On the Gelfand–Kirillov conjecture for the W-algebras attached to the minimal nilpotent orbits

Consider the W-algebra H   attached to the minimal nilpotent orbit in a simple Lie algebra gg over an algebraically closed field of characteristic 0. We show that if an analogue of the Gelfand–Kirillov conjecture holds for such a W-algebra, then it holds for the universal enveloping algebra U(g)U(g). This, together with a result of A. Premet, implies that the analogue of the Gelfand–Kirillov conjecture fails for some W  -algebras attached to the minimal nilpotent orbit in Lie algebras of types BnBn(n≥3)(n≥3), DnDn(n≥4)(n≥4), E6,E7,E8E6,E7,E8, and F4F4.