| Version 12 |
Version 11 |
| \begin{definition} |
\begin{definition} |
| A {\em locally compact quantum group} defined as in ref. \cite{LV2k3} is a \emph{quadruple $\G = (A, \Delta, \mu, \nu)$}, where $A$ is either a $C^*$-- or a |
A {\em locally compact quantum group} defined as in ref. \cite{LV2k3} is a \emph{quadruple $\G = (A, \Delta, \mu, \nu)$}, where $A$ is either a $C^*$-- or a |
| \PMlinkname{\em $W^*$-- algebra equipped with a co-associative comultiplication}{WeakHopfCAlgebra2} |
\PMlinkname{\em $W^*$-- algebra equipped with a co-associative comultiplication}{WeakHopfCAlgebra2} |
| $\Delta: A \to A \otimes A$ and two faithful semi-finite normal weights, $\mu$ and $\nu$ --{\em right and -left Haar measures}. |
$\Delta: A \to A \otimes A$ and two faithful semi-finite normal weights, $\mu$ and $\nu$ --{\em right and -left Haar measures}. |
| \end{definition} |
\end{definition} |
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\textbf{Examples} |
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| \begin{example} |
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| \begin{enumerate} |
\begin{enumerate} |
| \item An ordinary unimodular group $G$ with Haar measure $ \mu$. |
\item An ordinary unimodular group $G$ with Haar measure $ \mu$. |
| $A := L^{\infty}(G, \mu), \Delta: f(g) \mapsto f(gh)$, |
$A := L^{\infty}(G, \mu), \Delta: f(g) \mapsto f(gh)$, |
| $S: f(g) \mapsto f(g^{}-1), \phi(f) = \int_G f(g)d\mu (g)$, where $g, h \in G, f \in L^{\infty}(G, \mu)$. |
$S: f(g) \mapsto f(g^{}-1), \phi(f) = \int_G f(g)d\mu (g)$, where $g, h \in G, f \in L^{\infty}(G, \mu)$. |
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| \item A:= \L (G) is the von Neumann algebra generated by left-translations $L_g$ or by left convolutions |
\item 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(.) \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. |
$L_f :={ \int}_G f(g)L_g d \mu (g)$ with continuous functions $f(.) \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} |
\end{enumerate} |
| \end{example} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
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| \bibitem{LV2k3} |
\bibitem{LV2k3} |
| Leonid Vainerman. 2003. |
Leonid Vainerman. 2003. |
| \PMlinkexternal{Locally Compact Quantum Groups and Groupoids: Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 21-23, 2002.}{http://planetmath.org/?op=getobj&from=papers&id=471}, |
\PMlinkexternal{Locally Compact Quantum Groups and Groupoids: Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 21-23, 2002.}{http://planetmath.org/?op=getobj&from=papers&id=471}, |
| {\em Series in Mathematics and Theoretical Physics}, {\bf 2}, Series ed. V. Turaev., Walter de Gruyter Gmbh \& Co: Berlin. |
{\em Series in Mathematics and Theoretical Physics}, {\bf 2}, Series ed. V. Turaev., Walter de Gruyter Gmbh \& Co: Berlin. |
| \end{thebibliography} |
\end{thebibliography} |
