PlanetPhysics/L Compact Quantum Groups

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Definition 0.1

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

Examples

1. An ordinary unimodular [[../TrivialGroupoid/|group]] $G$ with Haar measure $\mu$:

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

1. A = \L (G) is the [[../QuantumOperatorAlgebra4/|von Neumann algebra]] generated by left-translations $L_g$ or by left [[../AssociatedGroupoidAlgebraRepresentations/|convolutions]] $L_f=\underset{G}{\int }(f(g)L_{g}d\mu (g))$ with continuous [[../Bijective/|functions]] $f(\dot{)}\in L^1(G,\mu )\mathrm{\Delta }:\mapsto L_g\otimes L_g\mapsto L_g^{-1},\varphi (f)=f(e)$, where $g\in G$, and $e$ is the unit of $G$.