
Journal of Lie Theory 32 (2022), No. 3, 601642 Copyright Heldermann Verlag 2022 On Weakly Complete Universal Enveloping Algebras: A PoincaréBirkhoffWitt Theorem Karl Heinrich Hofmann Fachbereich Mathematik, Technische Universität, Darmstadt, Germany hofmann@mathematik.tudarmstadt.de Linus Kramer Kramer Mathematisches Institut, Universität Münster, Germany linus.kramer@unimuenster.de [Abstractpdf] \def\C{\mathbb C} \def\G{\mathbb G} \def\P{\mathbb P} \def\R{\mathbb R} \def\g{\mathfrak g} \def\i{\mathfrak i} \def\UU{\mathop{\bf U\hphantom{}}\nolimits} The Poincar\'eBirkhoffWitt 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) [{\it On weakly complete group algebras of Compact Groups}, J. Lie Theory 30 (2020) 407426] recently introduced the weakly complete universal enveloping algebra $\UU(\g)$ of a profinitedimensional 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 $\UU(\g)$, algebraically generated by $\g\subseteq \UU(\g)$. It is further shown that, inspite of $\UU$ being a left adjoint functor, it nevertheless preserves projective limits in the form $\UU(\lim_\i \g/\i)\cong \lim_\i\UU(\g/\i)$, for profinitedimensional Lie algebras $\g$ represented as projective limits of their finitedimensional quotients. The required theory is presented in an appendix which is of independent interest.\par In a natural way, a weakly complete enveloping algebra $\UU(\g)$ is a weakly complete symmetric Hopf algebra with a Lie subalgebra $\P(\UU(\g))$ of {\it primitive} elements containing $\g$ (indeed properly if $\g\ne\{0\}$), and with a nontrivial multiplicative proLie group $\G(\UU(\g))$ of {\it grouplike} units, having $\P(\UU(\g))$ as its Lie algebra  in contrast with the classical Poincar\'eBirhoffWitt environment of $U(L)$, thus providing a new aspect of Lie's Third Fundamental Theorem: Indeed a canonical proLie subgroup $\Gamma^*(\g)$ of $\G(\UU(\g))$ is identified whose Lie algebra is naturally isomorphic to $\g$. The structure of $\UU(\g)$ is described in detail for $\dim\g=1$. The primitive and grouplike components and their mutual relationship are evaluated precisely.\par In (HK), cited above, and in the work of R.\,Dahmen and K.\,H.\,Hofmann [{\it The proLie group aspect of weakly complete algebras and weakly complete group Hopf algebras}, J. Lie Theory 29 (2019) 413455] the real weakly complete group Hopf algebra $\R[G]$ of a compact group $G$ was described. In particular, the set $\P(\R[G]))$ of primitive elements of $\R[G]$ 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 $\omega_\g\colon\UU(\g)\to\R[G]$, implementing the identity on $\g$ and inducing a morphism of proLie groups $\Gamma^*(G)\to\G(\R[G])\cong G$: yet another aspect of Sophus Lie's Third Fundamental Theorem\,! Keywords: Associative algebra, Lie algebra, universal enveloping algebra, weakly complete vector space, projective limit, proLie group, profinitedimensional Lie algebra, power series algebra, symmetric Hopf algebra, primitive element, grouplike element, Poincar\'eWitt theorem. MSC: 22E15, 22E65, 22E99. [ Fulltextpdf (296 KB)] for subscribers only. 