Center and derivations of generalized Weyl algebras over $\mathbb{Z}/p^n\mathbb{Z}$

Abstract

Let $A$ be either a classical generalized Weyl algebra (also known as a noncommutative deformation of type A Kleinian singularity) or the enveloping algebra $U(\mathfrak{sl}_{2})$ over $\mathbb{Z}/p^n\mathbb{Z}.$ In this paper we compute the center and derivations of $A.$ More specifically, we show that the center of $U(\mathfrak{sl}_2)$ is generated by the Casimir element over the ring of the Witt vectors (of length $n$) of its $p$-center. Our description of derivations of $A$ implies that if the ground ring is a field $k$ of characteristic $p>2,$ then the restriction homomorphism $HH^1_{k}(A)\to Der_{k}(Z(A), Z(A))$ from the first Hochschild cohomology of $A$ to $k$-derivations of the center is an isomorphism.