PlanetPhysics/L Compact Quantum Groups

\newcommand{\sqdiagram}[9]{Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

Definition 0.1

A locally compact quantum group defined as in ref. is a quadruple $QCG_{l}=(A,\Delta ,\mu ,\nu )$, where $A$ is either a $C^{*}$- or a $W^{*}$ - algebra equipped with a co-associative comultiplication $\Delta :A\to A\otimes A$ and two faithful semi-finite normal weights, $\mu$ and $\nu$ - right and -left Haar measures.

Examples

1. An ordinary unimodular group $G$ with Haar measure $\mu$:

$A=L^{\infty }(G,\mu ),\Delta :f(g)\mapsto f(gh)$, $S:f(g)\mapsto f(g^{}-1),\phi (f)=\int _{G}f(g)d\mu (g)$, where $g,h\in G,f\in L^{\infty }(G,\mu )$.

1. A = \L (G) is the von Neumann algebra generated by left-translations $L_{g}$ or by left convolutions $L_{f}=\int _{G}(f(g)L_{g}d\mu (g))$ with continuous functions $f({\dot {)}}\in L^{1}(G,\mu )\Delta :\mapsto L_{g}\otimes L_{g}\mapsto L_{g}^{-1},\phi (f)=f(e)$, where $g\in G$, and $e$ is the unit of $G$.