81R15-CompactQuantumGroup.tex

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\pmtitle{compact quantum group}
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\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{81R15}
\pmclassification{msc}{81R50}
\pmclassification{msc}{46L05}
\pmsynonym{quantum group}{CompactQuantumGroup}
\pmsynonym{compact matrix quantum group}{CompactQuantumGroup}
%\pmkeywords{compact quantum group}
%\pmkeywords{LocallyCompactQuantumGroup}
%\pmkeywords{C*-algebra}
%\pmkeywords{compact matrix quantum group}
%\pmkeywords{quantum group}
\pmrelated{CAlgebra3}
\pmrelated{QuantumOperatorAlgebrasInQuantumFieldTheories}
\pmrelated{FiniteQuantumGroup}
\pmrelated{DualityInMathematics}
\pmrelated{LocallyCompactQuantumGroup}
\pmrelated{QuantumGroups}
\pmrelated{GelfandTransform}

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\begin{document}
\begin{definition}
A \emph{compact quantum group, $Q_{CG}$} is defined as a particular case of a locally compact quantum group $Q_{LCG}$, that is, a quadruple $(A, \Delta, \mu, \nu)$, where $A$ is either a $C^*$-- or a 
\PMlinkname{$W^*$-- algebra equipped with a co-associative comultiplication}{WeakHopfCAlgebra2} 
$\Delta: A \to A \otimes A$, and two faithful semi-finite normal weights, $\mu$ and $\nu$ --right and -left Haar measures, and also when the object space $\mathbf{O}$ of the latter $Q_{LCG}$ is replaced by a compact topological space $Q^T_{CG}$, instead of being a locally compact topological space like $Q_{LCG}$. 


\end{definition}

\begin{thebibliography}{9}
\bibitem{M-V98}
A. Maes, and A. VanDaele. 1998. 
\PMlinkexternal{Notes on Compact Quantum Groups.}{http://arxiv.org/PS_cache/math/pdf/9803/9803122v1.pdf}, $arxiv.org.math-FA-9803122v1$, 43 pp.
\end{thebibliography}

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