  We study localizations (in the sense of J. L. Taylor) of the universal
enveloping algebra, U(g), of a complex Lie algebra g. Specifically, let f :
U(g) --> H be a homomorphism to some well-behaved topological Hopf algebra H.
We formulate some conditions on the dual algebra, H', that are sufficient for H
to be stably flat over U(g) (i.e., for f to be a localization). As an
application, we prove that the Arens-Michael envelope of U(g) is stably flat
over U(g) provided g admits a positive grading. We also show that Goodman's
weighted completions of U(g) are stably flat over U(g) for each nilpotent Lie
algebra g, and that Rashevskii's hyperenveloping algebra is stably flat over
U(g) for arbitrary g. Finally, Litvinov's algebra A(G) of analytic functionals
on the corresponding connected, simply connected complex Lie group G is shown
to be stably flat over U(g) precisely when g is solvable.
