Arens-Michael enveloping algebras and analytic smash products

Abstract:

Let $\mathfrak {g}$ be a finite-dimensional complex Lie algebra, and let $U(\mathfrak {g})$ be its universal enveloping algebra. We prove that if $\widehat {U}(\mathfrak {g})$, the Arens-Michael envelope of $U(\mathfrak {g})$ is stably flat over $U(\mathfrak {g})$ (i.e., if the canonical homomorphism $U(\mathfrak {g})\to \widehat {U}(\mathfrak {g})$ is a localization in the sense of Taylor (1972), then $\mathfrak {g}$ 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.