DG-models of Projective Modules and Nakajima Quiver Varieties

Details DG-models of Projective Modules and Nakajima Quiver Varieties DG-models of Projective Modules and Nakajima Quiver Varieties

2006-04-02

arxiv.org/abs/math/0604011v1

Mathematics

Quantum Algebra

27 pages

Scientific paper

Associated to each finite group $\Gamma$ in $SL_2(C)$ there is a family of noncommutative algebras which deforms the coordinate ring of the Kleinian singularity corresponding to that group. These algebras were defined by W. Crawley-Boevey and M. Holland, who also suggested a conjectural correspondence between the set of isomorphism classes of rank one projective modules over these algebras and associated Nakajima quiver varieties. In \cite{BGK}, V.Baranovski, V.Ginzburg and A.Kuznetsov proved the Crawley-Boevey-Holland conjecture using the methods of noncommutative projective geometry. In this paper we will state a refined ($G$-equivariant) version of this conjecture and, in the case of cyclic groups, give a new construction of this correspondence based on the notion of DG-model of a rank one projective module. This construction leads to a completely explicit description of ideals of the Crawley-Boevey-Holland algebras.

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