Stably flat completions of universal enveloping algebras

Let g be a complex Lie algebra, and let U(g) be its universal enveloping algebra. We study homo- logical properties of topological Hopf algebras containing U(g) as a dense subalgebra. Specifically, let θ:U(g) → H be a homomorphism to a topological Hopf algebra H. Assuming that H is either a nuclear Fréchet space or a nuclear (DF)-space, we formulate conditions on the dual algebra, H0, that are sufficient for H to be stably at over U(g) in the sense of A. Neeman and A. Ranicki (2001) (i.e., for θ to be a localization in the sense of J. L. Taylor (1972)). As an application, we prove that the Arens-Michael envelope, Ûb(g), of U(g) is stably at over U(g) provided g admits a positive grading. We also show that R. Goodman's (1979) weighted completions of U(g) are stably at over U(g) for each nilpotent Lie algebra g, and that P. K. Rashevskii's (1966) hyperenveloping algebra is stably at over U(g) for any g. Finally, the algebra A (G) of analytic functionals (introduced by G. L. Litvinov (1969)) on the corresponding connected, simply connected complex Lie group G is shown to be stably at over U(g) precisely when g is solvable.