Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities

Abstract

For any associative algebra A over a field K we define a family of algebras ${\Pi }^{\lambda }(A)$ for $\lambda \in K{\otimes }_{\mathbb{Z}}{\mathrm{K}}_0(A)$ . In case A is the path algebra of a quiver, one recovers the deformed preprojective algebra introduced by M. P. Holland and the author. In case A is the coordinate ring of a smooth curve, the family includes the ring of differential operators for A and the coordinate ring of the cotangent bundle for Spec A. In case A is quasi-free and ${\Omega }^1$ A is a finitely generated A-A-bimodule we prove that ${\Pi }^{\lambda }(A)$ is well-behaved under localization. We use this to prove a Conze embedding for deformations of Kleinian singularities.