Associated to each finite subgroup Γ of SL2(C) there is a family of noncommutative algebras Oτ(Γ), which is a deformation of the coordinate ring of the Kleinian singularity C2/Γ. We study finitely generated projective modules over these algebras. Our main result is a bijective correspondence between the set of isomorphism classes of rank one projective modules over Oτ and a certain class of quiver varieties associated to Γ. We show that this bijection is naturally equivariant under the action of a "large" Dixmier-type automorphism group G. Our construction leads to a completely explicit description of ideals of the algebras Oτ.
Homology, Homotopy and Applications, Vol. 9 (2007), No. 2, pp.177-208.