Smallness problem for quantum affine algebras and quiver varieties

Hernandez, David

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 2, pp. 271-306

The geometric small property (Borho-MacPherson [2]) of projective morphisms implies a description of their singularities in terms of intersection homology. In this paper we solve the smallness problem raised by Nakajima [37, 35] for certain resolutions of quiver varieties [37] (analogs of the Springer resolution): for Kirillov-Reshetikhin modules of simply-laced quantum affine algebras, we characterize explicitly the Drinfeld polynomials corresponding to the small resolutions. We use an elimination theorem for monomials of Frenkel-Reshetikhin $q$-characters that we establish for non necessarily simply-laced quantum affine algebras. We also refine results of [21] and extend the main result to general simply-laced quantum affinizations, in particular to quantum toroidal algebras (double affine quantum algebras).

La propriété géométrique de petitesse (Borho-MacPherson [2]) des morphismes projectifs implique une description de leurs singularités en termes d’homologie d’intersection. Dans cet article nous résolvons le problème de petitesse posé par Nakajima [37, 35] pour certaines résolutions de variétés carquois [37] (analogues de la résolution de Springer) : pour les modules de Kirillov-Reshetikhin des algèbres affines quantiques simplement lacées, nous caractérisons explicitement les polynômes de Drinfeld correspondant aux résolutions petites. Nous utilisons un théorème d’élimination pour les monômes des $q$-caractères de Frenkel-Reshetikhin, que nous établissons pour les algèbres affines quantiques non nécessairement simplement lacées. Nous raffinons également des résultats de [21] et étendons le résultat principal aux affinisées quantiques générales simplement lacées, en particulier aux algèbres toroïdales quantiques (algèbres quantiques doublement affines).

Classification : 17B37, 14L30, 81R50, 82B23, 17B67

[1] A. A. Beilinson, J. Bernstein & P. Deligne, Faisceaux pervers, in Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque 100, Soc. Math. France, 1982, 5-171. | Zbl | MR

[2] W. Borho & R. Macpherson, Partial resolutions of nilpotent varieties, in Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque 101, Soc. Math. France, 1983, 23-74. | Zbl | Numdam

[3] N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie. Chapitres 4 à 6, Paris, Hermann, 1968 ; Springer, 2007. | Zbl

[4] V. Chari & A. A. Moura, Characters and blocks for finite-dimensional representations of quantum affine algebras, Int. Math. Res. Not. 5 (2005), 257-298. | Zbl

[5] V. Chari & A. A. Moura, Characters of fundamental representations of quantum affine algebras, Acta Appl. Math. 90 (2006), 43-63. | Zbl

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

[7] V. Chari & A. Pressley, A guide to quantum groups, Cambridge University Press, 1994. | Zbl

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

[9] V. Chari & A. Pressley, Integrable and Weyl modules for quantum affine ${\mathrm{sl}}_2$, in Quantum groups and Lie theory (Durham, 1999), London Math. Soc. Lecture Note Ser. 290, Cambridge Univ. Press, 2001, 48-62. | Zbl

[10] G. W. Delius & N. J. Mackay, Affine quantum groups, in Encyclopedia of Mathematical Physics, Elsevier, 2006.

[11] V. G. DrinfelʼD, Quantum groups, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., 1987, 798-820. | Zbl | MR

[12] V. G. DrinfelʼD, A new realization of Yangians and of quantum affine algebras, Soviet Math. Dokl. 36 (1998), 212-216. | Zbl | MR

[13] E. Frenkel & E. Mukhin, Combinatorics of $q$-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phys. 216 (2001), 23-57. | 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. 248, Amer. Math. Soc., 1999, 163-205. | Zbl

[15] M. Goresky & R. Macpherson, Intersection homology theory, Topology 19 (1980), 135-162. | Zbl

[16] M. Goresky & R. Macpherson, Intersection homology. II, Invent. Math. 72 (1983), 77-129. | Zbl

[17] R. M. Hardt, Semi-algebraic local-triviality in semi-algebraic mappings, Amer. J. Math. 102 (1980), 291-302. | Zbl | MR

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

[19] D. Hernandez, Monomials of $q$ and $q,t$-characters for non simply-laced quantum affinizations, Math. Z. 250 (2005), 443-473. | Zbl | MR

[20] D. Hernandez, Representations of quantum affinizations and fusion product, Transform. Groups 10 (2005), 163-200. | Zbl | MR

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

[22] D. Hernandez, Drinfeld coproduct, quantum fusion tensor category and applications, Proc. Lond. Math. Soc. 95 (2007), 567-608. | Zbl | MR

[23] D. Hernandez, On minimal affinizations of representations of quantum groups, Comm. Math. Phys. 276 (2007), 221-259. | Zbl | MR

[24] M. Jimbo, A $q$-difference analogue of $U\left(𝔤\right)$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69. | Zbl | MR

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

[26] 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, J. Soviet Math 52 (1990), 3156-3164, translated from Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 160, Anal. Teor. Chisel i Teor. Funktsii. 8, 211-221, 301 (1987). | Zbl

[27] H. Knight, Spectra of tensor products of finite-dimensional representations of Yangians, J. Algebra 174 (1995), 187-196. | Zbl | MR

[28] A. Kuniba & J. Suzuki, Analytic Bethe ansatz for fundamental representations of Yangians, Comm. Math. Phys. 173 (1995), 225-264. | Zbl

[29] K. Miki, Representations of quantum toroidal algebra $U_q\left({\mathrm{sl}}_{n+1,\mathrm{tor}}\right)(n\ge 2)$, J. Math. Phys. 41 (2000), 7079-7098. | Zbl | MR

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

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

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

[33] H. Nakajima, $T$-analogue of the $q$-characters of finite dimensional representations of quantum affine algebras, in Physics and combinatorics, 2000 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, 196-219. | Zbl | MR

[34] H. Nakajima, Geometric construction of representations of affine algebras, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed. Press, 2002, 423-438. | Zbl | MR

[35] H. Nakajima, Problems on quiver varieties, in The 50th Geometry Symposium, Hokkaido Univ., 2003, http://www.math.kyoto-u.ac.jp/~nakajima/TeX/kika03.pdf.

[36] H. Nakajima, $t$-analogs of $q$-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Theory 7 (2003), 259-274 (electronic). | Zbl | MR

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

[38] O. Schiffmann, Nakajima's quiver varieties, Séminaire Bourbaki 2006/07, exposé no 976, Astérisque (2008), 295-344. | Zbl | MR | Numdam

[39] R. Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240-284. | Zbl | MR

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