Gelfand–Kirillov conjecture in positive characteristics

Let g be a finite-dimensional Lie algebra over an algebraically closed field of characteristic p>0 and let U(g) be the universal enveloping algebra of g. We show in this paper that the division ring of fractions of U(g) is isomorphic to the ring of fractions of a Weyl algebra in the following cases: for g=gln or sln 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 U(g) in the last two cases is a unique factorisation domain, in accordance with recent results of Premet, Tange, Braun and Hajarnavis.