Zassenhaus varieties of general linear Lie algebras

Let be a Lie algebra over an algebraically closed field of characteristic p>0 and let be the universal enveloping algebra of . We prove in this paper for and that the centre of is a unique factorisation domain and its field of fractions is rational. For our argument requires the assumption that p n while for 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.