Smallness problem for quantum affine algebras and quiver varieties

David Hernandez

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 2, page 271-306
  • ISSN: 0012-9593

Abstract

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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).

How to cite

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Hernandez, David. "Smallness problem for quantum affine algebras and quiver varieties." Annales scientifiques de l'École Normale Supérieure 41.2 (2008): 271-306. <http://eudml.org/doc/272159>.

@article{Hernandez2008,
abstract = {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).},
author = {Hernandez, David},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {quantum affine algebras; graded quiver varieties},
language = {eng},
number = {2},
pages = {271-306},
publisher = {Société mathématique de France},
title = {Smallness problem for quantum affine algebras and quiver varieties},
url = {http://eudml.org/doc/272159},
volume = {41},
year = {2008},
}

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
PB - Société mathématique de France
VL - 41
IS - 2
SP - 271
EP - 306
AB - 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 - eng
KW - quantum affine algebras; graded quiver varieties
UR - http://eudml.org/doc/272159
ER -

References

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