The Poincar-Birkhoff-Witt Theorem deals with the structure and universal property of the universal enveloping algebra U (L) of a Lie algebra L, e. g. , over R or C. K. H. Hofmann and L. Kramer (HK) On weakly complete group algebras of Compact Groups, J. Lie Theory 30 (2020) 407-426 recently introduced the weakly complete universal enveloping algebra U (g) of a profinite-dimensional topological Lie algebra g. Here it is shown that the classical universal enveloping algebra U (|g|) of the abstract Lie algebra underlying g is a dense subalgebra of U (g), algebraically generated by g U (g). It is further shown that, inspite of U being a left adjoint functor, it nevertheless preserves projective limits in the form U (lim i g/i) = lim i U (g/i), for profinite-dimensional Lie algebras g represented as projective limits of their finite-dimensional quotients. The required theory is presented in an appendix which is of independent interest. In a natural way, a weakly complete enveloping algebra U (g) is a weakly complete symmetric Hopf algebra with a Lie subalgebra P (U (g) ) of primitive elements containing g (indeed properly if g = 0), and with a nontrivial multiplicative pro-Lie group G (U (g) ) of grouplike units, having P (U (g) ) as its Lie algebra -in contrast with the classical Poincar-Birhoff-Witt environment of U (L), thus providing a new aspect of Lie's Third Fundamental Theorem: Indeed a canonical pro-Lie subgroup * (g) of G (U (g) ) is identified whose Lie algebra is naturally isomorphic to g. The structure of U (g) is described in detail for dim g = 1. The primitive and grouplike components and their mutual relationship are evaluated precisely. In (HK), cited above, and in the work of R. Dahmen and K. H. Hofmann The pro-Lie group aspect of weakly complete algebras and weakly complete group Hopf algebras, J. Lie Theory 29 (2019) 413-455 the real weakly complete group Hopf algebra RG of a compact group G was described. In particular, the set P (RG) ) of primitive elements of RG was identified as the Lie algebra g of G. It is now shown that for any compact group G with Lie algebra g there is a natural morphism of weakly complete symmetric Hopf algebras g: U (g) RG, implementing the identity on g and inducing a morphism of pro-Lie groups * (G) G (RG) = G: yet another aspect of Sophus Lie's Third Fundamental Theorem !
Hofmann et al. (Sat,) studied this question.