Arens-Michael enveloping algebras and analytic smash products

Let g be a finite-dimensional complex Lie algebra, and let Û(g) be its universal enveloping algebra. We prove that if Û(g), the Arens-Michael envelope of U(g) is stably flat over U(g) (i.e., if the canonical homomorphism U (g) → Û(g) is a localization in the sense of Taylor (1972), 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. © 2006 American Mathematical Society.