Zassenhaus varieties of general linear Lie algebras

Let g be a Lie algebra over an algebraically closed field of characteristic p>0 and let U(g) be the universal enveloping algebra of g. We prove in this paper for g=gln and g=sln that the centre of U(g) is a unique factorisation domain and its field of fractions is rational. For g=sln our argument requires the assumption that p∤n while for g=gln it works for any p. It turned out that our two main results are closely related to each other. The first one confirms in type A a recent conjecture of A. Braun and C. Hajarnavis while the second answers a question of J. Alev.