), p. 283-318. | DOI | MR | Zbl

[3] R. Bezrukavnikov, I. Mirković & D. Rumynin - Localization of modules for a semisimple Lie algebra in prime characteristic, à paraître dans Ann. of Math. | MR | Zbl

[4] A. Braverman & D. Gaitsgory - Crystals via the affine Grassmannian, Duke Math. J. 107 (2001), p. 561-575. | DOI | MR | Zbl

[5] J. Briançon - Description de Hilb n C{x,y}, Invent. Math. 41 (1977), p. 45-89. | DOI | EuDML | MR | Zbl

[6] V. Chari & A. Pressley - Quantum affine algebras and their representations, in Representations of groups (Banff, AB, 1994), CMS Conf. Proc., vol. 16, Amer. Math. Soc., 1995, p. 59-78. | MR | Zbl

[7] N. Chriss & V. Ginzburg - Representation theory and complex geometry, Birkhäuser, 1997. | MR | Zbl

[8] W. Crawley-Boevey - Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), p. 257-293. | DOI | MR | Zbl

[9] W. Crawley-Boevey, Normality of Marsden-Weinstein reductions for representations of quivers, Math. Ann. 325 (2003), p. 55-79. | DOI | MR | Zbl

[10] W. Crawley-Boevey & M. P. Holland - Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), p. 605-635. | DOI | MR | Zbl

[11] V. G. Drinfel'D - A new realization of Yangians and of quantum affine algebras, Dokl. Akad. Nauk SSSR 296 (1987), p. 13-17. | MR | Zbl

[12] G. Ellingsrud & S. A. Stromme - On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87 (1987), p. 343-352. | DOI | EuDML | MR | Zbl

[13] J. Fogarty - Algebraic families on an algebraic surface, Amer. J. Math 90 (1968), p. 511-521. | DOI | MR | Zbl

[14] E. Frenkel & N. Reshetikhin - The q-characters of representations of quantum affine algebras and deformations of 𝒲-algebras, in Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), Contemp. Math., vol. 248, Amer. Math. Soc;, 1999, p. 163-205. | DOI | MR | Zbl

[15] I. Frenkel, M. Khovanov & O. Schiffmann - Homological realization of Nakajima varieties and Weyl group actions, Compos. Math. 141 (2005), p. 1479-1503. | DOI | MR | Zbl

[16] V. Ginzburg - Deligne-Langlands conjecture and representations of affine Hecke algebras, prépublication Moscow Univ., 1985.

[17] V. Ginzburg & M. Kapranov - Hilbert schemes and Nakajima's quiver varieties, non publié, 1995.

[18] V. Ginzburg &, É. Vasserot - Langlands reciprocity for affine quantum groups of type A n , Internat. Math. Res. Notices 3 (1993), p. 67-85. | DOI | MR | Zbl

[19] G. Gonzalez-Sprinberg & J.-L. Verdier - Construction géométrique de la correspondance de McKay, Ann. Sci. École Norm. Sup. (4) 16 (1983), p. 409-449 (1984). | EuDML | Numdam | MR | Zbl

[20] L. Göttsche - The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990), p. 193-207. | DOI | EuDML | MR | Zbl

[21] I. Grojnowski - Instantons and affine algebras. I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), p. 275-291. | DOI | MR | Zbl

[22] G. Hatayama, A. Kuniba, M. Okado, T. Takagi & Z. Tsuboi - Paths, crystals and fermionic formulae, in MathPhys odyssey, 2001, Prog. Math. Phys., vol. 23, Birkhäuser, 2002, p. 205-272. | DOI | MR | Zbl

[23] T. Hausel - Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform, Proc. Natl. Acad. Sci. USA 103 (2006), p. 6120-6124. | DOI | MR | Zbl

[24] D. Hernandez - Algebraic approach to q,t-characters, Adv. Math. 187 (2004), p. 1-52. | DOI | MR | Zbl

[25] D. Hernandez, The Kirillov-Reshetikhin conjecture and solutions of T-systems, J. reine angew. Math. 596 (2006), p. 63-87. | MR | Zbl

[26] D. Hernandez & H. Nakajima - Level 0 monomial crystals, Nagoya Math. J. 184 (2006), p. 85-153. | DOI | MR | Zbl

[27] J. Hong & S.-J. Kang - Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, 2002. | DOI | MR | Zbl

[28] Y. Ito & I. Nakamura - McKay correspondence and Hilbert schemes, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), p. 135-138. | DOI | MR | Zbl

[29] A. Joseph - Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 29, Springer, 1995. | MR | Zbl

[30] V. Kac - Infinite-dimensional Lie algebras, third éd., Cambridge University Press, 1990. | MR | Zbl

[31] V. Kac & A. K. Raina - Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, vol. 2, World Scientific Publishing Co. Inc., 1987. | MR | Zbl

[32] J. Kamnitzer - Crystal structures on Mirković-Vilonen polytopes, prépublication arXiv:math.QA/0505398, 2005.

[33] M. Kapranov - Eisenstein series and quantum affine algebras, J. Math. Sci. (New York) 84 (1997), p. 1311-1360, Algebraic geometry, 7. | DOI | MR | Zbl

[34] M. Kashiwara - Bases cristallines des groupes quantiques, Cours Spécialisés, vol. 9, Société Mathématique de France, 2002, Edited by Charles Cochet. | MR | Zbl

[35] M. Kashiwara, On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), p. 117-175. | DOI | MR | Zbl

[36] M. Kashiwara, Realizations of crystals, in Combinatorial and geometric representation theory (Seoul, 2001), Contemp. Math., vol. 325, Amer. Math. Soc., 2003, p. 133-139. | DOI | MR | Zbl

[37] M. Kashiwara & Y. Saito - Geometric construction of crystal bases, Duke Math. J. 89 (1997), p. 9-36. | DOI | MR | Zbl

[38] D. Kazhdan & G. Lusztig - Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), p. 153-215. | DOI | EuDML | MR | Zbl

[39] A. N. Kirillov & N. Reshetikhin - Representations of Yangians and multiplicities of the inclusion of the irreducible components of the tensor product of representations of simple Lie algebras, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 160 (1987), p. 211-221, 301. | MR | Zbl

[40] P. Kronheimer - The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), p. 665-683. | DOI | MR | Zbl

[41] P. Kronheimer & H. Nakajima - Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), p. 263-307. | DOI | EuDML | MR | Zbl

[42] A. Kuniba, T. Nakanishi & J. Suzuki - Functional relations in solvable lattice models. I. Functional relations and representation theory, Internat. J. Modem Phys. A 9 (1994), p. 5215-5266. | DOI | MR | Zbl

[43] M. Lehn & C. Sorger - Symmetric groups and the cup product on the cohomology of Hilbert schemes, Duke Math. J. 110 (2001), p. 345-357. | DOI | MR | Zbl

[44] P. Littelmann - Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), p. 499-525. | DOI | MR | Zbl

[45] G. Lusztig - Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), p. 365-421. | DOI | MR | Zbl

[46] G. Lusztig, Affine quivers and canonical bases, Publ. Math. I.H.É.S. 76 (1992), p. 111-163. | DOI | EuDML | Numdam | MR | Zbl

[47] G. Lusztig, Bases in equivariant K-theory, Represent. Theory 2 (1998), p. 298-369. | DOI | MR | Zbl

[48] G. Lusztig, Fermionic forms and Betti numbers, prépublication arXiv:math.QA/0005010, 2000.

[49] G. Lusztig, Quiver varieties and Weyl group actions, Ann. Inst. Fourier (Grenoble) 50 (2000), p. 461-489. | DOI | EuDML | Numdam | MR | Zbl

[50] G. Lusztig, Remarks on quiver varieties, Duke Math. J. 105 (2000), p. 239-265. | DOI | MR | Zbl

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

[52] A. Maffei - Quiver varieties of type A, Comment. Math. Helv. 80 (2005), p. 1-27. | DOI | MR | Zbl

[53] A. Malkin - Tensor product varieties and crystals : the ADE case, Duke Math. J. 116 (2003), p. 477-524. | DOI | MR | Zbl

[54] I. Mirković & K. Vilonen - Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett. 7 (2000), p. 13-24. | DOI | MR | Zbl

[55] I. Mirković & M. Vybornov - On quiver varieties and affine Grassmannians of type A, C. R. Math. Acad. Sci. Paris 336 (2003), p. 207-212. | DOI | MR | Zbl

[56] S. Mozgovoy - Fermionic forms and quiver varieties, prépublication arXiv:math.QA/0610084, 2006.

[57] S. Mozgovoy, On the multiplicities of the irreducible highest weight modules over Kac-Moody algebras, prépublication arXiv:math.RT/0609349, 2006.

[58] D. Mumford, J. Fogarty & F. Kirwan - Geometric invariant theory, third éd., Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge, vol. 34, Springer, 1994. | MR | Zbl

[59] H. Nakajima - A geometric construction of algebras, Lectures at the University of Hong Kong http://www.math.kyoto-u.ac.jp/~nakajima/bibli.html.

[60] H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), p. 365-416. | MR | Zbl

[61] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2) 145 (1997), p. 379-388. | DOI | MR | Zbl

[62] H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), p. 515-560. | DOI | MR | Zbl

[63] H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, 1999. | DOI | MR | Zbl

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

[65] H. Nakajima, Quiver varieties and McKay correspondence, Lectures at Hokkaido University http://www.math.kyoto-u.ac.jp/~nakajima/bibli.html, 2001.

[66] H. Nakajima, Quiver varieties and tensor products, Invent. Math. 146 (2001), p. 399-449. | DOI | MR | Zbl

[67] H. Nakajima, Reflection functors for quiver varieties and Weyl group actions, Math. Ann. 327 (2003), p. 671-721. | DOI | MR | Zbl

[68] H. Nakajima, t-analogs of q-characters of quantum affine algebras of type An,Dn, in Combinatorial and geometric representation theory (Seoul, 2001), Contemp. Math., vol. 325, Amer. Math. Soc., 2003, p. 141-160. | MR | Zbl

[69] H. Nakajima, Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math. (2) 160 (2004), p. 1057-1097. | DOI | MR | Zbl

[70] C. Okonek, M. Schneider & H. Spindler - Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, 1980. | MR | Zbl

[71] C. M. Ringel - Hall algebras and quantum groups, Invent. Math. 101 (1990), p. 583-591. | DOI | EuDML | MR | Zbl

[72] Y. Saito - Crystal bases and quiver varieties, Math. Ann. 324 (2002), p. 675-688. | DOI | MR | Zbl

[73] O. Schiffmann - Noncommutative projective curves and quantum loop algebras, Duke Math. J. 121 (2004), p. 113-168. | DOI | MR | Zbl

[74] P. Slodowy - Simple singularities and simple algebraic groups, Lecture Notes in Mathematies, vol. 815, Springer, 1980. | MR | Zbl

[75] C. Vafa & E. Witten - A strong coupling test of S-duality, Nuclear Phys. B 431 (1994), p. 3-77. | DOI | MR | Zbl

[76] M. Varagnolo - Quiver varieties and Yangians, Lett. Math. Phys. 53 (2000), p. 273-283. | DOI | MR | Zbl

[77] M. Varagnolo & É. Vasserot - On the K-theory of the cyclic quiver variety, Internat. Math. Res. Notices 18 (1999), p. 1005-1028. | DOI | MR | Zbl

[78] M. Varagnolo & É. Vasserot, Standard modules of quantum affine algebras, Duke Math. J. 111 (2002), p. 509-533. | DOI | MR | Zbl

[79] M. Varagnolo & É. Vasserot, Canonical bases and quiver varieties, Represent. Theory 7 (2003), p. 227-258. | DOI | MR | Zbl

[80] M. Varagnolo & É. 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. | DOI | MR | Zbl

[81] É. Vasserot - Affine quantum groups and equivariant K-theory, Transform. Groups 3 (1998), p. 269-299. | DOI | MR | Zbl

[82] É. Vasserot, Sur l'anneau de cohomologie du schéma de Hilbert de 𝐂 2 , C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), p. 7-12. | DOI | MR | Zbl