0.1 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 𝔄 and 𝔅 denote C*–algebras equipped with a *–homomorphism η_s : 𝔅→𝔄, and a *–antihomomorphism η_t : 𝔅→𝔄 whose images in 𝔄 commute. A non–commutative Haar measure is defined as a completely positive map P : 𝔄→𝔅 which satisfies P(Aη_s(B)) = P(A)B . Alternatively, the composition ℰ = η_s ∘ P : 𝔄→η_s(B) ⊂ 𝔄 is a faithful conditional expectation.

0.2 Groupoids and quantum compact groupoids

Let us consider G to be a (topological) groupoid.

We denote by C_c(G) the space of smooth complex–valued functions with compact support on G . In particular, for all f,g ∈ C_c(G), the function defined via convolution

(0.1)

is again an element of C_c(G), where the convolution product defines the 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^∗(γ) = f(γ⁻¹) .

0.2.1 Groupoid representations

We recall that following Landsman (1998) a representation of a groupoid G, consists of a family (or field) of Hilbert spaces {ℋ_x}_x∈X indexed by X = Ob G, along with a collection of maps {U(γ)}_γ∈G, satisfying:

• U(γ) : ℋ_s(γ)→ℋ_r(γ), is unitary.
• U(γ₁γ₂) = U(γ₁)U(γ₂), whenever (γ₁,γ₂) ∈ G⁽²⁾  (the set of arrows).
• U(γ⁻¹) = U(γ)^∗, for all γ ∈ G .

0.2.2 Lie groupoids, their dual algebroids and representations on Hilbert space bundles

Suppose now G_lc is a Lie groupoid. Then the isotropy group G_x is a Lie group, and for a (left or right) Haar measure μ_x on G_x, we can consider the Hilbert spaces ℋ_x = L²(G _x,μ_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

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given by

(0.3)

for all γ ∈ G_x, and ξ ∈ℋ_x . For each x ∈ X = Ob G, π_x defines an involutive representation π_x : C_c(G)→ℋ_x . We can define a norm on C_c(G) given by

(0.4)

whereby the completion of C_c(G) in this norm, defines the reduced C*–algebra C_r^∗(G) of G _lc. It is perhaps the most commonly used C*–algebra for Lie groupoids (groups) in noncommutative geometry.

0.2.3 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 𝔅–action λ and a right 𝔅–action ρ on 𝔄 by λ(B)A = Aη_t(B) and ρ(B)A = Aη_s(B) . For the sake of localization of the intended Hilbert module, we implant a 𝔅–valued inner product on 𝔄 given by ⟨A,C⟩_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 𝔄 given by ∥A∥² = ∥P(A^∗A)∥_𝔅 . The completion of 𝔄 in this new norm is denoted by 𝔄⁻ leading then to a Hilbert module over 𝔅 .

The tensor product 𝔄⁻⊗_𝔅𝔄⁻ can be shown to be a Hilbert bimodule over 𝔅, which for i = 1, 2, leads to *–homorphisms φⁱ : 𝔄→ℒ_𝔅(𝔄⁻⊗ 𝔄⁻) . Next is to define the (unital) C*–algebra 𝔄 ⊗_𝔅𝔄 as the C*–algebra contained in ℒ_𝔅(𝔄⁻⊗ 𝔄⁻) that is generated by φ¹(𝔄) and φ²(𝔄) .

0.3 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 Δ : 𝔄→𝔄 ⊗_𝔅𝔄, together with a coinverse Q : 𝔄→𝔄 that it is both an algebra and bimodule antihomomorphism. Finally, the following axiomatic relationships are observed :

(0.5)

where τ is a flip map : τ(a ⊗ b) = (b ⊗ a) .

0.3.1 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 locally compact groupoid (instead of a compact groupoid), and then following the steps in the above construction with the topological groupoid G being replaced by G_lc. Additional integrability and Haar measure system conditions need however be also satisfied as in the general case of locally compact groupoid representations.

References