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 that for g=gl_n and g=sl_n the centre of U(g) is a unique factorisation domain and its field of fractions is rational. For g=sl_n our argument requires the assumption that p\nmid n while for g=gl_n 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 ${\rm A}$ a recent conjecture of A.Braun and C.Hajarnavis while the second answers a question of J.Alev.