  Let g be a finite-dimensional complex Lie algebra, and let U(g) be its
universal enveloping algebra. We prove that if \hat{U}(g), the Arens-Michael
envelope of U(g), is stably flat over U(g) (i.e., if the canonical homomorphism
U(g)-->\hat{U}(g) is a localization in the sense of Taylor), then g is
solvable. To this end, given a cocommutative Hopf algebra H and an H-module
algebra A, we explicitly describe the Arens-Michael envelope of the smash
product A#H as an ``analytic smash product'' of their completions w.r.t.
certain families of seminorms.
