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 [[../LocallyCompactGroupoid/|groupoid]] and a [[../ComultiplicationInAQuantumGroup/|quantum group]]. Since this construction is relevant to the definition of [[../UcLocallyCompactQuantumGroupoids/|locally compact quantum groupoids]] and their [[../CategoricalGroupRepresentation/|representations]] investigated here, its exposition is required before we can step up to the next level of generality. Firstly, let $𝔄$ and $𝔅$ 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 $𝔄$ [[../Commutator/|commute]]. A non--commutative [[../HigherDimensionalQuantumAlgebroid/|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 [[../Cod/|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 $𝖦$ to be a ([[../CoIntersections/|topological]]) groupoid.

We denote by $C_c(𝖦)$ the space of smooth complex--valued [[../Bijective/|functions]] with compact support on $𝖦$~. In particular, for all $f,g\in C_c(𝖦)$, the function defined via [[../AssociatedGroupoidAlgebraRepresentations/|convolution]]

$(f*g)(\gamma )=\underset{{\gamma }_1\circ {\gamma }_2=\gamma }{\int }f({\gamma }_1)g({\gamma }_2),$

is again an element of $C_c(𝖦)$, where the convolution product defines the [[../Identity2/|composition law]] on $C_c(𝖦)$~. We can turn $C_c(𝖦)$ 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 [[../CosmologicalConstant2/|field]]) of [[../NormInducedByInnerProduct/|Hilbert spaces]] $\{{\mathscr{H}}_x{\}}_{x\in X}$ 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 ${𝖦}_{lc}$ is a [[../LieAlgebroids/|Lie groupoid]]. Then the isotropy [[../TrivialGroupoid/|group]] ${𝖦}_x$ is a [[../BilinearMap/|Lie group]], and for a (left or right) Haar measure ${\mu }_x$ on ${𝖦}_x$, we can consider the Hilbert spaces ${\mathscr{H}}_x=L^2({𝖦}_x,{\mu }_x)$ 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 [[../QuantumSpinNetworkFunctor2/|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 $\gamma \in {𝖦}_x$, and $\xi \in {\mathscr{H}}_x$~. 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

$\Vert f\Vert =\underset{x\in X}{\sup }\Vert {\pi }_x(f)\Vert ,$

whereby the completion of $C_c(𝖦)$ in this norm, defines the reduced C*--algebra Failed to parse (syntax error): {\displaystyle C^*_r(\mathsf{G )} of ${𝖦}_{lc}$}. It is perhaps the most commonly used C*--algebra for Lie groupoids (groups) in [[../NoncommutativeGeometry4/|noncommutative geometry]].

Hilbert bimodules and tensor products

The next step requires a little familiarity with the theory of Hilbert [[../RModule/|modules]] (see e.g. Lance, 1995). We define a left $𝔅$--action $\lambda$ and a right $𝔅$--action $\rho$ on $𝔄$ 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 $𝔅$--valued [[../NormInducedByInnerProduct/|inner product]] on $𝔄$ given by $⟨A,C{⟩}_{𝔅}=P(A^{*}C)<math>.Letusrecallthat$P</math> is defined as a completely positive map . Since $P$ is faithful, we fit a new norm on $𝔄$ given by $\Vert A{\Vert }^2=\Vert P(A^{*}A){\Vert }_{𝔅}<math>.Thecompletionof$\mathfrak A</math> in this new norm is denoted by ${𝔄}^{-}$ leading then to a Hilbert module over $𝔅$~.

The [[../Tensor/|tensor]] product ${𝔄}^{-}{\otimes }_{𝔅}{𝔄}^{-}<math>canbeshowntobeaHilbertbimoduleover$\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 ${ℒ}_{𝔅}({𝔄}^{-}\otimes {𝔄}^{-})<math>thatisgeneratedby$\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/|coproduct]] [[../Cod/|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 [[../QuantumOperatorAlgebra5/|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 ${𝖦}_{lc}$ to be a [[../LocallyCompactGroupoid/|locally compact groupoid]] (instead of a compact groupoid), and then following the steps in the above construction with the [[../GroupoidHomomorphism2/|topological groupoid]] $𝖦$ being replaced by ${𝖦}_{lc}$. Additional integrability and Haar measure [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] conditions need however be also satisfied as in the general case of locally compact groupoid representations .