Gelfand–Kirillov conjecture in positive characteristics

Let be a finite-dimensional Lie algebra over an algebraically closed field of characteristic p>0 and let be the universal enveloping algebra of . We show in this paper that the division ring of fractions of is isomorphic to the ring of fractions of a Weyl algebra in the following cases: for or if p n, for the Witt algebra W1 and for some tensor product W1 A of W1 with a truncated polynomial ring. Furthermore we also show that the centre of in the last two cases is a unique factorisation domain, in accordance with recent results of Premet, Tange, Braun and Hajarnavis.