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%%% Primary Title: compact quantum groupoids
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%%% Filename: CompactQuantumGroupoids.tex
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\begin{document}
\subsection{Introduction and basic concepts}
Compact quantum groupoids were introduced in Landsman (1998) as a
simultaneous generalization of a compact \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} and a \htmladdnormallink{quantum group}{http://planetphysics.us/encyclopedia/ComultiplicationInAQuantumGroup.html}. Since this construction is relevant to the definition of \htmladdnormallink{locally compact quantum groupoids}{http://planetphysics.us/encyclopedia/UcLocallyCompactQuantumGroupoids.html} and their \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} investigated here, its exposition is required before we can step up to the next level of generality. Firstly, let $\mathfrak A$ and $\mathfrak B$ denote C*--algebras equipped with a *--homomorphism $\eta_s : \mathfrak B \lra \mathfrak A$, and a *--antihomomorphism $\eta_t : \mathfrak B \lra \mathfrak A$ whose images in $\mathfrak A$
\htmladdnormallink{commute}{http://planetphysics.us/encyclopedia/Commutator.html}. A non--commutative \htmladdnormallink{Haar measure}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html} is defined as a completely
positive map $P: \mathfrak A \lra \mathfrak B$ which satisfies
$P(A \eta_s (B)) = P(A) B$~. Alternatively, the \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} $\E = \eta_s \circ P : \mathfrak A \lra \eta_s (B) \subset \mathfrak A$ is a faithful conditional expectation.
\subsection{Groupoids and quantum compact groupoids}
Let us consider $\mathsf{G}$ to be a (\htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html}) groupoid.
We denote by $C_c(\mathsf{G})$ the space of smooth complex--valued \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} with compact support on $\mathsf{G}$~. In particular, for all $f,g \in C_c(\mathsf{G})$, the function defined via \htmladdnormallink{convolution}{http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html}
\begin{equation} (f ~*~g)(\gamma)
= \int_{\gamma_1 \circ \gamma_2 = \gamma} f(\gamma_1) g
(\gamma_2)~,
\end{equation}
is again an element of $C_c(\mathsf{G})$, where the convolution product
defines the \htmladdnormallink{composition law}{http://planetphysics.us/encyclopedia/Identity2.html} on $C_c(\mathsf{G})$~. We can turn
$C_c(\mathsf{G})$ into a *--algebra once we have defined the involution
$*$, and this is done by specifying $f^*(\gamma) = \overline{f(\gamma^{-1})}$~.
\subsubsection{Groupoid representations}
We recall that following Landsman (1998) a \emph{representation} of a groupoid $\grp$, consists of a family (or \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html}) of \htmladdnormallink{Hilbert spaces}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $\{\mathcal H_x \}_{x \in X}$
indexed by $X = \ob~ \grp$, along with a collection of maps $\{
U(\gamma)\}_{\gamma \in \grp}$, satisfying:
\begin{itemize}
\item[1.]
$U(\gamma) : \mathcal H_{s(\gamma)} \lra \mathcal H_{r(\gamma)}$,
is unitary.
\item[2.]
$U(\gamma_1 \gamma_2) = U(\gamma_1) U( \gamma_2)$, whenever
$(\gamma_1, \gamma_2) \in \grp^{(2)}$~ (the set of arrows).
\item[3.]
$U(\gamma^{-1}) = U(\gamma)^*$, for all $\gamma \in \grp$~.
\end{itemize}
\subsubsection{Lie groupoids, their dual algebroids and representations
on Hilbert space bundles}
Suppose now $\mathsf{G}_{lc}$ is a \htmladdnormallink{Lie groupoid}{http://planetphysics.us/encyclopedia/LieAlgebroids.html}. Then the isotropy \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $\mathsf{G}_x$ is a \htmladdnormallink{Lie group}{http://planetphysics.us/encyclopedia/BilinearMap.html}, and for a (left or right) Haar
measure $\mu_x$ on $\mathsf{G}_x$, we can consider the Hilbert
spaces $\mathcal H_x = L^2(\mathsf{G}_x, \mu_x)$ as exemplifying the
above sense of a representation. Putting aside some technical
details which can be found in Connes (1994) and Landsman (2006), the
overall idea is to define an \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} of Hilbert spaces
\begin{equation}\pi_x(f) : L^2(\mathsf{G_x},\mu_x) \lra L^2(\mathsf{G}_x, \mu_x)~,
\end{equation}
given by
\begin{equation}
(\pi_x(f) \xi)(\gamma) = \int f(\gamma_1) \xi (\gamma_1^{-1}
\gamma)~ d\mu_x~,
\end{equation}
for all $\gamma \in \mathsf{G}_x$, and
$\xi \in \mathcal H_x$~. For each $x \in X =\ob ~\mathsf{G}$, $\pi_x$
defines an involutive representation $\pi_x : C_c(\mathsf{G}) \lra
\mathcal H_x$~. We can define a \htmladdnormallink{norm}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} on $C_c(\mathsf{G})$ given by
\begin{equation}
\Vert f \Vert = \sup_{x \in X} \Vert \pi_x(f) \Vert~,
\end{equation}
whereby the completion of $C_c(\mathsf{G})$ in this norm, defines
\emph{the reduced C*--algebra $C^*_r(\mathsf{G})$ of $\mathsf{G}_{lc}$}. It is
perhaps the most commonly used C*--algebra for Lie groupoids
(groups) in \htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}.
\subsubsection{Hilbert bimodules and tensor products}
The next step requires a little familiarity with the theory of
Hilbert \htmladdnormallink{modules}{http://planetphysics.us/encyclopedia/RModule.html} (see e.g. Lance, 1995). We define a left
$\mathfrak B$--action $\lambda$ and a right $\mathfrak B$--action
$\rho$ on $\mathfrak A$ by $\lambda(B)A = A \eta_t (B)$ and
$\rho(B)A = A \eta_s(B)$~. For the sake of localization of the
intended Hilbert module, we implant a $\mathfrak B$--valued \htmladdnormallink{inner product}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} on $\mathfrak A$ given by $\langle A, C \rangle_{\mathfrak
B} = P(A^* C)$ ~. Let us recall that $P$ is defined as a \emph{completely positive map}.
Since $P$ is faithful, we fit a new norm on $\mathfrak A$ given by $\Vert A \Vert^2 = \Vert P(A^* A)
\Vert_{\mathfrak B}$~. The completion of $\mathfrak A$ in this new
norm is denoted by $\mathfrak A^{-}$ leading then to a Hilbert
module over $\mathfrak B$~.
The \htmladdnormallink{tensor}{http://planetphysics.us/encyclopedia/Tensor.html} product $\mathfrak A^{-} \otimes_{\mathfrak B}\mathfrak
A^{-}$ can be shown to be a Hilbert bimodule over $\mathfrak B$,
which for $i=1,2$, leads to *--homorphisms $\vp^{i} : \mathfrak A
\lra \mathcal L_{\mathfrak B}(\mathfrak A^{-} \otimes \mathfrak
A^{-})$~. Next is to define the (unital) C*--algebra $\mathfrak A
\otimes_{\mathfrak B} \mathfrak A$ as the C*--algebra contained in
$ \mathcal L_{\mathfrak B}(\mathfrak A^{-} \otimes \mathfrak
A^{-})$ that is generated by $\vp^1(\mathfrak A)$ and
$\vp^2(\mathfrak A)$~.
\subsection{Definition of compact quantum groupoids: axioms, coproducts, and bimodule antihomomorphism}
The last stage of the recipe for defining a
compact quantum groupoid entails considering a certain \htmladdnormallink{coproduct}{http://planetphysics.us/encyclopedia/Coproduct.html} \htmladdnormallink{operation}{http://planetphysics.us/encyclopedia/Cod.html} $\Delta : \mathfrak A \lra \mathfrak A
\otimes_{\mathfrak B} \mathfrak A$, together with a coinverse $Q :
\mathfrak A \lra \mathfrak A$ that it is both an algebra and
bimodule \htmladdnormallink{antihomomorphism}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}. Finally, the following axiomatic
relationships are observed~:
\begin{equation}
\begin{aligned}
(\ID \otimes_{\mathfrak B} \Delta) \circ \Delta &= (\Delta
\otimes_{\mathfrak B} \ID) \circ \Delta \\ (\ID \otimes_{\mathfrak
B} P) \circ \Delta &= P \\ \tau \circ (\Delta \otimes_{\mathfrak
B} Q) \circ \Delta &= \Delta \circ Q
\end{aligned}
\end{equation}
where $\tau$ is a flip map : $\tau(a \otimes b) = (b \otimes a)$~.
\subsubsection{Locally compact quantum groupoids (LCQG)}
There is a natural extension of the above definition of quantum compact groupoids to \textit{locally compact quantum groupoids} by taking $\mathsf{G}_{lc}$ to be a \htmladdnormallink{locally compact groupoid}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} (instead of a compact groupoid), and then following the steps in the above construction with the \htmladdnormallink{topological groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} $\mathsf{G}$ being replaced by $\mathsf{G}_{lc}$. Additional integrability and Haar measure \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} conditions need however be also satisfied as in the general case of locally compact groupoid \textit{representations}.
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E. M. Alfsen and F. W. Schultz: \emph{Geometry of State Spaces of Operator Algebras}, Birkh\"auser, Boston--Basel--Berlin (2003).
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M. R. Buneci.: \emph{Groupoid Representations}, Ed. Mirton: Timishoara (2003).
\bibitem{Fell}
J. M. G. Fell.: The Dual Spaces of C*--Algebras., \emph{Transactions of the American Mathematical Society}, \textbf{94}: 365--403 (1960).
\bibitem{GR02}
R. Gilmore: \emph{Lie Groups, Lie Algebras and Some of Their Applications.},
Dover Publs., Inc.: Mineola and New York, 2005.
\bibitem{Hahn1}
P. Hahn: Haar measure for measure groupoids, \textit{Trans. Amer. Math. Soc}. \textbf{242}: 1--33(1978).
\bibitem{Hahn2}
P. Hahn: The regular representations of measure groupoids., \textit{Trans. Amer. Math. Soc}. \textbf{242}:34--72(1978).
\end{thebibliography}
\end{document}
