l-compact quantum groupsLCompactQuantumGroups2008-12-16 05:57:032008-12-16 05:57:03Definitionlocally compact quantum groupcompact quantum group% this is the default PlanetPhysics preamble. as your knowledge
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% define commands here\textbf{Definition 0.1}
A \emph{locally compact quantum group} defined as in ref. \cite{LV2k3} is a quadruple $QG_{lc} =(A, \Delta, \mu, \nu)$, 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.
\textbf{Examples}
\begin{enumerate}
\item 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
$g, h \in G, f \in L^{\infty} (G, \mu)$.
\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.
\end{enumerate}
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\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.
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