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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

Linus Kramer
Mathematisches Institut, Universität Münster, 48149 Münster, Germany


\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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