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compact quantum groupoids (Topic)

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 $\eta_s : \mathfrak{B} {\longrightarrow}\mathfrak{A}$, and a *–antihomomorphism $\eta_t : \mathfrak{B} {\longrightarrow}\mathfrak{A}$ whose images in $\mathfrak{A}$ commute. A non–commutative Haar measure is defined as a completely positive map $P: \mathfrak{A} {\longrightarrow}\mathfrak{B}$ which satisfies $P(A \eta_s (B)) = P(A) B$ . Alternatively, the composition $\mathcal E= \eta_s \circ P : \mathfrak{A} {\longrightarrow}\eta_s (B) \subset \mathfrak{A}$ 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

$\displaystyle (f ~*~g)(\gamma) = \int_{\gamma_1 \circ \gamma_2 = \gamma} f(\gamma_1) g (\gamma_2)~,$ (0.1)

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 ${\mathsf{G}}$, consists of a family (or field) of Hilbert spaces $\{\mathcal H_x \}_{x \in X}$ indexed by $X = {\rm Ob}~ {\mathsf{G}}$, along with a collection of maps $\{ U(\gamma)\}_{\gamma \in {\mathsf{G}}}$, satisfying:
1.   $U(\gamma) : \mathcal H_{s(\gamma)} {\longrightarrow}\mathcal H_{r(\gamma)}$, is unitary.
2.   $U(\gamma_1 \gamma_2) = U(\gamma_1) U( \gamma_2)$, whenever $(\gamma_1, \gamma_2) \in {\mathsf{G}}^{(2)}$  (the set of arrows).
3.   $U(\gamma^{-1}) = U(\gamma)^*$, for all $\gamma \in {\mathsf{G}}$ .

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

$\displaystyle \pi_x(f) : L^2(\mathsf{G_x},\mu_x) {\longrightarrow}L^2(\mathsf{G}_x, \mu_x)~,$ (0.2)
given by
$\displaystyle (\pi_x(f) \xi)(\gamma) = \int f(\gamma_1) \xi (\gamma_1^{-1} \gamma)~ d\mu_x~,$ (0.3)
for all $\gamma \in \mathsf{G}_x$, and $\xi \in \mathcal H_x$ . For each $x \in X ={\rm Ob}~\mathsf{G}$, $\pi_x$ defines an involutive representation $\pi_x : C_c(\mathsf{G}) {\longrightarrow} \mathcal H_x$ . We can define a norm on $C_c(\mathsf{G})$ given by
$\displaystyle \Vert f \Vert = \sup_{x \in X} \Vert \pi_x(f) \Vert~,$ (0.4)
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 $\langle A, C \rangle_{\mathfrak{B}} = P(A^* C)$ . Let us recall that $P$ is defined as a completely positive map. Since $P$ is faithful, we fit a new norm on $\mathfrak{A}$ given by $\Vert A \Vert^2 = \Vert P(A^* A) \Vert_{\mathfrak{B}}$ . The completion of $\mathfrak{A}$ in this new norm is denoted by $\mathfrak{A}^{-}$ leading then to a Hilbert module over $\mathfrak{B}$ .

The tensor product $\mathfrak{A}^{-} \otimes_{\mathfrak{B}}\mathfrak{A}^{-}$ can be shown to be a Hilbert bimodule over $\mathfrak{B}$, which for $i=1,2$, leads to *–homorphisms $\varphi ^{i} : \mathfrak{A} {\longrightarrow}\mathcal L_{\mathfrak{B}}(\mathfrak{A}^{-} \otimes \mathfrak{A}^{-})$ . 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 $\varphi ^1(\mathfrak{A})$ and $\varphi ^2(\mathfrak{A})$ .

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 $\Delta : \mathfrak{A} {\longrightarrow}\mathfrak{A} \otimes_{\mathfrak{B}} \mathfrak{A}$, together with a coinverse $Q : \mathfrak{A} {\longrightarrow}\mathfrak{A}$ that it is both an algebra and bimodule antihomomorphism. Finally, the following axiomatic relationships are observed :
\begin{equation*}\begin{aligned} ({\rm id}\otimes_{\mathfrak{B}} \Delta) \circ \... ...es_{\mathfrak{B}} Q) \circ \Delta &= \Delta \circ Q \end{aligned}\end{equation*}
where $\tau$ is a flip map : $\tau(a \otimes b) = (b \otimes a)$ .

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.

Bibliography

1
E. M. Alfsen and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkhäuser, Boston–Basel–Berlin (2003).
2
M. R. Buneci.: Groupoid Representations, Ed. Mirton: Timishoara (2003).
3
J. M. G. Fell.: The Dual Spaces of C*–Algebras., Transactions of the American Mathematical Society, 94: 365–403 (1960).
4
R. Gilmore: Lie Groups, Lie Algebras and Some of Their Applications., Dover Publs., Inc.: Mineola and New York, 2005.
5
P. Hahn: Haar measure for measure groupoids, Trans. Amer. Math. Soc. 242: 1–33(1978).
6
P. Hahn: The regular representations of measure groupoids., Trans. Amer. Math. Soc. 242:34–72(1978).



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See Also: uniform continuity over locally compact quantum groupoids

Other names:  quantum compact groupoids, CQGs, QCGs
Also defines:  groupoid C*-algebra
Keywords:  groupoid C*-algebra, compact quantum groupoids (CQGDs)

Cross-references: system, topological groupoid, locally compact groupoid, antihomomorphism, operation, coproduct, tensor, inner product, modules, noncommutative geometry, norm, operator, Lie group, group, Lie groupoid, Hilbert spaces, field, composition law, convolution, functions, topological, composition, Haar measure, commute, representations, locally compact quantum groupoids, quantum group, groupoid
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This is version 4 of compact quantum groupoids, born on 2009-01-10, modified 2009-01-26.
Object id is 370, canonical name is CompactQuantumGroupoids.
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Physics Classification: 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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