Factoriality for the reductive Zassenhaus variety and quantum enveloping algebra

Let U(g) be the enveloping algebra of a finite dimensional reductive Lie algebra g over an algebraically closed field of prime characteristic. Let Uϵ,P(s:) be the simply connected quantum enveloping algebra at the root of unity ϵ, of a complex semi-simple finite dimensional Lie algebra s:. We show, by similar proofs, that the centers of both are factorial. While the first result was established by R. Tange [32] (by different methods), the second one confirms a conjecture in [4]. We also provide a general criterion for the factoriality of the centers of enveloping algebras in prime characteristic.