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 -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 -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).
@article{ASENS_2008_4_41_2_271_0, author = {Hernandez, David}, title = {Smallness problem for quantum affine algebras and quiver varieties}, journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, pages = {271--306}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {Ser. 4, 41}, number = {2}, year = {2008}, doi = {10.24033/asens.2068}, mrnumber = {2468483}, zbl = {1189.17014}, language = {en}, url = {http://www.numdam.org/articles/10.24033/asens.2068/} }
TY - JOUR AU - Hernandez, David TI - Smallness problem for quantum affine algebras and quiver varieties JO - Annales scientifiques de l'École Normale Supérieure PY - 2008 SP - 271 EP - 306 VL - 41 IS - 2 PB - Société mathématique de France UR - http://www.numdam.org/articles/10.24033/asens.2068/ DO - 10.24033/asens.2068 LA - en ID - ASENS_2008_4_41_2_271_0 ER -
%0 Journal Article %A Hernandez, David %T Smallness problem for quantum affine algebras and quiver varieties %J Annales scientifiques de l'École Normale Supérieure %D 2008 %P 271-306 %V 41 %N 2 %I Société mathématique de France %U http://www.numdam.org/articles/10.24033/asens.2068/ %R 10.24033/asens.2068 %G en %F ASENS_2008_4_41_2_271_0
Hernandez, David. Smallness problem for quantum affine algebras and quiver varieties. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 2, pp. 271-306. doi : 10.24033/asens.2068. http://www.numdam.org/articles/10.24033/asens.2068/
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