# PlanetPhysics/L Compact Quantum Groups

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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 $G$ .