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

### William Crawley-Boevey

University of Leeds, UK

## Abstract

For any associative algebra A over a field K we define a family of algebras $\Pi^\lambda(A)$ for $\lambda \in K \,\otimes_{\Bbb Z}\,{\rm K_0}(A)$ . 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.

## Cite this article

William Crawley-Boevey, Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities. Comment. Math. Helv. 74 (1999), no. 4, pp. 548–574

DOI 10.1007/S000140050105