Smallness problem for quantum affine algebras and quiver varieties

Details Smallness problem for quantum affine algebras and quiver varieties Smallness problem for quantum affine algebras and quiver varieties

2006-07-21

arxiv.org/abs/math/0607526v2

Annales Scientifiques de l'Ecole Normale Superieure (4) 41 (2008), No. 2, 271-306

Mathematics

Quantum Algebra

33 pages; accepted for publication in Annales Scientifiques de l'Ecole Normale Superieure

Scientific paper

The geometric small property (Borho-MacPherson) 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 (math.QA/0105173) for certain resolutions of quiver varieties (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 (math.QA/0501202) and extend the main result to general simply-laced quantum affinizations, in particular to quantum toroidal algebras (double affine quantum algebras).

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