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 Failed to parse (unknown function "\lra"): {\displaystyle \eta_s : \mathfrak B \lra \mathfrak A} , and a *--antihomomorphism Failed to parse (unknown function "\lra"): {\displaystyle \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 Failed to parse (unknown function "\lra"): {\displaystyle P: \mathfrak A \lra \mathfrak B} which satisfies $P(A\eta _{s}(B))=P(A)B$~. Alternatively, the composition Failed to parse (unknown function "\E"): {\displaystyle \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 Failed to parse (unknown function "\grp"): {\displaystyle \grp} , consists of a family (or field) of Hilbert spaces $\{{\mathcal {H}}_{x}\}_{x\in X}$ indexed by Failed to parse (unknown function "\ob"): {\displaystyle X = \ob~ \grp} , along with a collection of maps Failed to parse (unknown function "\grp"): {\displaystyle \{ U(\gamma)\}_{\gamma \in \grp}} , satisfying:

 \item[1.] Failed to parse (unknown function "\lra"): {\displaystyle 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 Failed to parse (unknown function "\grp"): {\displaystyle (\gamma_1, \gamma_2) \in \grp^{(2)}}
~ (the set of arrows). \item[3.] $U(\gamma ^{-1})=U(\gamma )^{*}$, for all Failed to parse (unknown function "\grp"): {\displaystyle \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

Failed to parse (unknown function "\lra"): {\displaystyle \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 Failed to parse (unknown function "\ob"): {\displaystyle x \in X =\ob ~\mathsf{G}} , $\pi _{x}$ defines an involutive representation Failed to parse (unknown function "\lra"): {\displaystyle \pi_x : C_c(\mathsf{G}) \lra \mathcal H_x<math>~. We can define a [[../NormInducedByInnerProduct/|norm]] on } C_c(\mathsf{G})</math> 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 Failed to parse (syntax error): {\displaystyle 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 $\langle A,C\rangle _{\mathfrak {B}}=P(A^{*}C)<math>~.Letusrecallthat$P</math> is defined as a completely positive map . Since $P$ is faithful, we fit a new norm on ${\mathfrak {A}}$ given by $\Vert A\Vert ^{2}=\Vert P(A^{*}A)\Vert _{\mathfrak {B}}<math>~.Thecompletionof$\mathfrak A</math> in this new norm is denoted by ${\mathfrak {A}}^{-}$ leading then to a Hilbert module over ${\mathfrak {B}}$~.

The tensor product ${\mathfrak {A}}^{-}\otimes _{\mathfrak {B}}{\mathfrak {A}}^{-}<math>canbeshowntobeaHilbertbimoduleover$\mathfrak B</math>, which for $i=1,2$, leads to *--homorphisms Failed to parse (unknown function "\vp"): {\displaystyle \vp^{i} : \mathfrak A \lra \mathcal L_{\mathfrak B}(\mathfrak A^{-} \otimes \mathfrak A^{-})<math>~. Next is to define the (unital) C*--algebra } \mathfrak A \otimes_{\mathfrak B} \mathfrak A</math> as the C*--algebra contained in ${\mathcal {L}}_{\mathfrak {B}}({\mathfrak {A}}^{-}\otimes {\mathfrak {A}}^{-})<math>thatisgeneratedby$\vp^1(\mathfrak A)</math> and Failed to parse (unknown function "\vp"): {\displaystyle \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 Failed to parse (unknown function "\lra"): {\displaystyle \Delta : \mathfrak A \lra \mathfrak A \otimes_{\mathfrak B} \mathfrak A<math>, together with a coinverse } Q : \mathfrak A \lra \mathfrak A</math> that it is both an algebra and bimodule antihomomorphism. Finally, the following axiomatic relationships are observed~:

Failed to parse (unknown function "\ID"): {\displaystyle (\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 .