Factorial properties of the enveloping algebra of a nilpotent Lie algebra in prime characteristic

Let U(L) be the enveloping algebra of a finite-dimensional nilpotent Lie algebra L, over a prime characteristic field. We prove that its center Z(U(L)) is a unique factorization (UFD). We also show that U(L) has a non-commutative UFD property, namely, each height one prime ideal in U(L) is generated by a central element. We prove both results simultaneously, using non-commutative (PI, maximal order) technique. Our results are prime characteristic analogues of similar ones in characteristic zero, which are due to Dixmier [J. Dixmier, Sur lʹalgèbre enveloppante dʹune algebra de Lie nilpotente, Arch. Math. 10 (1959) 321–32] and Moeglin [C. Moeglin, Factorialité dans les algèbres enveloppantes, C. R. Acad. Sci. Paris (A) 282 (1976) 1269–1272]. We have recently applied these results to show that U(L) is a Calabi–Yau algebra.