Farkhod Eshmatov

August 2007 Ph.D.

First Position

Postdoc at the University of Michigan


The Calogero-Moser Correspondence for Noncommutative Deformations of Kleinian Singularities


Research Area

Noncommutative algebraic geometry


Associated to each finite subgroup Γ of SL2(C) ther 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.

First, we establish a bijective correspondence between the set of isomorphism classes of rank one projective modules over Oτ(Γ) 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. This construction leads to a completely explicit description of ideals of the algebras Oτ(Γ).

Second, we construct functors between the category of Oτ(Γ)-modules and that of deformed preprojective algebras of a certain quiver associated to Γ. We show that the restriction of these functors to simple modules over preprojective algebras give another description of Oτ(Γ)-ideals.