Journal of Lie Theory 30 (2020), No. 2, 407--424
Copyright Heldermann Verlag 2020
On Weakly Complete Group Algebras of Compact Groups
Karl Heinrich Hofmann
Fachbereich Mathematik, Technische Universität, 64289 Darmstadt, Germany
hofmann@mathematik.tu-darmstadt.de
Linus Kramer
Mathematisches Institut, Universität Münster, 48149 Münster, Germany
linus.kramer@uni-muenster.de
[Abstract-pdf]
\def\hats #1{\hat{\hat{\hbox{$#1$}}}} \def\K{\mathbb K} \newcommand{\R}{\mathbb{R}} \def\C{\mathbb C} \def\Hom{\mathop{\rm Hom}\nolimits} \def\SS{{\mathbb S}} \def\A{{\hat G}} \def\UU{\mathop{\bf U\hphantom{}}\nolimits} \def\L{\mathfrak L} A topological vector space over the real or complex field $\K$ is {\it weakly complete} if it is isomorphic to a power $\K^J$. For each topological group $G$ there is a {\it weakly complete topological group Hopf algebra} $\K[G]$ over $\K=\R$ or $\C$, for which three insights are contributed:\\[1mm] Firstly, {\it there is a comprehensive structure theorem saying that the topological algebra $\K[G]$ is the cartesian product of its finite dimensional minimal ideals whose structure is clarified.}\\[1mm] Secondly, {\it for a compact {\rm abelian} group $G$ and its character group $\A$, the weakly complete {\rm complex} Hopf algebra $\C[G]$ is the product algebra $\C^\A$ with the comultiplication $c\colon\C^\A\to\C^{\A\times\A}\cong\C^\A\otimes\C^\A$, $c(F)(\chi_1,\chi_2) = F(\chi_1+\chi_2)$ for $F\colon\A\to\C$ in $\C^\A$. The subgroup $\Gamma(\C^\A)$ of grouplike elements of the group of units of the algebra $\C^\A$ is $\Hom(\A,(\C\setminus\{0\},.))$ while the vector subspace of primitive elements is $\Hom(\A,(\C,+))$.} This forces the group $\Gamma(\R[G])\subseteq\Gamma(\C[G])$ to be $\smash{\Hom(\A,\SS^1)\cong\hats G\cong G}$ with the complex circle group $\SS^1$. While the relation $\Gamma(\R[G])\cong G$ remains true for {\it any} compact group, $\Gamma(\C[G])\cong G$ holds for a compact abelian group $G$ if and only if it is profinite.\\[1mm] Thirdly, for each pro-Lie algebra $L$ a weakly complete universal enveloping Hopf algebra $\UU_\K(L)$ over $\K$ exists such that {\it for each {\em connected} compact group $G$ the weakly complete real group Hopf algebra $\R[G]$ is a quotient Hopf algebra of $\UU_\R(\L(G))$ with the (pro-)Lie algebra $\L(G)$ of $G$. The group $\Gamma(\UU_\R(\L(G)))$ of grouplike elements of the weakly complete enveloping algebra of $\L(G)$ maps onto $\Gamma(\R[G])\cong G$ and is therefore nontrivial} in contrast to the case of the discrete classical enveloping Hopf algebra of an abstract Lie algebra.
Keywords: Weakly complete vector space, weakly complete algebra, group algebra, Hopf algebra, compact group, Lie algebra, universal enveloping algebra.
MSC: 22E15, 22E65, 22E99.
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