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%%% Primary Title: locally compact quantum group
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\begin{document}
\textbf{Definition 0.1}
A \emph{locally compact quantum group} defined as in ref. \cite{LV2k3} 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 \htmladdnormallink{Haar measures}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html}.
\textbf{Examples}
\begin{enumerate}
\item An ordinary unimodular \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $G$ 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
$g, h \in G, f \in L^{\infty} (G, \mu)$.
\item A = \L (G) is the \htmladdnormallink{von Neumann algebra}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} generated by left-translations $L_g$ or by left \htmladdnormallink{convolutions}{http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} $L_f = \int_G (f(g)L_g d \mu (g))$ with continuous \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} $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 $G$.
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{LV2k3}
Leonid Vainerman. 2003.\emph{Locally Compact Quantum Groups and Groupoids:} Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 21-23, 2002 \emph{Series in Mathematics and Theoretical Physics}, \textbf{2}, Series ed. V. Turaev., Walter de Gruyter Gmbh et Co: Berlin.
\end{thebibliography}
\end{document}
