PlanetPhysics/Compact Quantum Groupoids

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Introduction and basic concepts
Compact quantum groupoids were introduced in Landsman (1998) as a simultaneous generalization of a compact groupoid and a quantum group. Since this construction is relevant to the definition of locally compact quantum groupoids and their representations 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 $$\mathfrak A$$ commute. A non--commutative Haar measure 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 composition $$\E = \eta_s \circ P : \mathfrak A \lra \eta_s (B) \subset \mathfrak A$$ is a faithful conditional expectation.

Groupoids and quantum compact groupoids
Let us consider $$\mathsf{G}$$ to be a (topological) groupoid.

We denote by $$C_c(\mathsf{G})$$ the space of smooth complex--valued functions with compact support on $$\mathsf{G}$$~. In particular, for all $$f,g \in C_c(\mathsf{G})$$, the function defined via convolution $$ (f ~*~g)(\gamma) = \int_{\gamma_1 \circ \gamma_2 = \gamma} f(\gamma_1) g (\gamma_2)~, $$

is again an element of $$C_c(\mathsf{G})$$, where the convolution product defines the composition law 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 $$f^*(\gamma) = \overline{f(\gamma^{-1})}$$~.

Groupoid representations
We recall that following Landsman (1998) a representation of a groupoid $$\grp$$, consists of a family (or field) of Hilbert spaces $$\{\mathcal H_x \}_{x \in X}$$ indexed by $$X = \ob~ \grp$$, along with a collection of maps $$\{ U(\gamma)\}_{\gamma \in \grp}$$, satisfying:

\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 $$\gamma \in \grp$$~.

\subsubsection{Lie groupoids, their dual algebroids and representations on Hilbert space bundles}

Suppose now $$\mathsf{G}_{lc}$$ is a Lie groupoid. Then the isotropy group $$\mathsf{G}_x$$ is a Lie group, 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 operator of Hilbert spaces $$\pi_x(f) : L^2(\mathsf{G_x},\mu_x) \lra L^2(\mathsf{G}_x, \mu_x)~, $$ given by $$ (\pi_x(f) \xi)(\gamma) = \int f(\gamma_1) \xi (\gamma_1^{-1} \gamma)~ d\mu_x~, $$ 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 $$\pi_x : C_c(\mathsf{G}) \lra \mathcal H_xC_c(\mathsf{G})$$ given by $$ \Vert f \Vert = \sup_{x \in X} \Vert \pi_x(f) \Vert~, $$ whereby the completion of $$C_c(\mathsf{G})$$ in this norm, defines 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 noncommutative geometry.

Hilbert bimodules and tensor products
The next step requires a little familiarity with the theory of Hilbert modules (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 inner product on $$\mathfrak A$$ given by $$\langle A, C \rangle_{\mathfrak B} = P(A^* C)P$$ is defined as a completely positive map. Since $$P$$ is faithful, we fit a new norm on $$\mathfrak A$$ given by $$\Vert A \Vert^2 = \Vert P(A^* A) \Vert_{\mathfrak B}\mathfrak A$$ in this new norm is denoted by $$\mathfrak A^{-}$$ leading then to a Hilbert module over $$\mathfrak B$$~.

The tensor product $$\mathfrak A^{-} \otimes_{\mathfrak B}\mathfrak A^{-}\mathfrak B$$, which for $$i=1,2$$, leads to *--homorphisms $$\vp^{i} : \mathfrak A \lra \mathcal L_{\mathfrak B}(\mathfrak A^{-} \otimes \mathfrak A^{-})\mathfrak A \otimes_{\mathfrak B} \mathfrak A$$ as the C*--algebra contained in $$ \mathcal L_{\mathfrak B}(\mathfrak A^{-} \otimes \mathfrak A^{-})\vp^1(\mathfrak A)$$ and $$\vp^2(\mathfrak A)$$~.

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 coproduct operation $$\Delta : \mathfrak A \lra \mathfrak A \otimes_{\mathfrak B} \mathfrak AQ : \mathfrak A \lra \mathfrak A$$ that it is both an algebra and bimodule antihomomorphism. Finally, the following axiomatic relationships are observed~: $$

(\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

$$ where $$\tau$$ is a flip map : $$\tau(a \otimes b) = (b \otimes a)$$~.

Locally compact quantum groupoids (LCQG)
There is a natural extension of the above definition of quantum compact groupoids to locally compact quantum groupoids by taking $$\mathsf{G}_{lc}$$ to be a locally compact groupoid (instead of a compact groupoid), and then following the steps in the above construction with the topological groupoid $$\mathsf{G}$$ being replaced by $$\mathsf{G}_{lc}$$. Additional integrability and Haar measure system conditions need however be also satisfied as in the general case of locally compact groupoid representations.