Journal of Lie Theory 32 (2022), No. 3, 601--642
Copyright Heldermann Verlag 2022
On Weakly Complete Universal Enveloping Algebras: A Poincaré-Birkhoff-Witt Theorem
Karl Heinrich Hofmann
Fachbereich Mathematik, Technische Universität, Darmstadt, Germany
hofmann@mathematik.tu-darmstadt.de
Linus Kramer Kramer
Mathematisches Institut, Universität Münster, Germany
linus.kramer@uni-muenster.de
[Abstract-pdf]
\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\'e-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) [{\it On weakly complete group algebras of Compact Groups}, J. Lie Theory 30 (2020) 407--426] recently introduced the weakly complete universal enveloping algebra $\UU(\g)$ 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 $\UU(\g)$, 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 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.\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 pro-Lie group $\G(\UU(\g))$ of {\it grouplike} units, having $\P(\UU(\g))$ as its Lie algebra -- in contrast with the classical Poincar\'e-Birhoff-Witt environment of $U(L)$, thus providing a new aspect of Lie's Third Fundamental Theorem: Indeed a canonical pro-Lie 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 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 $\R[G]$ 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 pro-Lie 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, pro-Lie group, profinite-dimensional Lie algebra, power series algebra, symmetric Hopf algebra, primitive element, grouplike element, Poincar\'e-Witt theorem.
MSC: 22E15, 22E65, 22E99.
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