2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71592Let 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.11 pagesFunctional AnalysisRings and Algebras46M18, 46H05, 16S30, 16S40, 18G25text