Extensions of locally compact quantum groups and the bicrossed product construction

Stefaan Vaes^{1} Department of Mathematics K.U.Leuven Celestijnenlaan 200 B B-3001 Leuven Belgium

e-mail : Stefaan.Vaes@wis.kuleuven.ac.be

Leonid Vainerman Max-Planck-Institut für Mathematik Vivatsgasse 7 D-53111 Bonn Germany

e-mail : vainerma@mpim-bonn.mpg.de

Introduction

The major motivation for our paper is the fundamental work [18] of G.I. Kac on extensions of finite groups which are, in modern terms, finite dimensional Kac algebras (or Hopf ∗-algebras). Being invented in early sixties (see [19]) in order to explain in a symmetric way duality for locally compact (l.c.) groups, Kac algebras gave historically the first wide class of quantum groups that included besides usual groups and their duals also nontrivial (i.e., non-commutative and non-cocommutative) objects [20], [21]. In fact, one of the goals of [18] was to give a systematic approach to the construction of such objects. The general theory of Kac algebras was completed independently on the one hand by G.I. Kac and the second author [22] and on the other hand by M. Enock and J.-M. Schwartz (for a survey see [14]).

After the appearence of quantum groups in the eighties a lot of efforts were spent in order to construct their general theory in operator algebraic framework that would cover known examples and that would be as elegant and symmetric as the one of Kac algebras. Important steps in this direction were made by S. Baaj and G. Skandalis [3], S.L. Woronowicz [51], [52], T. Masuda and Y. Nakagami [31] and A. Van Daele [50]. The general theory of l.c. quantum groups was proposed by J. Kustermans and the first author [24], [25] (see [26] for an overview).

It is natural to expect that the main ideas of [18] still work in this much more general framework. Indeed, generalizing the classical group extension theory, G.I. Kac explained there that in order to construct an extension of finite groups G_{1} and G_{2}, it is necessary and sufficient: 1) To define a pair of compatible actions of G_{1} and G_{2} on each other (as on sets) or, equivalently, G_{1} and G_{2} must be subgroups of a certain group G such that G_{1} ∩ G_{2} = {e} and any g ∈ G can be written as g = g_{1}g_{2} (g_{1} ∈ G_{1}, g_{2} ∈ G_{2}); so G_{1} and G_{2} must form a matched pair (this term was introduced later by W.M. Singer [42] and M. Takeuchi [44]). 2) To define a pair of compatible 2-cocycles for these actions, so G_{1} and G_{2} must form a cocycle matched pair. Then an arbitrary extension has a structure of their cocycle bicrossed product.

The last construction was studied intensively by S. Majid both in algebraic and in analytic aspects [27], [28], [29], [30]. In particular, in [27] he defines a matched pair of Hopf algebras and studies their bicrossed product, later in [29] and [30], 6.3 he considers cocycle bicrossed product of Hopf algebras. In [28] S. Majid defines a matched pair of l.c. groups with continuous mutual actions and constructs the corresponding bicrossed product Hopf-von Neumann algebra, which is a Kac algebra under some additional assumption (modularity). T. Yamanouchi proved [53] that one always gets a quasi Woronowicz algebra. Our more close analysis, involving measurable and almost everywhere defined actions, shows that in general this bicrossed product is a l.c. quantum group. Maijid also gave very interesting concrete examples of bicrossed products (without cocycles).

1Research Assistant of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.)

S. Baaj and G. Skandalis considered in [3] the bicrossed product construction in the framework of regular multiplicative unitaries; to keep themselves in this framework, these authors considered G_{1} and G_{2} as a matched pair of l.c. groups if they are closed subgroups of a l.c. group G such that the map (g_{1}, g_{2}) ↦ g_{1}g_{2} is a homeomorphism of G_{1} × G_{2} onto G. In [4] they extended this notion requiring the above map to be a homeomorphism of G_{1} × G_{2} only onto an open dense subset of G, which is very close to what we use in this paper. Let us mention also the algebraic papers [1], [2], [17], [32], [40], [42], [44] on extensions of Hopf algebras and especially [33], where A. Masuoka established interesting connections with extensions of Lie bialgebras.

All this, as well as the recent paper [38], served as a motivation for our work aimed at the construction of new examples of l.c. quantum groups. Let us explain its structure.

In Preliminaries we establish notations and briefly summarize the main definitions and results of the theory of l.c. quantum groups in the von Neumann algebraic setting.

Section 1 is in a sense preparatory, although we believe its results are of independent interest. Here we define and study cocycle actions of l.c. quantum groups on von Neumann algebras generalizing twisted group actions [35], [8], [37] and ordinary (without cocycles) l.c. quantum group actions [45]. We construct the von Neumann cocycle crossed product algebra which is an operator algebra version of the Hopf algebraic cocycle crossed product [34], Chapter 7, we perform the dual weight construction and obtain a canonical unitary implementation for these cocycle actions. Finally, we explain relations between cocycle crossed products and cleft extensions of von Neumann algebras, which is also motivated by the Hopf algebraic results [34], Chapter 7. Some facts concerning operator spaces [5]-[7], [12] are needed for this discussion.

Section 2 starts with the most general (and inevitably quite complicated) definition of a cocycle matched pair of l.c. quantum groups. Then we construct the corresponding cocycle bicrossed product (which is an operator algebra version of the Hopf algebraic construction described in [30], 6.3), show that it is a l.c. quantum group and compute its invariant weights, fundamental unitary and the dual l.c. quantum group. Finally, we investigate when the above cocycle bicrossed product is compact and discrete.

In Section 3 the notions of extensions and the cleft extensions of l.c. quantum groups are introduced and oneto-one correspondence between cleft extensions and cocycle bicrossed products is established. We also define isomorphic extensions and give necessary and sufficient conditions when two cleft extensions are isomorphic. These results are similar to Hopf algebraic results obtained in [2].

Section 4 is devoted to the study of the special, but the most important, case when the cocycle matched pair is formed by usual l.c. groups. When both groups are discrete, we extend the result obtained by G.I. Kac [18] for finite groups and show that every corresponding extension is cleft and so has a cocycle bicrossed product structure. Then for a matched pair of l.c. groups we describe the corresponding bicrossed product and compute all the ingredients of the corresponding l.c. quantum group characterizing the Kac algebra case (here we extend [28], Theorem 2.12). Finally, we introduce the group of extensions related to a matched pair of l.c. groups and discuss relations with cocycles in the sense of S. Baaj and G. Skandalis.

Discussing examples of matched pairs of l.c. groups and corresponding extensions in Section 5, we start with a brief survey of known examples distinguishing the cases when all extensions are or are not Kac algebras. Then we discuss the relation between l.c. quantum groups obtained from matched pairs of Lie groups and the corresponding infinitesimal objects: Hopf algebras and Lie bialgebras. Finally, we construct new examples of matched pairs of Lie groups and compute the corresponding extensions getting this way new examples of l.c. quantum groups. The description of examples presented in Subsections 5.3 and 5.4 was announced in [46]. In Subsection 5.5 we present the first, up to our knowledge, series of extensions with nontrivial cocycles which are not Kac algebras. For this we compute explicitly a family of corresponding 2-cocycles.

Acknowledgements. Both authors are grateful to Professor A. Van Daele for stimulating discussions. The first author wants to thank the research group in Leuven for the nice working atmosphere. The second author is grateful to the Katholieke Universiteit Leuven and to the Max-Planck-Institut für Mathematik in Bonn for kind hospitality during his work on this paper and to Professor D. Gurevich for helpful discussions.

Preliminaries

We denote by ⊗ tensor product of Hilbert spaces or von Neumann algebras and by Σ (resp., σ) the flip map on it. We also use the leg-numbering notation. For example, if H, K and L are Hilbert spaces and X ∈ B(H ⊗L), we denote by X_{13} (resp., X_{12}, X_{23}) the operator (1⊗Σ^{∗})(X ⊗1)(1⊗Σ) (resp., X ⊗1, 1⊗X) defined on H ⊗K ⊗L. If now H = H_{1}⊗H_{2} itself is a tensor product of two Hilbert spaces, then we sometimes switch from the leg-numbering notation with respect to H ⊗K ⊗L to the one with respect to the finer tensor product H_{1} ⊗ H_{2} ⊗ K ⊗ L, for example, from X_{13} to X_{124}. There is no confusion here, because the number of legs changes.

We denote the σ-strong^{∗} closure of a subset A of a von Neumann algebra N by A^{− σ-strong^{∗}} and we use [43] as a general reference to the modular theory of normal semifinite faithful (n.s.f. ) weights on von Neumann algebras. If θ is a weight on a von Neumann algebra N, we use the notations

M_{θ}^{+} = {x ∈ N^{+} | θ(x) < +∞}, N_{θ} = {x ∈ N | x^{∗}x ∈ M_{θ}^{+}} and M_{θ} = span M_{θ}^{+} .

If N_{0} is a von Neumann subalgebra of N and if T is an operator valued weight from N to N_{0} [43], Section 11.5, we denote by N_{T} the set of elements x ∈ N such that T (x^{∗}x) is a bounded operator. Recall that if T is a n.s.f. operator valued weight and θ_{0} is a n.s.f. weight on N_{0}, then θ := θ_{0} T defines a n.s.f. weight on N. The modular automorphism group of θ_{0} is the restriction of the modular automorphism group of θ to the von Neumann algebra N_{0}, [43], Section 11.9.

If θ is a n.s.f. weight on a von Neumann algebra N and if (H_{θ}, π_{θ}, Λ_{θ}) is a GNS-construction for θ, recall that the GNS-map Λ_{θ} is σ-strong^{∗}– norm closed. This expression means that the map Λ_{θ} is closed for the σ-strong^{∗} topology on N and the norm topology on H_{θ}. We denote by T_{θ} the Tomita ^{∗}-algebra, consisting of elements x ∈ N analytic w.r.t. the modular group σ^{θ} of θ and such that σ_{z}^{θ}(x) ∈ N_{θ} ∩ N_{θ}^{∗} for all z ∈ ℂ.

In this paper we use systematically l.c. quantum groups in von Neumann algebraic setting [25] including special types of morphisms in the definition of extensions in Section 3. However it should be mentioned that a comprehensive definition of morphisms and then of a category of l.c. quantum groups can be done only in the universal C^{∗}-algebraic setting [23]; this was known already for Kac algebras [14], Chapter 5.

A pair (M, Δ) is called a (von Neumann algebraic) l.c. quantum group [25] when

• M is a von Neumann algebra and Δ : M → M ⊗ M is a normal and unital ∗-homomorphism satisfying the coassociativity relation : (Δ ⊗ ι)Δ = (ι ⊗ Δ)Δ.

• There exist n.s.f. weights φ and ψ on M such that

– φ is left invariant in the sense that φ((ω ⊗ ι)Δ(x)) = φ(x)ω(1) for all x ∈ M_{φ}^{+} and ω ∈ M_{∗}^{+},

– ψ is right invariant in the sense that ψ((ι ⊗ ω)Δ(x)) = ψ(x)ω(1) for all x ∈ M_{ψ}^{+} and ω ∈ M_{∗}^{+}.

From [24], Theorem 7.14 we know that a left invariant weight on (M, Δ) is unique up to a positive scalar and the same holds for a right invariant weight.

Let us fix a left invariant n.s.f. weight φ on (M, Δ) and represent M on the GNS-space of φ such that (H, ι, Λ) is a GNS-construction for φ. Then we can define a unitary W on H ⊗ H by

W^{∗}(Λ(a) ⊗ Λ(b)) = (Λ ⊗ Λ)(Δ(b)(a ⊗ 1)) for all a, b ∈ N_{φ} .

Here Λ ⊗ Λ denotes the canonical GNS-map for the tensor product weight φ ⊗ φ. One proves that W satisfies the pentagonal equation: W_{12}W_{13}W_{23} = W_{23}W_{12}. We say that W is a multiplicative unitary. The comultiplication can be given in terms of W by the formula Δ(x) = W^{∗}(1 ⊗ x)W for all x ∈ M. Also the von Neumann algebra M can be written in terms of W as

M = {(ι ⊗ ω)(W) | ω ∈ B(H)_{∗}}^{− σ-strong^{∗}} .

Next the l.c. quantum group (M, Δ) has an antipode S, which is the unique σ-strong^{∗} closed linear map from M to M satisfying (ι⊗ω)(W) ∈ D(S) for all ω ∈ B(H)_{∗} and S(ι⊗ω)(W) = (ι⊗ω)(W^{∗}) and such that the elements (ι ⊗ ω)(W) form a σ-strong^{∗} core for S. S has a polar decomposition S = Rτ_{−i/2} where R is an anti-automorphism of M and (τ_{t}) is a strongly continuous one-parameter group of automorphisms of M. We call R the unitary antipode and (τ_{t}) the scaling group of (M, Δ). From [24], Proposition 5.26 we know that σ(R ⊗ R)Δ = ΔR. So φR is a right invariant weight on (M, Δ) and we take ψ := φR. Let us denote by (σ_{t}) the modular automorphism groups of φ. From [24], Proposition 6.8 we get the existence of a number ν > 0 such that ψ σ_{t} = ν^{−t} ψ for all t ∈ ℝ. Hence we get the existence of a unique positive, self-adjoint operator δ_{M} affiliated to M, such that σ_{t}(δ_{M}) = ν^{t} δ_{M} for all t ∈ ℝ and ψ = φ_{δ_{M}}, see [24], Definition 7.1. Formally this means that ψ(x) = φ(δ_{M}^{1/2}xδ_{M}^{1/2}), and for a precise definition of φ_{δ_{M}} we refer to [47]. The operator δ_{M} is called the modular element of (M, Δ). If δ_{M} = 1 we call (M, Δ) unimodular.

We use the notation Δ^{op} to denote the opposite comultiplication defined by Δ^{op} := σΔ.

From [24], Notation 7.2 we get the canonical GNS-construction (H, ι, Γ) for the n.s.f. weight ψ, formally given by Γ(x) = Λ(xδ_{M}^{1/2}). Then we can define another multiplicative unitary V by

V (Γ(a) ⊗ Γ(b)) = (Γ ⊗ Γ)(Δ(a)(1 ⊗ b)) for all a, b ∈ N_{ψ} .

The comultiplication can be expressed by Δ(x) = V (x ⊗ 1)V^{∗} for all x ∈ M.

The dual l.c. quantum group (M̂, Δ̂) is defined in [24], Section 8. Its von Neumann algebra M̂ is

M̂ = {(ω ⊗ ι)(W) | ω ∈ B(H)_{∗}}^{− σ-strong^{∗}}

and the comultiplication Δ̂(x) = ΣW (x ⊗ 1)W^{∗}Σ for all x ∈ M̂. If we turn the predual M_{∗} into a Banach algebra with product ω μ = (ω ⊗ μ)Δ and define

λ : M_{∗} → M̂ : λ(ω) = (ω ⊗ ι)(W),

then λ is a homomorphism and λ(M_{∗}) is a σ-strong^{∗} dense subalgebra of M̂. To construct explicitly a left invariant n.s.f. weight φ̂ with GNS-construction (H, ι, Λ̂), first introduce the space

I = {ω ∈ M_{∗} | there exists a vector ξ(ω) ∈ H such that ω(x^{∗}) = ⟨ξ(ω), Λ(x)⟩ for all x ∈ N_{φ}} .

If ω ∈ I then such a vector ξ(ω) clearly is uniquely determined. Next one proves that there exists a unique n.s.f. weight φ̂ on M̂ with GNS-construction (H, ι, Λ̂) such that λ(I) is a σ-strong^{∗}– norm core for Λ̂ and

Λ̂(λ(ω)) = ξ(ω) for all ω ∈ I .

One proves that the weight φ̂ is left invariant, and the associated multiplicative unitary is denoted by Ŵ. From [24], Proposition 8.16 it follows that Ŵ = ΣW^{∗}Σ.

Since (M̂, Δ̂) is again a l.c. quantum group, we can introduce the antipode Ŝ, the unitary antipode R̂ and the scaling group (τ̂_{t}) exactly as we did it for (M, Δ). Also, we can again construct the dual of (M̂, Δ̂), starting from the left invariant weight φ̂ with GNS-construction (H, ι, Λ̂). From [24], Theorem 8.29 we have that the bidual l.c. quantum group (M̂ ˆ , Δ̂ ˆ ) is isomorphic to (M, Δ). Also, defining Î and ξ̂ similarly to the definition of I and ξ, it follows from [24], Proposition 8.30 that, for all ω ∈ Î,

Λ((ι ⊗ ω)(W^{∗})) = ξ̂(ω) .

We denote by (σ̂_{t}) the modular automorphism groups of the weight φ̂. The modular conjugations of the weights φ and φ̂ will be denoted by J and Ĵ respectively. Then it is worthwhile to mention that

R(x) = Ĵx^{∗}Ĵ for all x ∈ M  and  R̂(y) = Jy^{∗}J for all y ∈ M̂ .

From [25], Proposition 2.15 we know that

V = (Ĵ ⊗ Ĵ)ΣW^{∗}Σ(Ĵ ⊗ Ĵ)

and in particular V ∈ M̂′ ⊗ M.

Let us mention important special cases of l.c. quantum groups.

a) If (τ_{t}) is trivial and the modular element δ_{M} is affiliated to the center of M, (M, Δ) becomes a Kac algebra [14]. Let us explain this in more detail. From [14] we know that (M, Δ) is a Kac algebra if and only if (τ_{t}) is trivial and σ_{t} R = R σ_{−t} for all t ∈ ℝ. Now denote by (σ′_{t}) the modular automorphism group of ψ. Because ψ = φR we get that σ′_{t} R = R σ_{−t} for all t ∈ ℝ. Hence (M, Δ) is a Kac algebra if and only if (τ_{t}) is trivial and σ′ = σ. From [47] we know that σ′_{t}(x) = δ_{M}^{it}σ_{t}(x)δ_{M}^{−it} for all x ∈ M and t ∈ ℝ. Hence σ′ = σ if and only if δ_{M} is affiliated to the center of M.

In particular, (M, Δ) is a Kac algebra if M is commutative. Then (M, Δ) is generated by a usual l.c. group G : M = L^{∞}(G), (Δf)(g, h) = f(gh), (Sf)(g) = f(g^{−1}), φ(f) = ∫ f(g) dg, where f ∈ L^{∞}(G), g, h ∈ G and we integrate with respect to the left Haar measure dg on G. The right invariant weight ψ is given by ψ(f) = ∫ f(g^{−1}) dg. The modular element δ_{M} is given by the strictly positive function g ↦ δ_{G}(g)^{−1}.

M acts on H = L^{2}(G) by multiplication and

(W ξ)(g, h) = ξ(g, g^{−1}h)

for all ξ ∈ H ⊗ H = L^{2}(G × G). Then M̂ = L(G) is the group von Neumann algebra generated by the operators (λ_{g})_{g∈G} of the left regular representation of G and Δ̂(λ_{g}) = λ_{g} ⊗ λ_{g}. Clearly, Δ̂^{op} := σΔ̂ = Δ̂, so Δ̂ is cocommutative.

b) A l.c. quantum group is called compact if its Haar measure is finite: φ(1) < +∞, which is equivalent to the fact that the norm closure of {(ι ⊗ ω)(W)|ω ∈ B(H)_{∗}} is a unital C^{∗}-algebra. A l.c. quantum group (M, Δ) is called discrete if (M̂, Δ̂) is compact.

Sometimes we refer to Hopf algebra theory, and then we use the Sweedler notation for Δ:

Δ(a) = ∑ a_{(1)} ⊗ a_{(2)} for all a ∈ M.

1 Cocycle crossed products and the dual weight construction

1.1 Cocycle actions, crossed products and the dual action

Definition 1.1. We call a pair (α, U) a cocycle action of a l.c. quantum group (M, Δ) on a von Neumann algebra N if

α : N → M ⊗ N

is a normal, injective and unital ∗-homomorphism,

U ∈ M ⊗ M ⊗ N

is a unitary, and if α and U satisfy

(ι ⊗ α)α(x) = U (Δ ⊗ ι)α(x) U^{∗} for all x ∈ N
(ι ⊗ ι ⊗ α)(U) (Δ ⊗ ι ⊗ ι)(U) = (1 ⊗ U) (ι ⊗ Δ ⊗ ι)(U) .

Consider some special cases. First, let (α, u) be a twisted action of a (separable) l.c. group G on a (σ-finite) von Neumann algebra N. This means that G → Aut N : s ↦ α_{s} and u : G × G → N are Borel maps, such that u takes values in the set of unitaries of N and

α_{s} α_{t} = Ad u(s, t) α_{st}
α_{r}(u(s, t)) u(r, st) = u(r, s) u(rs, t)

nearly everywhere. Putting M = L^{∞}(G), one can identify M ⊗ N with L^{∞}(G, N) and hence one can define α : N → L^{∞}(G) ⊗ N by (α(x))(s) = α_{s^{−1}}(x) for x ∈ N and s ∈ G. We also identify M ⊗ M ⊗ N with L^{∞}(G × G, N) and define U ∈ L^{∞}(G) ⊗ L^{∞}(G) ⊗ N by U(s, t) = u(t^{−1}, s^{−1}). Then (α, U) is a cocycle action of the commutative l.c. quantum group (L^{∞}(G), Δ_{G}) on N.

Second, if U = 1, we obtain the definition of an ordinary action α of (M, Δ) on N [45], Definition 1.1.

Third, let us explain the link with the Hopf algebra theory [34], Definition 4.1.1 and Lemma 7.1.2. Let (H, Δ, S, ε) be a Hopf algebra and A a unital algebra equipped with a linear map H ⊗ A → A : h ⊗ a → h · a such that

h · (ab) = ∑(h_{(1)} · a) (h_{(2)} · b) and h · 1 = ε(h)1

and an invertible linear map σ : H ⊗ H → A such that

h · (k · a) = ∑ σ(h_{(1)}, k_{(1)}) ((h_{(2)}k_{(2)}) · a) σ^{−1}(h_{(3)}, k_{(3)})
∑(h_{(1)} · σ(k_{(1)}, m_{(1)})) σ(h_{(2)}, k_{(2)}m_{(2)}) = ∑ σ(h_{(1)}, k_{(1)}) σ(h_{(2)}k_{(2)}, m) .

Here the invertibility of σ means that there exists a linear map σ^{−1} from H ⊗ H to A for which

∑ σ(h_{(1)}, k_{(1)}) σ^{−1}(h_{(2)}, k_{(2)}) = ε(h)ε(k)1 = ∑ σ^{−1}(h_{(1)}, k_{(1)}) σ(h_{(2)}, k_{(2)}) .

If now H is finite dimensional, the linear dual Ĥ is a Hopf algebra by (ωμ)(h) = ∑ ω(h_{(1)})μ(h_{(2)}) and ∑ ω_{(1)}(h)ω_{(2)}(k) = ω(kh) for all ω, μ ∈ Ĥ and h, k ∈ H. Observe that we use the opposite comultiplication on Ĥ. Identifying Ĥ ⊗ A with the space of linear maps from H to A one defines α : A → Ĥ ⊗ A by α(a)(h) = h · a. Then α is a unital homomorphism. Further identifying Ĥ ⊗ Ĥ ⊗ A with the space of linear maps from H ⊗ H to A we define U ∈ Ĥ ⊗ Ĥ ⊗ A by U(h, k) = σ(k, h). Then U is invertible and

(ι ⊗ α)α(a) = U (Δ ⊗ ι)α(a) U^{−1}
(ι ⊗ ι ⊗ α)(U) (Δ ⊗ ι ⊗ ι)(U) = (1 ⊗ U) (ι ⊗ Δ ⊗ ι)(U) .

In this setting without involution it is natural to replace U^{∗} by U^{−1}, so the Hopf algebraic definition of a cocycle action agrees with our definition.

Let us now start with the operator algebraic theory of cocycle actions and their crossed products.

Notation 1.2. If (α, U) is a cocycle action of (M, Δ) on a von Neumann algebra N we denote

W̃ = (W ⊗ 1)U^{∗}

the unitary in M ⊗ B(H) ⊗ N.

Definition 1.3. Given a cocycle action (α, U) of a l.c. quantum group (M, Δ) on a von Neumann algebra N, the cocycle crossed product M _{α,U}⋉ N is the von Neumann subalgebra of B(H) ⊗ N generated by

α(N) and {(ω ⊗ ι ⊗ ι)(W̃) | ω ∈ M_{∗}} .

As in the case of ordinary actions [45], Proposition 2.2 we can define a dual action α̂ of (M̂, Δ̂^{op}) on M _{α,U}⋉ N with trivial cocycle.

Proposition 1.4. There exists a unique action α̂ of (M̂, Δ̂^{op}) on M _{α,U}⋉ N such that

α̂(α(x)) = 1 ⊗ α(x) for all x ∈ N
(ι ⊗ α̂)(W̃) = W_{12}W̃_{134} .

Moreover denoting by Ṽ the unitary (J ⊗ J)ΣW Σ(J ⊗ J) we have

α̂(z) = (Ṽ ⊗ 1)(1 ⊗ z)(Ṽ^{∗} ⊗ 1) for all z ∈ M _{α,U}⋉ N.

Proof. Observe that Ṽ ∈ M̂ ⊗ M′ and Ṽ (1 ⊗ x)Ṽ^{∗} = Δ̂^{op}(x) for all x ∈ M̂. So one gets for every x ∈ N

(Ṽ ⊗ 1)(1 ⊗ α(x))(Ṽ^{∗} ⊗ 1) = 1 ⊗ α(x) .

Next we have

Ṽ_{23}W̃_{134}Ṽ_{23}^{∗} = Ṽ_{23}W_{13}U_{134}^{∗}Ṽ_{23}^{∗} = Ṽ_{23}W_{13}Ṽ_{23}^{∗} U_{134}^{∗}
= (ι ⊗ Δ̂^{op})(W)_{123} U_{134}^{∗} = W_{12}W_{13}U_{134}^{∗} = W_{12}W̃_{134} .

So it is clear that we can define a normal, unital and injective ∗-homomorphism

α̂ : M _{α,U}⋉ N → M̂ ⊗ (M _{α,U}⋉ N)

by α̂(z) = (Ṽ ⊗ 1)(1 ⊗ z)(Ṽ^{∗} ⊗ 1) for all z ∈ M _{α,U}⋉ N, and then α̂ satisfies α̂(α(x)) = 1 ⊗ α(x) for x ∈ N and (ι ⊗ α̂)(W̃) = W_{12}W̃_{134}. It is obvious that

(ι ⊗ α̂)α̂(α(x)) = (Δ̂^{op} ⊗ ι)α̂(α(x))

for all x ∈ N. Further we have

(ι ⊗ ι ⊗ α̂)(ι ⊗ α̂)(W̃) = (ι ⊗ ι ⊗ α̂)(W_{12}W̃_{134})
= W_{12}W_{13}W̃_{145} = (ι ⊗ Δ̂^{op})(W)_{123}W̃_{145}
= (ι ⊗ Δ̂^{op} ⊗ ι)(W_{12}W̃_{134}) = (ι ⊗ Δ̂^{op} ⊗ ι)(ι ⊗ α̂)(W̃) .

Both statements together give (ι ⊗ α̂)α̂(z) = (Δ̂^{op} ⊗ ι)α̂(z) for all z ∈ M _{α,U}⋉ N. Hence α̂ is indeed an action of (M̂, Δ̂^{op}) on the von Neumann algebra M _{α,U}⋉ N. The uniqueness statement is obvious. Let us get another formula for α̂. Since M _{α,U}⋉ N ⊂ B(H) ⊗ N, the following proposition makes sense. Proposition 1.5. For all z ∈ M _{α,U}⋉ N we have

α̂(z) = W̃ (ι ⊗ α)(z)W̃^{∗} .

Proof. By definition (ι ⊗ α)α(x) = W̃^{∗}(1 ⊗ α(x))W̃, so

W̃ (ι ⊗ α)α(x)W̃^{∗} = 1 ⊗ α(x) = α̂(α(x)) .

Next we have

W̃_{234}(ι ⊗ ι ⊗ α)(W̃)W̃_{234}^{∗} = W̃_{234}W_{12}(ι ⊗ ι ⊗ α)(U^{∗})W̃_{234}^{∗}
= W̃_{234}W_{12} (Δ ⊗ ι ⊗ ι)(U)(ι ⊗ Δ ⊗ ι)(U^{∗})(1 ⊗ U^{∗}) W̃_{234}^{∗}
= W_{23}W_{12}W_{23}^{∗}U_{134}^{∗} = W_{12}W_{13}U_{134}^{∗}
= W_{12}W̃_{134} = (ι ⊗ α̂)(W̃) .

Both computations together give the formula stated in the proposition. The following result will be needed in Section 5. Proposition 1.6. Let x ∈ N. Then α(x) commutes with the elements {(ω ⊗ ι ⊗ ι)(W̃) | ω ∈ B(H)_{∗}} if and only if α(x) = 1 ⊗ x. Proof. Since (ι ⊗ α)α(x) = W̃^{∗}(1 ⊗ α(x))W̃, α(x) commutes with the elements {(ω ⊗ ι ⊗ ι)(W̃) | ω ∈ B(H)_{∗}} if and only if (ι⊗α)α(x) = 1⊗α(x). Since ι⊗α is faithful this equality is valid if and only if α(x) = 1⊗x.

1.2 Stabilization of cocycle actions

Definition 1.7. A cocycle action (α, U) of (M, Δ) on a von Neumann algebra N is said to be stabilizable with a unitary X ∈ M ⊗ N if

(1 ⊗ X)(ι ⊗ α)(X) = (Δ ⊗ ι)(X)U^{∗} .

Proposition 1.8. Let (α, U) be a cocycle action of (M, Δ) on N which is stabilizable with a unitary X ∈ M ⊗ N. Then the formulas

β : N → M ⊗ N : β(x) = Xα(x)X^{∗} and Φ : z ↦ X^{∗}zX

define respectively an action of (M, Δ) on N and a ∗-isomorphism from M _{β}⋉ N onto M _{α,U}⋉ N satisfying

α̂Φ = (ι ⊗ Φ)β̂ .

Proof. Define β as above. Then one has for all x ∈ N

(ι ⊗ β)β(x) = (1 ⊗ X)(ι ⊗ α)(X) (ι ⊗ α)α(x) (ι ⊗ α)(X^{∗})(1 ⊗ X^{∗})
= (Δ ⊗ ι)(X)U^{∗} U(Δ ⊗ ι)α(x)U^{∗} U(Δ ⊗ ι)(X^{∗}) = (Δ ⊗ ι)β(x) .

Hence β is an action of (M, Δ) on N. Considering Φ as an isomorphism of B(H) ⊗ N, we clearly have Φ(β(x)) = α(x) for all x ∈ N and

(ι ⊗ Φ)(W ⊗ 1) = (1 ⊗ X^{∗})(W ⊗ 1)(1 ⊗ X) = (W ⊗ 1)(Δ ⊗ ι)(X^{∗})(1 ⊗ X)
= (W ⊗ 1)U^{∗}(ι ⊗ α)(X^{∗}) = W̃ (ι ⊗ α)(X^{∗}) .

So Φ(M _{β}⋉ N) ⊂ M _{α,U}⋉ N; similarly one proves the converse inclusion. Hence Φ is an isomorphism of M _{β}⋉ N onto M _{α,U}⋉ N.

Recalling the definition of the dual actions α̂ and β̂ and the unitary Ṽ implementing them, we only have to observe that 1 ⊗ X and Ṽ ⊗ 1 commute to get the formula (ι ⊗ Φ)β̂ = α̂Φ.

The next proposition shows that many cocycle actions are stabilizable.

Proposition 1.9. Let (α, U) be a cocycle action of (M, Δ) on N. Then (α ⊗ ι, U ⊗ 1) is a cocycle action of (M, Δ) on N ⊗ B(H) which is stabilizable with the unitary

X = V_{31}^{∗}U_{312}^{∗} .

Proof. It is easy to check that X ∈ M ⊗ N ⊗ B(H). Next we have to prove that

(1 ⊗ X)(ι ⊗ α ⊗ ι)(X) = (Δ ⊗ ι ⊗ ι)(X)(U^{∗} ⊗ 1) .

So we have to prove that

X_{341}(ι ⊗ ι ⊗ α)(X_{231}) = (ι ⊗ Δ ⊗ ι)(X_{231})U_{234}^{∗} .

Starting the computation on the left hand side we get

X_{341}(ι ⊗ ι ⊗ α)(X_{231}) = V_{13}^{∗}U_{134}^{∗} V_{12}^{∗}(ι ⊗ ι ⊗ α)(U^{∗})
= V_{13}^{∗}U_{134}^{∗} V_{12}^{∗} (Δ ⊗ ι ⊗ ι)(U)(ι ⊗ Δ ⊗ ι)(U^{∗})U_{234}^{∗}
= V_{13}^{∗}U_{134}^{∗} U_{134}V_{12}^{∗} (ι ⊗ Δ ⊗ ι)(U^{∗})U_{234}^{∗}
= (ι ⊗ Δ)(V^{∗})_{123}(ι ⊗ Δ ⊗ ι)(U^{∗})U_{234}^{∗}
= (ι ⊗ Δ ⊗ ι)(X_{231})U_{234}^{∗} .

This computation proves our result.

Let us give now a definition of the fixed point algebra of a cocycle action.

Definition 1.10. Let (α, U) be a cocycle action of (M, Δ) on N. Then

N^{α} = {x ∈ N | α(x) = 1 ⊗ x}

is called the fixed point algebra of (α, U). It is clear that N^{α} is a von Neumann subalgebra of N.

Theorem 1.11. Let (α, U) be a cocycle action of a l.c. quantum group (M, Δ) on a von Neumann algebra N and α̂ the dual action of (M̂, Δ̂^{op}) on the cocycle crossed product M _{α,U}⋉ N. Then

1. (M _{α,U}⋉ N)α̂ = α(N) ,

2. (M _{α,U}⋉ N ∪ M′ ⊗ ℂ)′′ = B(H) ⊗ N ,

3. M _{α,U}⋉ N = ⟨(ω ⊗ ι ⊗ ι)(W̃)α(x) | x ∈ N, ω ∈ M_{∗}⟩^{−σ-strong^{∗}} .

Proof. First let (α, U) be stabilizable with a unitary X. Using the action β and the isomorphism Φ of Proposition 1.8 we get the first two statements, because they are valid for ordinary actions [45], Theorems 2.6 and 2.7. Recalling that (ι ⊗ Φ)(W ⊗ 1) = W̃ (ι ⊗ α)(X^{∗}) and using that

M _{β}⋉ N = ⟨((ω ⊗ ι)(W) ⊗ 1) β(x) | x ∈ N, ω ∈ M_{∗}⟩^{− σ-strong^{∗}}

we get

M _{α,U}⋉ N = ⟨(ω ⊗ ι ⊗ ι)(W̃ (ι ⊗ α)(X^{∗})) α(x) | x ∈ N, ω ∈ M_{∗}⟩^{−σ-strong^{∗}} .

Now we define the following linear subspace of M _{α,U}⋉ N :

O := ⟨(ω ⊗ ι ⊗ ι)(W̃)α(x) | x ∈ N, ω ∈ M_{∗}⟩^{−σ-strong^{∗}} .

Choose ξ, η ∈ H, x ∈ N and an orthonormal basis (e_{i})_{i∈I} for H. Then we have with σ-strong^{∗} convergence

(ω_{ξ,η} ⊗ ι ⊗ ι)(W̃ (ι ⊗ α)(X^{∗}))α(x) = ∑_{i∈I}(ω_{e_{i},η} ⊗ ι ⊗ ι)(W̃)α((ω_{ξ,e_{i}} ⊗ ι)(X^{∗})x) ∈ O .

So M _{α,U}⋉ N ⊂ O and hence O = M _{α,U}⋉ N. This proves the result for stabilizable actions. From Proposition 1.9 the general result now follows.

1.3 The dual weight construction

Theorem 1.11 shows that α(N) is the fixed point algebra of the dual action α̂ of (M̂, Δ̂^{op}) on M _{α,U}⋉ N, so the formula T := (φ̂ ⊗ ι ⊗ ι)α̂ defines a normal and faithful operator valued weight from M _{α,U}⋉ N to α(N) [45], Proposition 1.3. To define then the dual weight θ̃ for a given weight θ on N by the formula θ̃ = θα^{−1}T, we have to prove first that T is semifinite.

Lemma 1.12. We still write T := (φ̂ ⊗ ι ⊗ ι)α̂. For every ω ∈ I and x ∈ N we have

(ω ⊗ ι ⊗ ι)(W̃)α(x) ∈ N_{T}

and

T (α(x^{∗})(ω ⊗ ι ⊗ ι)(W̃)^{∗}(ω ⊗ ι ⊗ ι)(W̃)α(x)) = ∥ξ(ω)∥^{2}α(x^{∗}x) .

Proof. Let N act on K. Choose ω ∈ I, x ∈ N, a vector ξ ∈ H ⊗ K and an orthonormal basis (e_{i})_{i∈I} for H ⊗ K. Define

z := α(x^{∗})(ω ⊗ ι ⊗ ι)(W̃)^{∗}(ω ⊗ ι ⊗ ι)(W̃)α(x) .

Then we get

(ι ⊗ ω_{ξ,ξ})α̂(z) = (ι ⊗ ω_{α(x)ξ,α(x)ξ})((ω ⊗ ι ⊗ ι ⊗ ι)(W_{12}W̃_{134})^{∗}(ω ⊗ ι ⊗ ι ⊗ ι)(W_{12}W̃_{134}))

= ∑_{i∈I}((ι ⊗ ω_{α(x)ξ,e_{i}})(W̃)ω ⊗ ι)(W)^{∗}((ι ⊗ ω_{α(x)ξ,e_{i}})(W̃)ω ⊗ ι)(W) .

Because φ̂ is normal we get that

φ̂(ι ⊗ ω_{ξ,ξ})α̂(z) = ∑_{i∈I}∥ξ((ι ⊗ ω_{α(x)ξ,e_{i}})(W̃)ω)∥^{2}

= ∑_{i∈I} ∥(ι ⊗ ω_{α(x)ξ,e_{i}})(W̃)ξ(ω)∥^{2}

= ∥ξ(ω) ⊗ α(x)ξ∥^{2} = ∥ξ(ω)∥^{2}ω_{ξ,ξ}(α(x^{∗}x)) .

This final computation proves the lemma.

Combining this lemma with Theorem 1.11 one gets that the operator valued weight T is semifinite. So one can use the theory of operator valued weights [43] to give the following definition.

Definition 1.13. Let (α, U) be a cocycle action of (M, Δ) on N. Given a n.s.f. weight θ on N we can define the dual n.s.f. weight θ̃ on M _{α,U}⋉ N by the formula

θ̃ = θα^{−1} (φ̂ ⊗ ι ⊗ ι)α̂ .

Let us fix a n.s.f. weight θ on N with GNS-construction (K, π_{θ}, Λ_{θ}). Let θ̃ be the dual weight of θ. Lemma 1.12 implies by polarization the following formula for θ̃ on a dense subset of M _{α,U}⋉ N.

Corollary 1.14. For all ω, μ ∈ I and x, y ∈ N_{θ} the element (ω ⊗ ι ⊗ ι)(W̃)α(x) belongs to Nθ̃ and

θ̃(α(y^{∗})(μ ⊗ ι ⊗ ι)(W̃)^{∗}(ω ⊗ ι ⊗ ι)(W̃)α(x)) = ⟨ξ(ω) ⊗ Λ_{θ}(x), ξ(μ) ⊗ Λ_{θ}(y)⟩.

The application of the dual weight θ̃ is based on a concrete GNS-construction for it. The corollary above suggests to define a GNS-map Λ̃ for θ̃ by the formula

Λ̃((ω ⊗ ι ⊗ ι)(W̃)α(x)) = ξ(ω) ⊗ Λ_{θ}(x) ,

but it is not enough to know that the elements (ω ⊗ ι ⊗ ι)(W̃)α(x) span a dense set of M _{α,U}⋉ N, we need to prove that they form a core for a GNS-map for θ̃.

Proposition 1.15. Let (α, U) be a cocycle action of (M, Δ) on N and θ a n.s.f. weight on N with GNSconstruction (K, π_{θ}, Λ_{θ}). Then there exists a unique GNS-construction (H ⊗ K, ι ⊗ π_{θ}, Λ̃) for the dual weight θ̃ satisfying

1. ⟨(ω_{η,Λ(b)} ⊗ ι ⊗ ι)(W̃)α(x) | η ∈ H, b ∈ T_{φ}, x ∈ N_{θ}⟩ is a σ-strong^{∗}– norm core for Λ̃.

2. Λ̃((ω ⊗ ι ⊗ ι)(W̃)α(x)) = ξ(ω) ⊗ Λ_{θ}(x) for all ω ∈ I and x ∈ N_{θ}.

Proof. Let N act on K by π_{θ}, hence we may assume that π_{θ} = ι. Suppose first that (α, U) is stabilizable with the unitary X. Then we can construct the action β and the isomorphism Φ as in Proposition 1.8. Because (ι ⊗ Φ)β̂ = α̂Φ we get immediately that θ̃Φ = θ̃_{β}, where θ̃_{β} is the dual weight of θ on M _{β}⋉ N. Let (H ⊗ K, ι, Λ̃_{β}) be the canonical GNS-construction for θ̃_{β} as defined in [45], Definition 3.4. Define for z ∈ Nθ̃

Λ̃(z) = X^{∗}Λ̃_{β}(Φ^{−1}(z)).

Then (H ⊗ K, ι, Λ̃) is a GNS-construction for θ̃. The isomorphism Φ and [45], Propositions 3.10 and 7.1 give

⟨(ω_{η,Λ(b)} ⊗ ι ⊗ ι)(W̃ (ι ⊗ α)(X^{∗})) α(x) | η ∈ H, b ∈ T_{φ}, x ∈ N_{θ}⟩

is a σ-strong^{∗}– norm core for Λ̃ and

Λ̃((ω ⊗ ι ⊗ ι)(W̃ (ι ⊗ α)(X^{∗})) α(x)) = X^{∗}(ξ(ω) ⊗ Λ_{θ}(x))  (1.1)

for all ω ∈ I and x ∈ N_{θ}. By Corollary 1.14 we can define a unique isometry V : H ⊗ K → H ⊗ K such that

V(ξ(ω) ⊗ Λ_{θ}(x)) = Λ̃((ω ⊗ ι ⊗ ι)(W̃)α(x))

for all ω ∈ I and x ∈ N_{θ}. Later we will show that V = 1 to get the second statement of the proposition. Let

D_{0} := ⟨(ω_{η,Λ(b)} ⊗ ι ⊗ ι)(W̃)α(x) | η ∈ H, b ∈ T_{φ}, x ∈ N_{θ}⟩

and D the domain of the σ-strong^{∗}– norm closure of the restriction of Λ̃ to D_{0}. Choose η ∈ H, b ∈ T_{φ}, x ∈ N_{θ} and an orthonormal basis (e_{i})_{i∈I} in H. Define

z := (ω_{η,Λ(b)} ⊗ ι ⊗ ι)(W̃ (ι ⊗ α)(X^{∗})) α(x)

and for every finite subset I_{0} ⊂ I

z_{I_{0}} := ∑_{i∈I_{0}}(ω_{e_{i},Λ(b)} ⊗ ι ⊗ ι)(W̃) α((ω_{η,e_{i}} ⊗ ι)(X^{∗})x) .

Then every z_{I_{0}} belongs to D_{0} and, using [45], Proposition 7.1, we get

Λ̃(z_{I_{0}}) = V(∑_{i∈I_{0}} ξ(ω_{e_{i},Λ(b)}) ⊗ (ω_{η,e_{i}} ⊗ ι)(X^{∗})Λ_{θ}(x))

= V(∑_{i∈I_{0}} Jσ_{i/2}(b)Je_{i} ⊗ (ω_{η,e_{i}} ⊗ ι)(X^{∗})Λ_{θ}(x)) .

So we observe that (Λ̃(z_{I_{0}}))_{I_{0}⊂I} converges in norm to

V (Jσ_{i/2}(b)J ⊗ 1) X^{∗} (η ⊗ Λ_{θ}(x)) = V X^{∗} (Jσ_{i/2}(b)Jη ⊗ Λ_{θ}(x)) = V X^{∗} (ξ(ω_{η,Λ(b)}) ⊗ Λ_{θ}(x)) .

Further (z_{I_{0}})_{I_{0}⊂I} converges in σ-strong^{∗} topology to z. So we may conclude that z ∈ D and

Λ̃(z) = V X^{∗} (ξ(ω_{η,Λ(b)}) ⊗ Λ_{θ}(x)) .

In the beginning of the proof we saw that the elements z span a σ-strong^{∗}– norm core for Λ̃, so D_{0} is a σ-strong^{∗}– norm core for Λ̃ and by equation (1.1)

Λ̃(z) = X^{∗}(ξ(ω_{η,Λ(b)}) ⊗ Λ_{θ}(x)),

so V = 1. Hence we have also proved the second statement of the proposition for stabilizable cocycle actions.

Consider now the general case. Choose a n.s.f. trace Tr on B(H). Then θ ⊗Tr is a n.s.f. weight on N ⊗B(H), and (α ⊗ ι, U ⊗ 1) is a stabilizable cocycle action of (M, Δ) on N ⊗ B(H). For β = α ⊗ ι it is clear that

M _{β,U⊗1}⋉ (N ⊗ B(H)) = (M _{α,U}⋉ N) ⊗ B(H)

and β̂ = α̂ ⊗ ι. Then it is clear that the dual weight of θ ⊗ Tr equals to the tensor product weight θ̃ ⊗ Tr.

Choose now a GNS-construction (L, π_{Tr}, Λ_{Tr}) for Tr and let (K ⊗ L, ι ⊗ π_{Tr}, Λ_{θ} ⊗ Λ_{Tr}) be the canonical GNS-construction for the tensor product weight θ ⊗ Tr. By the previous part of the proof we get a GNSconstruction (H ⊗ K ⊗ L, ι ⊗ ι ⊗ π_{Tr}, Λ̃_{1}) for the dual weight θ̃ ⊗ Tr, such that

1. Λ̃_{1}((ω ⊗ ι ⊗ ι)(W̃)α(x) ⊗ y) = ξ(ω) ⊗ Λ_{θ}(x) ⊗ Λ_{Tr}(y) for all ω ∈ I, x ∈ N_{θ} and y ∈ N_{Tr}.

2. ⟨(ω_{η,Λ(b)} ⊗ ι ⊗ ι)(W̃)α(x) ⊗ y | η ∈ H, b ∈ T_{φ}, x ∈ N_{θ}, y ∈ N_{Tr}⟩ is a σ-strong^{∗}– norm core for Λ̃_{1}.

To deduce the second statement above from the first part of the proof, observe that the elements x ⊗ y with x ∈ N_{θ} and y ∈ N_{Tr} span a σ-strong^{∗}– norm core for Λ_{θ} ⊗ Λ_{Tr}.

Choose now an arbitrary GNS-construction (K_{2}, π, Λ̃_{2}) for the weight θ̃. The lemma following this proposition shows that

⟨(ω_{η,Λ(b)} ⊗ ι ⊗ ι)(W̃)α(x) | η ∈ H, b ∈ T_{φ}, x ∈ N_{θ}⟩

is a σ-strong^{∗}– norm core for Λ̃_{2}. By Corollary 1.14 there is a unique isometry V : H ⊗ K → K_{2} such that

V(ξ(ω) ⊗ Λ_{θ}(x)) = Λ̃_{2}((ω ⊗ ι ⊗ ι)(W̃)α(x))

for all ω ∈ I and x ∈ N_{θ}. Because the elements (ω ⊗ ι ⊗ ι)(W̃)α(x) span a core for Λ̃_{2}, V is unitary and we can define a new GNS-construction (H ⊗ K, ρ, Λ̃) for θ̃ by ρ(z) = V^{∗}π(z)V and Λ̃(z) = V^{∗}Λ̃_{2}(z). Of course the elements (ω ⊗ ι ⊗ ι)(W̃)α(x) still span a core of Λ̃ and

Λ̃((ω ⊗ ι ⊗ ι)(W̃)α(x)) = ξ(ω) ⊗ Λ_{θ}(x)

for all ω ∈ I and x ∈ N_{θ}. Let (H ⊗ K ⊗ L, ρ ⊗ π_{Tr}, Λ̃ ⊗ Λ_{Tr}) be the canonical GNS-construction for the weight θ̃ ⊗ Tr. Then Λ̃ ⊗ Λ_{Tr} and Λ̃_{1} agree on a core for both Λ̃ ⊗ Λ_{Tr} and Λ̃_{1}. Hence Λ̃ ⊗ Λ_{Tr} = Λ̃_{1}, from where ρ ⊗ π_{Tr} = ι ⊗ ι ⊗ π_{Tr}. Then finally ρ = ι, which concludes the proof.

Now we still have to prove the following lemma.

Lemma 1.16. Let N_{1} and N_{2} be von Neumann algebras and φ_{i} a n.s.f. weight on N_{i} (i = 1, 2). Suppose that (K_{1}, π_{1}, Λ_{1}) is a GNS-construction for φ_{1} and that (K, π, Λ) is a GNS-construction for the tensor product weight φ_{1} ⊗ φ_{2}. Suppose further that D is a linear subspace of N_{φ_{1}} such that the algebraic tensor product D ⊙ N_{φ_{2}} is a σ-strong^{∗}– norm core for Λ. Then D is a σ-strong^{∗}– norm core for Λ_{1}.

Proof. Choose a GNS-construction (K_{2}, π_{2}, Λ_{2}) for φ_{2}. Because all GNS-constructions for φ_{1} ⊗ φ_{2} are isomorphic, we may assume that

(K, π, Λ) = (K_{1} ⊗ K_{2}, π_{1} ⊗ π_{2}, Λ_{1} ⊗ Λ_{2}) .

Choose now a ∈ N_{φ_{1}}. Fix an element b ∈ N_{φ_{2}}, b ≠ 0 and fix elements c, d ∈ T_{φ_{2}} such that φ_{2}(d^{∗}bc) = 1.

Choose then a net (x_{α})_{α} in D ⊙ N_{φ_{2}} such that

x_{α} → a ⊗ b σ-strong^{∗} and (Λ_{1} ⊗ Λ_{2})(x_{α}) → Λ_{1}(a) ⊗ Λ_{2}(b) in norm .

Then ((ι ⊗ ω_{Λ_{2}(c),Λ_{2}(d)})(x_{α}))_{α} will be a net in D, converging σ-strong^{∗} to a. Further it is easy to verify that

Λ_{1}((ι ⊗ ω_{Λ_{2}(c),Λ_{2}(d)})(x_{α})) = (1 ⊗ θ_{Λ_{2}(dσ_{i}^{2}(c)^{∗})}^{∗})(Λ_{1} ⊗ Λ_{2})(x_{α})

and this converges in norm to

(1 ⊗ θ_{Λ_{2}(dσ_{i}^{2}(c)^{∗})}^{∗})(Λ_{1}(a) ⊗ Λ_{2}(b)) = Λ_{1}(a) .

So, indeed D is a σ-strong^{∗}– norm core for Λ_{1}.

Terminology 1.17. If (α, U) is a cocycle action of (M, Δ) on N and θ is a n.s.f. weight on N with GNSconstruction (K, π_{θ}, Λ_{θ}), then the GNS-construction (H ⊗ K, ι ⊗ π_{θ}, Λ̃) for θ̃ obtained in Proposition 1.15 is called the canonical GNS-construction for θ̃.

1.4 The unitary implementation of a cocycle action and the biduality theorem

The results of this subsection follow from Propositions 1.8 and 1.9 and from similar results for ordinary actions [45], so we omit their proofs.

Definition 1.18. Let (α, U) be a cocycle action of (M, Δ) on N and let θ be a n.s.f. weight on N with GNS-construction (K, π_{θ}, Λ_{θ}). Let (H ⊗ K, ι ⊗ π_{θ}, Λ̃) be the canonical GNS-construction for the dual weight θ̃. Denote with J_{θ} and J̃ the modular conjugations of θ and θ̃ in these GNS-constructions. Then

U_{α} = J̃(Ĵ ⊗ J_{θ})

is called the unitary implementation of (α, U) obtained by θ.

To justify this terminology let us state the following proposition.

Proposition 1.19. With the notation of the previous definition we have

1. (ι ⊗ π_{θ})α(x) = U_{α}(1 ⊗ π_{θ}(x))U_{α}^{∗} for all x ∈ N .

2. U_{α} ∈ M ⊗ B(K) .

3. (Δ ⊗ ι)(U_{α}) = (ι ⊗ ι ⊗ π_{θ})(U^{∗}) U_{α 23}U_{α 13} (Ĵ ⊗ Ĵ ⊗ J_{θ})(ι ⊗ ι ⊗ π_{θ})(U_{213})(Ĵ ⊗ Ĵ ⊗ J_{θ}) .

4. (Ĵ ⊗ J_{θ})U_{α} = U_{α}^{∗}(Ĵ ⊗ J_{θ}) .

In fact, the unitary implementation U_{α} of (α, U) does not depend on the choice of a weight θ on N. If θ and θ′ are both n.s.f. weights on N with GNS-constructions (K, π_{θ}, Λ_{θ}) and (K′, π′_{θ}, Λ′_{θ}) and u is the unitary from K onto K′ intertwining π_{θ} and π′_{θ} and mapping the positive cone of K onto the one of K′, then

U′_{α} = (1 ⊗ u)U_{α}(1 ⊗ u^{∗}) .

Finally we state the biduality theorem.

Proposition 1.20. Let (α, U) be a cocycle action of (M, Δ) on N. Then the formula

Φ : z ↦ W̃ (ι ⊗ α)(z)W̃^{∗}

defines a ∗-isomorphism of B(H) ⊗ N onto M̂ α̂⋉ (M _{α,U}⋉ N) satisfying

Φ(z) = α̂(z) for all z ∈ M _{α,U}⋉ N ,
Φ(x ⊗ 1) = x ⊗ 1 ⊗ 1 for all x ∈ M′ .

Defining

μ : B(H) ⊗ N → M ⊗ B(H) ⊗ N : μ(z) = (ΣV^{∗} ⊗ 1)U^{∗}(ι ⊗ α)(z)U(V Σ ⊗ 1)

we have (J ⊗ Φ)μ = α̂ ˆ Φ, where J (x) = JĴxĴJ for all x ∈ M.

Let us mention that up to the obvious isomorphism between B(H)⊗N and N ⊗B(H) the action μ precisely agrees with the stabilized action of (M, Δ) on N ⊗ B(H) given by combining Propositions 1.8 and 1.9. Combining the previous proposition with the formula

α̂(M _{α,U}⋉ N) = (M̂ α̂⋉ (M _{α,U}⋉ N))^{α̂} ˆ

we get the following corollary.

Corollary 1.21. Let (α, U) be a cocycle action of (M, Δ) on N. Then we have

M _{α,U}⋉ N = {z ∈ B(H) ⊗ N | (V^{∗} ⊗ 1)U^{∗}(ι ⊗ α)(z)U(V ⊗ 1) = z_{13}} .

1.5 Cocycle crossed products and cleft extensions of von Neumann algebras

Here we answer the following question. Let (M, Δ) be a l.c. quantum group and let θ : N → M̂ ⊗ N be an action of (M̂, Δ̂^{op}) on the von Neumann algebra N. When does N have the structure of a cocycle bicrossed product such that θ agrees with the dual action?

Proposition 1.22. Let N be a von Neumann algebra and (M, Δ) a l.c. quantum group. Suppose that θ : N → M̂ ⊗ N is an action of (M̂, Δ̂^{op}) on N. Then the following assertions are equivalent.

1. There exists a cocycle action (α, U) of (M, Δ) on N^{θ} such that

(N, θ) ≅ (M _{α,U}⋉ N^{θ}, α̂) .

2. There exists a unitary X ∈ M ⊗ N such that

(ι ⊗ θ)(X) = W_{12}X_{13} .

If the second condition is satisfied the formulas

α(z) = X^{∗}(1 ⊗ z)X for all z ∈ N^{θ} and U = X_{23}^{∗}X_{13}^{∗}(Δ ⊗ ι)(X)

define a cocycle action (α, U) of (M, Δ) on N^{θ}. The formula

π : N → M _{α,U}⋉ N^{θ} : π(z) = X^{∗}θ(z)X

defines a ∗-isomorphism satisfying

(ι ⊗ π)(X) = W_{12}U^{∗},  π(x) = α(x) for all x ∈ N^{θ}  and  α̂π = (ι ⊗ π)θ .

Proof. Let us first suppose that the second statement is true and take a corresponding unitary X. Define

α : N^{θ} → M ⊗ N : α(z) = X^{∗}(1 ⊗ z)X .

Then we get for all z ∈ N^{θ}

(ι ⊗ θ)α(z) = (ι ⊗ θ)(X^{∗})(1 ⊗ θ(z))(ι ⊗ θ)(X) = X_{13}^{∗}W_{12}^{∗}(1 ⊗ 1 ⊗ z)W_{12}X_{13} = α(z)_{13} .

So α(N^{θ}) ⊂ M ⊗ N^{θ}. Next we define the unitary U in M ⊗ M ⊗ N by

U := X_{23}^{∗}X_{13}^{∗}(Δ ⊗ ι)(X) ,

from where

(ι ⊗ ι ⊗ θ)(U) = (ι ⊗ θ)(X^{∗})_{234} (ι ⊗ θ)(X^{∗})_{134} (Δ ⊗ θ)(X) = X_{24}^{∗}W_{23}^{∗} X_{14}^{∗}W_{13}^{∗} (Δ ⊗ ι ⊗ ι)(W_{12}X_{13})
= X_{24}^{∗}X_{14}^{∗}(Δ ⊗ ι)(X)_{124} = U_{124} .

Hence U ∈ M ⊗ M ⊗ N^{θ}. To show that (α, U) is a cocycle action of (M, Δ) on N^{θ} one has for all z ∈ N^{θ}:

(Δ ⊗ ι)α(z) = (Δ ⊗ ι)(X^{∗})(1 ⊗ 1 ⊗ z)(Δ ⊗ ι)(X) = U^{∗}X_{23}^{∗}X_{13}^{∗}(1 ⊗ 1 ⊗ z)X_{13}X_{23}U = U^{∗} (ι ⊗ α)α(z) U .

Further we have

(ι ⊗ ι ⊗ α)(U) (Δ ⊗ ι ⊗ ι)(U) = X_{34}^{∗} X_{24}^{∗}X_{14}^{∗}(Δ ⊗ ι)(X)_{124} X_{34} X_{34}^{∗}(Δ ⊗ ι)(X^{∗})_{124}((Δ ⊗ ι)Δ ⊗ ι)(X)
= X_{34}^{∗}X_{24}^{∗}X_{14}^{∗}((Δ ⊗ ι)Δ ⊗ ι)(X) .

On the other hand,

(1 ⊗ U) (ι ⊗ Δ ⊗ ι)(U) = X_{34}^{∗}X_{24}^{∗}(Δ ⊗ ι)(X)_{234} (Δ ⊗ ι)(X^{∗})_{234}X_{14}^{∗}((ι ⊗ Δ)Δ ⊗ ι)(X)
= X_{34}^{∗}X_{24}^{∗}X_{14}^{∗}((Δ ⊗ ι)Δ ⊗ ι)(X) .

Hence we obtain the equality

(ι ⊗ ι ⊗ α)(U) (Δ ⊗ ι ⊗ ι)(U) = (1 ⊗ U) (ι ⊗ Δ ⊗ ι)(U)

so (α, U) is a cocycle action of (M, Δ) on N. We claim that the map

π : N → B(H) ⊗ N : π(z) = X^{∗}θ(z)X

is such that π(N) ⊂ M _{α,U}⋉ N^{θ}. To show this we use Corollary 1.21. First let us check that π(N) ⊂ B(H) ⊗ N^{θ}. We have for all z ∈ N:

(ι ⊗ θ)π(z) = (ι ⊗ θ)(X^{∗}) (ι ⊗ θ)θ(z) (ι ⊗ θ)(X) = X_{13}^{∗}W_{12}^{∗} (Δ̂^{op} ⊗ ι)θ(z) W_{12}X_{13}
= X_{13}^{∗} θ(z)_{13} X_{13} = π(z)_{13} .

Hence we get π(N) ⊂ B(H) ⊗ N^{θ}. To prove our claim it now suffices to observe that

(V^{∗} ⊗ 1)U^{∗} (ι ⊗ α)π(z) U(V ⊗ 1) = (V^{∗} ⊗ 1)U^{∗} X_{23}^{∗}X_{13}^{∗} θ(z)_{13} X_{13}X_{23} U(V ⊗ 1)
= (V^{∗} ⊗ 1) (Δ ⊗ ι)(X^{∗}) θ(z)_{13} (Δ ⊗ ι)(X) (V ⊗ 1)
= X_{13}^{∗}V_{12}^{∗} θ(z)_{13} V_{12}X_{13} = X_{13}^{∗}θ(z)_{13}X_{13} = π(z)_{13}

for all z ∈ N, where we used the fact that V ∈ M̂′ ⊗ M in the last line of the computation.

Next, for z ∈ N^{θ} we clearly have

π(z) = X^{∗}θ(z)X = X^{∗}(1 ⊗ z)X = α(z)

and further

(ι ⊗ π)(X) = X_{23}^{∗}(ι ⊗ θ)(X)X_{23} = X_{23}^{∗}W_{12}X_{13}X_{23} = W_{12}U^{∗}

so we may conclude that M _{α,U}⋉ N^{θ} ⊂ π(N). Thus, we have proved that π(N) = M _{α,U}⋉ N^{θ}.

To finish the first part of the proof we only have to verify the formula α̂π = (ι ⊗ π)θ, but for z ∈ N we get

(ι ⊗ π)θ(z) = X_{23}^{∗} (ι ⊗ θ)θ(z) X_{23} = X_{23}^{∗} (Δ̂^{op} ⊗ ι)θ(z) X_{23}
= X_{23}^{∗}(Ṽ ⊗ 1)θ(z)_{23}(Ṽ^{∗} ⊗ 1)X_{23}
= (Ṽ ⊗ 1)(1 ⊗ π(z))(Ṽ^{∗} ⊗ 1) = α̂(π(z)) .

Here we used the notation Ṽ = (J ⊗ J)ΣW Σ(J ⊗ J) and observe that Ṽ (1 ⊗ z)Ṽ^{∗} = Δ̂^{op}(z) for all z ∈ M̂ and Ṽ ∈ M̂ ⊗ M′. So we have proved that the second statement of the proposition implies the first one.

To prove the converse statement we have to show that for every cocycle action (α, U) of (M, Δ) on N there exists a unitary X ∈ M ⊗ (M _{α,U}⋉ N) such that

(ι ⊗ α̂)(X) = W_{12}X_{13} .

Taking X = W_{12}U^{∗} an easy computation proves the result.

From algebraic point of view this proposition is related to the notion of a cleft extension [34], Definition 7.2.1; the definition below is given for the case of Hopf ∗-algebra actions:

Let (M̂, Δ̂, Ŝ, ε̂) be a Hopf ∗-algebra and let θ : N → M̂ ⊗ N be an action of a (M̂, Δ̂^{op}, Ŝ^{−1}, ε̂) on a unital ∗-algebra N, then the extension N^{θ} ⊂ N is called a cleft extension, if there exists a linear map γ : M̂ → N such that

• γ is a comodule map, which means that (ι ⊗ γ)Δ̂^{op} = θγ .

• γ is convolution unitary, i.e., m(γ^{∗} ⊗ γ)Δ̂^{op}(z) = ε̂(z)1 = m(γ ⊗ γ^{∗})Δ̂^{op}(z) for all z ∈ M̂, where m is the multiplication map, γ^{∗}(z) = γ(Ŝ(z^{∗}))^{∗}.

For a similar definition in the framework of l.c. quantum groups we use the predual space M_{∗} of M. When we turn M_{∗} into an operator space (see e.g. [5]), the map λ : M_{∗} → M̂ : λ(ω) = (ω ⊗ ι)(W) will be completely contractive. Next, it is possible to define the comultiplication Δ̂ on the level of M_{∗} using the extended Haagerup tensor product ⊗_{eh} [12]. Loosely speaking one can give a meaning in M_{∗} ⊗_{eh} M_{∗} to expressions like

∑_{i∈I} ω_{i} ⊗ μ_{i}

where I is an index set, (ω_{i}) (resp., (μ_{i})) is a bounded infinite row (resp., column) over M_{∗}. Then one can show that there exists a unique complete contraction from M_{∗} to M_{∗} ⊗_{eh} M_{∗}, also denoted by Δ̂^{op}, such that

(λ ⊗_{eh} λ)Δ̂^{op} = Δ̂^{op}λ

where λ ⊗_{eh} λ is the obvious completely contractive embedding of M_{∗} ⊗_{eh} M_{∗} into M̂ ⊗ M̂. We then have

Δ̂^{op}(ω_{ξ,η}) = ∑_{i∈I} ω_{e_{i},η} ⊗ ω_{ξ,e_{i}}

for all ξ, η ∈ H and for an orthonormal basis (e_{i})_{i∈I} in H. The following definition looks very much like the Hopf algebraic definition above, but has an exact operator algebraic meaning.

Definition 1.23. Let N be a von Neumann algebra, (M, Δ) a l.c. quantum group and θ : N → M̂ ⊗ N an action of (M̂, Δ̂^{op}) on N. Then the extension N^{θ} ⊂ N is called a cleft extension of von Neumann algebras if there exists a complete contraction

γ : M_{∗} → N

such that

• γ is a comodule map, which means that

(λ ⊗_{eh} γ)Δ̂^{op}(ω) = θ(γ(ω)) for all ω ∈ M_{∗}

where λ ⊗_{eh} γ is the natural complete contraction from M_{∗} ⊗_{eh} M_{∗} to M̂ ⊗ N.

• γ is convolution unitary, which means that

m(γ^{∗} ⊗_{eh} γ)Δ̂^{op}(ω) = ω(1) 1 = m(γ ⊗_{eh} γ^{∗})Δ̂^{op}(ω) for all ω ∈ M_{∗}

where γ^{∗}(ω) = γ(ω̄)^{∗} for all ω ∈ M_{∗}, γ^{∗} ⊗_{eh} γ maps M_{∗} ⊗_{eh} M_{∗} into N ⊗_{eh} N completely contractively and where m : N ⊗_{eh} N → N is the completely contractive multiplication map.

The following result sheds some light on Proposition 1.22 from algebraic point of view.

Proposition 1.24. Let N be a von Neumann algebra, (M, Δ) a l.c. quantum group and θ : N → M̂ ⊗ N an action of (M̂, Δ̂^{op}) on N. Then the extension N^{θ} ⊂ N is a cleft extension of von Neumann algebras if and only if one of the equivalent statements of Proposition 1.22 is satisfied.

Proof. Denoting by ⊗_{max} the maximal operator space tensor product, it follows from [7], Proposition 5.4 that the completely bounded maps from M_{∗} to N can be identified with (M_{∗} ⊗_{max} N_{∗})^{∗}. Combining this with [6], Theorem 2.5 it follows that the completely bounded maps γ from M_{∗} to N are in (completely isometric) one-to-one correspondence with elements X ∈ M ⊗ N by the formula

γ(ω) = (ω ⊗ ι)(X) for all ω ∈ M_{∗} .

Let first one of the statements of Proposition 1.22 be satisfied. Then take such a unitary X ∈ M ⊗ N that (ι ⊗ θ)(X) = W_{12}X_{13}. Define γ(ω) = (ω ⊗ ι)(X) for all ω ∈ M_{∗}. Let (e_{i})_{i∈I} be an orthonormal basis for H and ξ, η ∈ H. Then

(λ ⊗_{eh} γ)Δ̂^{op}(ω_{ξ,η}) = ∑_{i∈I} λ(ω_{e_{i},η}) ⊗ γ(ω_{ξ,e_{i}})

= ∑_{i∈I}(ω_{e_{i},η} ⊗ ι)(W) ⊗ (ω_{ξ,e_{i}} ⊗ ι)(X)

= (ω_{ξ,η} ⊗ ι ⊗ ι)(W_{12}X_{13}) = θ((ω_{ξ,η} ⊗ ι)(X))

= θ(γ(ω_{ξ,η})) ,

so

(λ ⊗_{eh} γ)Δ̂^{op}(ω) = θ(γ(ω)) for all ω ∈ M_{∗} .

Further we have

m(γ^{∗} ⊗_{eh} γ)Δ̂^{op}(ω_{ξ,η}) = ∑_{i∈I} γ^{∗}(ω_{e_{i},η})γ(ω_{ξ,e_{i}})

= ∑_{i∈I}(ω_{e_{i},η} ⊗ ι)(X^{∗})(ω_{ξ,e_{i}} ⊗ ι)(X)

= (ω_{ξ,η} ⊗ ι)(X^{∗}X) = ω_{ξ,η}(1) 1 .

Thus,

m(γ^{∗} ⊗_{eh} γ)Δ̂^{op}(ω) = ω(1) 1 for all ω ∈ M_{∗},

which concludes the first part of the proof, because the remaining equality can be obtained in a similar way.

Vice versa, let us have a map γ. Then there exists an element X ∈ M ⊗ N such that γ(ω) = (ω ⊗ ι)(X) for all ω ∈ M_{∗}. The same computations as above prove that (ι ⊗ θ)(X) = W_{12}X_{13}, and that X is a unitary.

2 Cocycle bicrossed products of l.c. quantum groups

We introduce cocycle matched pairs of l.c. quantum groups, give the cocycle bicrossed product construction for them and show that the result is again a l.c. quantum group.

2.1 Cocycle matched pairs of l.c. quantum groups

Definition 2.1. A pair (M_{1}, Δ_{1}), (M_{2}, Δ_{2}) is said to be a matched pair of l.c. quantum groups if a triple (τ, U, V) (called a cocycle matching) satisfies the following conditions:

U ∈ M_{1} ⊗ M_{1} ⊗ M_{2} and V ∈ M_{1} ⊗ M_{2} ⊗ M_{2} are both unitaries and
τ : M_{1} ⊗ M_{2} → M_{1} ⊗ M_{2} is a faithful ∗-homomorphism.

Denoting by σ the flip map and defining

α : M_{2} → M_{1} ⊗ M_{2} : α(y) = τ(1 ⊗ y) and β : M_{1} → M_{1} ⊗ M_{2} : β(x) = τ(x ⊗ 1)

we have

• (α, U) is a cocycle action of (M_{1}, Δ_{1}) on the von Neumann algebra M_{2} :

(ι ⊗ α)α(y) = U(Δ_{1} ⊗ ι)α(y)U^{∗}
(ι ⊗ ι ⊗ α)(U)(Δ_{1} ⊗ ι ⊗ ι)(U) = (1 ⊗ U)(ι ⊗ Δ_{1} ⊗ ι)(U) .

• (σβ, V_{321}) is a cocycle action of (M_{2}, Δ_{2}) on the von Neumann algebra M_{1} :

(β ⊗ ι)β(x) = V(ι ⊗ Δ_{2}^{op})β(x)V^{∗}
(β ⊗ ι ⊗ ι)(V)(ι ⊗ ι ⊗ Δ_{2}^{op})(V) = (V ⊗ 1)(ι ⊗ Δ_{2}^{op} ⊗ ι)(V) .

• (α, U) and (β, V) are matched in the following sense:

τ_{13}(α ⊗ ι)Δ_{2}(y) = V_{132}(ι ⊗ Δ_{2})α(y)V_{132}^{∗}
τ_{23}σ_{23}(β ⊗ ι)Δ_{1}(x) = U(Δ_{1} ⊗ ι)β(x)U^{∗}
(Δ_{1} ⊗ ι ⊗ ι)(V)(ι ⊗ ι ⊗ Δ_{2}^{op})(U^{∗}) = (U^{∗} ⊗ 1)(ι ⊗ τσ ⊗ ι)((β ⊗ ι ⊗ ι)(U^{∗})(ι ⊗ ι ⊗ α)(V))(1 ⊗ V) .

If both (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}) are commutative, i.e., are generated by l.c. groups, the above conditions split into two parts: 1) we have two ordinary actions α and β which are matched:

(ι ⊗ α)α(y) = (Δ_{1} ⊗ ι)α(y)
(β ⊗ ι)β(x) = (ι ⊗ Δ_{2}^{op})β(x)
τ_{13}(α ⊗ ι)Δ_{2}(y) = (ι ⊗ Δ_{2})α(y)
τ_{23}σ_{23}(β ⊗ ι)Δ_{1}(x) = (Δ_{1} ⊗ ι)β(x)

2) we have two cocycles U and V which are matched:

(ι ⊗ ι ⊗ α)(U)(Δ_{1} ⊗ ι ⊗ ι)(U) = (1 ⊗ U)(ι ⊗ Δ_{1} ⊗ ι)(U)
(β ⊗ ι ⊗ ι)(V)(ι ⊗ ι ⊗ Δ_{2}^{op})(V) = (V ⊗ 1)(ι ⊗ Δ_{2}^{op} ⊗ ι)(V)
(Δ_{1} ⊗ ι ⊗ ι)(V)(ι ⊗ ι ⊗ Δ_{2}^{op})(U^{∗}) = (U^{∗} ⊗ 1)(ι ⊗ τσ ⊗ ι)((β ⊗ ι ⊗ ι)(U^{∗})(ι ⊗ ι ⊗ α)(V))(1 ⊗ V) .

So to construct examples of cocycle matched pairs of l.c. groups, we first have to construct matched pairs without cocycles, and then to solve the above cocycle equations. Sections 4 and 5 are devoted to the analysis of this special case. Let us also stress that Definition 2.1 is an operator algebraic version of the Hopf algebraic definition [30], Chapter 6.

Suppose that (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}) are l.c. quantum groups and that τ : M_{1} ⊗ M_{2} → M_{1} ⊗ M_{2} is a faithful ∗-homomorphism. Then we put τ′ = τσ. It is easy to check that (τ, 1, 1) is a cocycle matching of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}) (with trivial cocycles) if and only if

(τ′ ⊗ ι)(ι ⊗ τ′)(Δ_{2}^{op} ⊗ ι) = (ι ⊗ Δ_{2}^{op})τ′  and  (ι ⊗ τ′)(τ′ ⊗ ι)(ι ⊗ Δ_{1}) = (Δ_{1} ⊗ ι)τ′ .

So, a matching τ of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}) with trivial cocycles is a von Neumann algebraic version of the notion of an inversion of (M_{2}, Δ_{2}^{op}) and (M_{1}, Δ_{1}), introduced in [3], Définition 8.1. Moreover, in the notation of [3], (M_{2}, Δ_{2}^{op}) is the von Neumann algebraic counterpart of (SŴ_{2}, δŴ_{2}) and (M_{1}, Δ_{1}) is the von Neumann algebraic counterpart of (Ŝ_{W_{1}}, δ̂_{W_{1}}), and hence the multiplicative unitary T of [3], Proposition 8.7 agrees with the multiplicative unitary Ŵ that we will define in Definition 2.2. Observe that our framework is wider than the one of [3], so we do not need to impose extra conditions on τ as it was done in [3].

2.2 Cocycle bicrossed products

Let (τ, U, V) be a cocycle matching of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}), α and β as in Definition 2.1. W_{1},(H_{1}, ι, Λ_{1}) and W_{2}, (H_{2}, ι, Λ_{2}) denote the multiplicative unitaries and GNS-constructions of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}) respectively. We denote H = H_{1} ⊗ H_{2} and Σ the flip map on H ⊗ H. As in Notation 1.2, we also denote W̃ = (W_{1} ⊗ 1)U^{∗} . Let us define the major ingredients of the cocycle bicrossed product (M, Δ).

Definition 2.2. Define unitaries W and Ŵ on H ⊗ H by

Ŵ = (β ⊗ ι ⊗ ι)((W_{1} ⊗ 1)U^{∗}) (ι ⊗ ι ⊗ α)(V(1 ⊗ Ŵ_{2})) and W = ΣŴ^{∗}Σ .

Let M = M_{1} _{α,U}⋉ M_{2} be the von Neumann subalgebra of B(H_{1}) ⊗ M_{2} generated by

α(M_{2}) and {(ω ⊗ ι ⊗ ι)((W_{1} ⊗ 1)U^{∗}) | ω ∈ M_{1 ∗}}.

Further we define the faithful ∗-homomorphism

Δ : M → B(H ⊗ H) : Δ(z) = W^{∗}(1 ⊗ z)W .

In the following propositions we will gradually prove that (M, Δ) is a l.c. quantum group, W is the associated multiplicative unitary and we will describe the dual l.c. quantum group (M̂, Δ̂). The proof of the following formula is obvious. Lemma 2.3. We have M ⊂ B(H_{1}) ⊗ M_{2} and for all z ∈ M we have

Δ^{op}(z) = (β ⊗ ι ⊗ ι)(W̃) (ι ⊗ ι ⊗ α)(V(ι ⊗ Δ_{2}^{op})(z)V^{∗}) (β ⊗ ι ⊗ ι)(W̃^{∗}) .

The definition of M makes it natural to give formulas for Δ(α(x)) and (ι ⊗ Δ)(W̃). Proposition 2.4. For all x ∈ M_{2} we have

Δ(α(x)) = (α ⊗ α)Δ_{2}(x) .

Proof. Choose x ∈ M_{2}. We will prove that Δ^{op}(α(x)) = (α ⊗ α)Δ_{2}^{op}(x). Observe that

(ι ⊗ ι ⊗ α)(V (ι ⊗ Δ_{2}^{op})α(x) V^{∗}) = (ι ⊗ ι ⊗ α)σ_{23}τ_{13}(α ⊗ ι)Δ_{2}(x)
= (τ ⊗ ι ⊗ ι)((ι ⊗ α ⊗ ι)(α ⊗ ι)Δ_{2}(x)_{1342})
= (τ ⊗ ι ⊗ ι)( ((W̃^{∗} ⊗ 1) (1 ⊗ (α ⊗ ι)Δ_{2}(x)) (W̃ ⊗ 1))_{1342} )
= (β ⊗ ι ⊗ ι)(W̃^{∗}) (α ⊗ α)Δ_{2}^{op}(x) (β ⊗ ι ⊗ ι)(W̃) .

Using the previous lemma we immediately get that Δ^{op}(α(x)) = (α ⊗ α)Δ_{2}^{op}(x). Proposition 2.5. We have

(ι ⊗ Δ^{op})(W̃) = (W̃ ⊗ 1 ⊗ 1) ((ι ⊗ α)β ⊗ ι ⊗ ι)(W̃) (ι ⊗ α ⊗ α)(V) .

Proof. Using the lemma we get (ι ⊗ Δ^{op})(W̃) = (1 ⊗ (β ⊗ ι ⊗ ι)(W̃)) (ι ⊗ ι ⊗ ι ⊗ α)((1 ⊗ V)(ι ⊗ ι ⊗ Δ_{2}^{op})(W̃)(1 ⊗ V^{∗})) (1 ⊗ (β ⊗ ι ⊗ ι)(W̃^{∗})) . The definition of W̃ and the fact that U and V are matched imply

(1 ⊗ V)(ι ⊗ ι ⊗ Δ_{2}^{op})(W̃)(1 ⊗ V^{∗})
= (1 ⊗ V)(W_{1} ⊗ 1 ⊗ 1)(ι ⊗ ι ⊗ Δ_{2}^{op})(U^{∗})(1 ⊗ V^{∗})
= (W_{1} ⊗ 1 ⊗ 1)(Δ_{1} ⊗ ι ⊗ ι)(V)(ι ⊗ ι ⊗ Δ_{2}^{op})(U^{∗})(1 ⊗ V^{∗})
= (W_{1} ⊗ 1 ⊗ 1)(U^{∗} ⊗ 1) (ι ⊗ τσ ⊗ ι)((β ⊗ ι ⊗ ι)(U^{∗})(ι ⊗ ι ⊗ α)(V)) .

From this we may conclude that

(ι ⊗ ι ⊗ ι ⊗ α)((1 ⊗ V)(ι ⊗ ι ⊗ Δ_{2}^{op})(W̃)(1 ⊗ V^{∗}))
= (W̃ ⊗ 1 ⊗ 1) τ_{23}σ_{23}((β ⊗ ι ⊗ α)(U^{∗})(ι ⊗ ι ⊗ (ι ⊗ α)α)(V))
= (W̃ ⊗ 1 ⊗ 1) τ_{23}σ_{23}((β ⊗ ι ⊗ α)(U^{∗})(1 ⊗ 1 ⊗ W̃^{∗})(ι ⊗ ι ⊗ α)(V)_{1245}(1 ⊗ 1 ⊗ W̃))

= (W̃ ⊗ 1 ⊗ 1) τ_{23}σ_{23}(β ⊗ ι ⊗ ι ⊗ ι)((ι ⊗ ι ⊗ α)(U^{∗})(1 ⊗ W̃^{∗})) (ι ⊗ α ⊗ α)(V) (1 ⊗ (β ⊗ ι ⊗ ι)(W̃)) . Using the first formula of the proof we get

(ι ⊗ Δ^{op})(W̃) = (1 ⊗ (β ⊗ ι ⊗ ι)(W̃)) (W̃ ⊗ 1 ⊗ 1) τ_{23}σ_{23}(β ⊗ ι ⊗ ι ⊗ ι)((ι ⊗ ι ⊗ α)(U^{∗})(1 ⊗ W̃^{∗}))
(ι ⊗ α ⊗ α)(V) .  (2.1)

Because for all x ∈ M_{1} we have

W̃^{∗}(1 ⊗ β(x))W̃ = τ_{23}σ_{23}(β ⊗ ι)Δ_{1}(x),

we get

(1 ⊗ (β ⊗ ι ⊗ ι)(W̃)) (W̃ ⊗ 1 ⊗ 1) = (W̃ ⊗ 1 ⊗ 1) τ_{23}σ_{23}(β ⊗ ι ⊗ ι ⊗ ι)(Δ_{1} ⊗ ι ⊗ ι)(W̃) .

Combining this with equation (2.1) we get (ι ⊗ Δ^{op})(W̃) = (W̃ ⊗ 1 ⊗ 1) τ_{23}σ_{23}(β ⊗ ι ⊗ ι ⊗ ι)((Δ_{1} ⊗ ι ⊗ ι)(W̃)(ι ⊗ ι ⊗ α)(U^{∗})(1 ⊗ W̃^{∗})) (ι ⊗ α ⊗ α)(V) . Finally

(Δ_{1} ⊗ ι ⊗ ι)(W̃)(ι ⊗ ι ⊗ α)(U^{∗})(1 ⊗ W̃^{∗}) = W_{1,13}W_{1,23}(Δ_{1} ⊗ ι ⊗ ι)(U^{∗})(ι ⊗ ι ⊗ α)(U^{∗})(1 ⊗ W̃^{∗})
= W_{1,13}W_{1,23}(ι ⊗ Δ_{1} ⊗ ι)(U^{∗})(1 ⊗ U^{∗}W̃^{∗})
= W_{1,13}W_{1,23}(ι ⊗ Δ_{1} ⊗ ι)(U^{∗})W_{1,23}^{∗}
= W_{1,13}U_{134}^{∗} = W̃_{134}

and hence we get

(ι ⊗ Δ^{op})(W̃) = (W̃ ⊗ 1 ⊗ 1) ((ι ⊗ α)β ⊗ ι ⊗ ι)(W̃) (ι ⊗ α ⊗ α)(V) .

Corollary 2.6. • Δ(M) ⊂ M ⊗ M.

• (Δ ⊗ ι)(W) = W_{13}W_{23}.

• (ι ⊗ Δ)Δ = (Δ ⊗ ι)Δ.

• The unitaries W and Ŵ defined on H ⊗ H are multiplicative. Proof. The first inclusion follows from Propositions 2.4 and 2.5 and from the definition of M. The second equality is equivalent to (ι ⊗ Δ^{op})(Ŵ) = Ŵ_{12}Ŵ_{13}. Propositions 2.4 and 2.5 imply

(ι ⊗ ι ⊗ Δ^{op})(Ŵ) = (β ⊗ Δ^{op})(W̃) (ι ⊗ ι ⊗ α ⊗ α)(ι ⊗ ι ⊗ Δ_{2}^{op})(V(1 ⊗ Ŵ_{2})) = (β ⊗ ι ⊗ ι)(W̃)_{1234} ((β ⊗ α)β ⊗ ι ⊗ ι)(W̃) (β ⊗ α ⊗ α)(V) (ι ⊗ ι ⊗ α ⊗ α)((ι ⊗ ι ⊗ Δ_{2}^{op})(V)Ŵ_{2,23}Ŵ_{2,24}) But we know that

(β ⊗ ι ⊗ ι)(V) (ι ⊗ ι ⊗ Δ_{2}^{op})(V) Ŵ_{2,23} Ŵ_{2,24} = (V ⊗ 1) (ι ⊗ Δ_{2}^{op} ⊗ ι)(V) Ŵ_{2,23} Ŵ_{2,24}
= (V ⊗ 1) Ŵ_{2,23} V_{124} Ŵ_{2,24}

so (ι ⊗ ι ⊗ Δ^{op})(Ŵ) = (β ⊗ ι ⊗ ι)(W̃)_{1234} ((β ⊗ α)β ⊗ ι ⊗ ι)(W̃) (ι ⊗ ι ⊗ α)(V(1 ⊗ Ŵ_{2}))_{1234} (ι ⊗ ι ⊗ α)(V(1 ⊗ Ŵ_{2}))_{1256} . (2.2) Finally we use the equality

((β ⊗ ι)β ⊗ ι ⊗ ι)(W̃) = (V ⊗ 1 ⊗ 1)Ŵ_{2,23}(β ⊗ ι ⊗ ι)(W̃)_{1245}Ŵ_{2,23}^{∗}(V^{∗} ⊗ 1 ⊗ 1)

to conclude that

((β ⊗ α)β ⊗ ι ⊗ ι)(W̃) = (ι ⊗ ι ⊗ α)(V(1 ⊗ Ŵ_{2}))_{1234}(β ⊗ ι ⊗ ι)(W̃)_{1256}(ι ⊗ ι ⊗ α)(V(1 ⊗ Ŵ_{2}))_{1234}^{∗} . Combining this with equation (2.2) we get

(ι ⊗ Δ^{op})(Ŵ) = Ŵ_{12}Ŵ_{13} .

The third and fourth statement of the corollary immediately follow from the second one.

So at the moment we have at our disposal a von Neumann algebra M, a coassociative comultiplication Δ : M → M ⊗ M and a multiplicative unitary W. Next we define a left invariant weight φ on M and show that W is the associated multiplicative unitary (the left regular representation).

Definition 2.7. We denote by φ the dual weight on the cocycle crossed product M = M_{1} _{α,U}⋉ M_{2} of the weight φ_{2} on M_{2}, as defined in Definition 1.13 and by (H, ι, Λ) the canonical GNS-construction of φ as defined in Terminology 1.17, given the GNS-construction (H_{2}, ι, Λ_{2}) for φ_{2}. This means that

• ⟨(ω_{η,Λ_{1}(b)} ⊗ ι ⊗ ι)(W̃)α(x) | η ∈ H_{1}, b ∈ T_{φ_{1}}, x ∈ N_{φ_{2}}⟩ is a σ-strong^{∗}– norm core for Λ.

• Λ((ω ⊗ ι ⊗ ι)(W̃)α(x)) = ξ_{1}(ω) ⊗ Λ_{2}(x) for all ω ∈ I_{1} and x ∈ N_{φ_{2}}.

The following result is important.

Proposition 2.8. The weight φ is left invariant, which means that

(ι ⊗ φ)Δ(z) = φ(z) 1 for all z ∈ M_{φ}^{+} .

Further we have, for all z ∈ N_{φ} and ω ∈ M_{∗}:

Λ((ω ⊗ ι)Δ(z)) = (ω ⊗ ι)(W^{∗})Λ(z).

Proof. It suffices to prove the second equality, then the first one follows as usual. Choose μ ∈ H_{1}, b ∈ T_{φ_{1}} and x ∈ N_{φ_{2}}. Let (e_{i})_{i∈I} be an orthonormal basis for H_{1}. It follows from Propositions 2.4 and 2.5 that, with σ-strong^{∗} convergence

Δ^{op}((ω_{μ,Λ_{1}(b)} ⊗ ι ⊗ ι)(W̃)α(x)) = ∑_{i∈I}((ω_{e_{i},Λ_{1}(b)} ⊗ ι ⊗ ι)(W̃) ⊗ 1 ⊗ 1)
(α ⊗ ι ⊗ ι)((ω_{μ,e_{i}} ⊗ ι ⊗ ι ⊗ ι)((β ⊗ ι ⊗ ι)(W̃)(ι ⊗ ι ⊗ α)(V))(ι ⊗ α)Δ_{2}^{op}(x)) .

Next choose η, ρ ∈ H and an orthonormal basis (f_{j})_{j∈J} for H. Then with σ-strong^{∗} convergence we have

(ι ⊗ ι ⊗ ω_{η,ρ})Δ^{op}((ω_{μ,Λ_{1}(b)} ⊗ ι ⊗ ι)(W̃)α(x))

= ∑_{(i,j)∈I×J}(ω_{e_{i},Λ_{1}(b)} ⊗ ι ⊗ ι)(W̃) α((ω_{μ,e_{i}} ⊗ ι ⊗ ω_{f_{j},ρ})((β ⊗ ι ⊗ ι)(W̃)(ι ⊗ ι ⊗ α)(V))(ι ⊗ ω_{η,f_{j}}α)Δ_{2}^{op}(x)) .

Denote with z_{I_{0}×J_{0}} the above expression summed over I_{0} × J_{0} where I_{0} ⊂ I and J_{0} ⊂ J are finite subsets. Then it follows from Definition 2.7 and [45], Proposition 7.1 that all z_{I_{0}×J_{0}} belong to N_{φ} and

Λ(z_{I_{0}×J_{0}}) = ∑_{(i,j)∈I_{0}×J_{0}} J_{1}σ_{i/2}^{1}(b)J_{1}e_{i} ⊗(ω_{μ,e_{i}} ⊗ι⊗ω_{f_{j},ρ})((β ⊗ι⊗ι)(W̃)(ι⊗ι⊗α)(V)) (ι⊗ω_{η,f_{j}}α)(Ŵ_{2})Λ_{2}(x) .

Taking the limit over J_{0} ⊂ J and using the fact that Λ is σ-strong^{∗}– norm closed we get that for all finite subsets I_{0} ⊂ I the element z_{I_{0}} defined by

z_{I_{0}} := ∑_{i∈I_{0}}(ω_{e_{i},Λ_{1}(b)} ⊗ ι ⊗ ι)(W̃) α((ω_{μ,e_{i}} ⊗ ι ⊗ ω_{η,ρ})((β ⊗ ι ⊗ ι)(W̃)(ι ⊗ ι ⊗ α)(V)(1 ⊗ (ι ⊗ α)Δ_{2}^{op}(x))))

belongs to N_{φ} and

Λ(z_{I_{0}}) = ∑_{i∈I_{0}} J_{1}σ_{i/2}^{1}(b)J_{1}e_{i} ⊗ (ω_{μ,e_{i}} ⊗ ι ⊗ ω_{η,ρ})(Ŵ)Λ_{2}(x) .

Taking the limit over I_{0} ⊂ I we get that the element z defined by

z := (ι ⊗ ι ⊗ ω_{η,ρ})Δ^{op}((ω_{μ,Λ_{1}(b)} ⊗ ι ⊗ ι)(W̃)α(x))

belongs to N_{φ} and

Λ(z) = (J_{1}σ_{i/2}^{1}(b)J_{1} ⊗ 1)(ι ⊗ ι ⊗ ω_{η,ρ})(Ŵ)(μ ⊗ Λ_{2}(x)) .

Because Ŵ ∈ M_{1} ⊗ B(H_{2} ⊗ H) we get

Λ(z) = (ι ⊗ ι ⊗ ω_{η,ρ})(Ŵ)(ξ_{1}(ω_{μ,Λ_{1}(b)}) ⊗ Λ_{2}(x)) = (ι ⊗ ι ⊗ ω_{η,ρ})(Ŵ)Λ((ω_{μ,Λ_{1}(b)} ⊗ ι ⊗ ι)(W̃)α(x)).

By Definition 2.7 the elements (ω_{μ,Λ_{1}(b)} ⊗ ι ⊗ ι)(W̃)α(x) span a core for Λ. So we can conclude that for all η, ρ ∈ H and all z ∈ N_{φ} the element (ι ⊗ ω_{η,ρ})Δ^{op}(z) belongs to N_{φ} and

Λ((ι ⊗ ω_{η,ρ})Δ^{op}(z)) = (ι ⊗ ω_{η,ρ})(Ŵ)Λ(z) .

From this it follows that for all z ∈ N_{φ} and ω ∈ M_{∗} we have (ω ⊗ ι)Δ(z) ∈ N_{φ} and

Λ((ω ⊗ ι)Δ(z)) = (ω ⊗ ι)(W^{∗})Λ(z) .

If we would already know that (M, Δ) is a l.c. quantum group there would exist a ∗-anti-automorphism R : z ↦ Ĵz^{∗}Ĵ of M (where Ĵ is the modular conjugation of the left invariant weight on the dual l.c. quantum group (M̂, Δ̂)) satisfying

ΔR = σ(R ⊗ R)Δ .  (2.3)

So, because φ is left invariant, we get that φR is right invariant. Thus we find first this dual l.c. quantum group and its modular conjugation Ĵ and then prove equation (2.3).

For a cocycle matching (τ, U, V) of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}) define τ̃ = στσ, Ũ = V_{321} and Ṽ = U_{321}. Then one can check that (τ̃, Ũ, Ṽ) is a cocycle matching of (M_{2}, Δ_{2}) and (M_{1}, Δ_{1}). As above we can then define a von Neumann algebra with comultiplication, a multiplicative unitary (both acting on H_{2} ⊗ H_{1}) and construct a left invariant weight and a GNS-construction. Using the obvious isomorphism from H_{2} ⊗ H_{1} onto H = H_{1} ⊗ H_{2} one has the following result.

Proposition 2.9. Define M̂ as the von Neumann subalgebra of M_{1} ⊗ B(H_{2}) generated by

β(M_{1}) and {(ι ⊗ ι ⊗ ω)(V(1 ⊗ Ŵ_{2})) | ω ∈ M_{2 ∗}} .

For all z ∈ M̂ we define

Δ̂(z) = Ŵ^{∗}(1 ⊗ z)Ŵ

and then Δ̂ is a normal and faithful ∗-homomorphism of M̂ into M̂ ⊗ M̂ satisfying

(ι ⊗ Δ̂)Δ̂ = (Δ̂ ⊗ ι)Δ̂ .

There exists on M̂ a n.s.f. weight φ̂ with GNS-construction (H, ι, Λ̂) such that

• ⟨(ι ⊗ ι ⊗ ω_{μ,Λ_{2}(b)})((1 ⊗ Ŵ_{2}^{∗})V^{∗}) β(x) | μ ∈ H_{2}, b ∈ T_{φ_{2}}, x ∈ N_{φ_{1}}⟩ is a σ-strong^{∗}– norm core for Λ̂.

• Λ̂((ι ⊗ ι ⊗ ω)((1 ⊗ Ŵ_{2}^{∗})V^{∗}) β(x)) = Λ_{1}(x) ⊗ ξ_{2}(ω) for all ω ∈ I_{2} and x ∈ N_{φ_{1}}.

• φ̂ is left invariant on (M̂, Δ̂) and we have, for all z ∈ Nφ̂ and ω ∈ M̂_{∗}:

Λ̂((ω ⊗ ι)Δ̂(z)) = (ι ⊗ ω)(W)Λ̂(z).

Later on we will prove that (M̂, Δ̂) is the dual of (M, Δ). The following two lemmas are known to be valid when (M, Δ) is indeed a l.c. quantum group, see [25], Corollary 2.2 and [24], equation (4.2).

Lemma 2.10. Let J and Ĵ be the modular conjugations of the weights φ and φ̂ in the GNS-constructions (H, ι, Λ) and (H, ι, Λ̂). Then

(Ĵ ⊗ J)W = W^{∗}(Ĵ ⊗ J) .

Proof. Let us introduce the notations ∇ and ˆ∇ for the modular operators of the weights φ and φ̂. We first claim that for all ω ∈ B(H)_{∗}

(ω ⊗ ι)(W^{∗})J∇^{1/2} ⊂ J∇^{1/2}(ω̄ ⊗ ι)(W^{∗}) .  (2.4)

To prove this claim, choose ω ∈ B(H)_{∗} and z ∈ N_{φ} ∩ N_{φ}^{∗}. Then, using Proposition 2.8, one gets

(ω̄ ⊗ ι)(W^{∗})Λ(z) = Λ((ω̄ ⊗ ι)Δ(z)) .

Because (ω̄ ⊗ ι)Δ(z) belongs to N_{φ} ∩ N_{φ}^{∗} we have that (ω̄ ⊗ ι)(W^{∗})Λ(z) belongs to the domain of ∇^{1/2} and

J∇^{1/2}(ω̄ ⊗ ι)(W^{∗})Λ(z) = Λ((ω ⊗ ι)Δ(z^{∗})) = (ω ⊗ ι)(W^{∗})Λ(z^{∗}) = (ω ⊗ ι)(W^{∗})J∇^{1/2}Λ(z) .

Because Λ(N_{φ} ∩ N_{φ}^{∗}) is a core for ∇^{1/2} we get the first claim.

Using Proposition 2.9 rather then 2.8 one has, for all ω ∈ B(H)_{∗}:

(ι ⊗ ω)(W)Ĵˆ∇^{1/2} ⊂ Ĵˆ∇^{1/2}(ι ⊗ ω̄)(W).  (2.5)

Rewriting equation (2.4) we get, for all ξ, η ∈ H, μ ∈ D(∇^{1/2}) and ρ ∈ D(∇^{−1/2}):

⟨W^{∗}(ξ ⊗ J∇^{1/2}μ), η ⊗ ρ⟩ = ⟨W (ξ ⊗ ∇^{1/2}Jρ), η ⊗ μ⟩.  (2.6)

Rewriting equation (2.5) we get, for all μ, ρ ∈ H, ξ ∈ D( ˆ∇^{1/2}) and η ∈ D( ˆ∇^{−1/2}):

⟨W (Ĵˆ∇^{1/2}ξ ⊗ μ), η ⊗ ρ⟩ = ⟨W^{∗}( ˆ∇^{1/2}Ĵη ⊗ μ), ξ ⊗ ρ⟩.  (2.7)

Next we claim that W ( ˆ∇ ⊗ ∇) = ( ˆ∇ ⊗ ∇)W. To prove this, choose ξ, η ∈ D( ˆ∇) and μ, ρ ∈ D(∇). Using the two previous equations one has

⟨W (ξ ⊗ μ), ˆ∇η ⊗ ∇ρ⟩ = ⟨W^{∗}(Ĵˆ∇^{1/2}η ⊗ μ), Ĵˆ∇^{1/2}ξ ⊗ ∇ρ⟩
= ⟨W (Ĵˆ∇^{1/2}η ⊗ J∇^{1/2}ρ), Ĵˆ∇^{1/2}ξ ⊗ J∇^{1/2}μ⟩
= ⟨W^{∗}( ˆ∇ξ ⊗ J∇^{1/2}ρ), η ⊗ J∇^{1/2}μ⟩
= ⟨W ( ˆ∇ξ ⊗ ∇μ), η ⊗ ρ⟩ .

From this chain of equalities the second claim follows. Then W ( ˆ∇^{1/2} ⊗ ∇^{1/2}) = ( ˆ∇^{1/2} ⊗ ∇^{1/2})W. Using this and equations (2.6) and (2.7) we get

⟨W (ξ ⊗ μ), η ⊗ ρ⟩ = ⟨W^{∗}( ˆ∇^{1/2}Ĵη ⊗ μ), ˆ∇^{−1/2}Ĵξ ⊗ ρ⟩
= ⟨W^{∗}( ˆ∇^{1/2} ⊗ ∇^{1/2})(Ĵη ⊗ ∇^{−1/2}μ), ˆ∇^{−1/2}Ĵξ ⊗ ρ⟩
= ⟨W^{∗}(Ĵη ⊗ ∇^{−1/2}μ), Ĵξ ⊗ ∇^{1/2}ρ⟩
= ⟨W (Ĵη ⊗ Jρ), Ĵξ ⊗ Jμ⟩
= ⟨(Ĵ ⊗ J)W^{∗}(Ĵ ⊗ J)(ξ ⊗ μ), η ⊗ ρ⟩ .

for every ξ ∈ D( ˆ∇^{1/2}), η ∈ D( ˆ∇^{−1/2}), μ ∈ D(∇^{−1/2}) and ρ ∈ D(∇^{1/2}). This means that

(Ĵ ⊗ J)W^{∗}(Ĵ ⊗ J) = W .

Lemma 2.11. We have

{(ι ⊗ ω)(W) | ω ∈ B(H)_{∗}}^{− σ-strong^{∗}} = M .

Proof. Define the following subspaces of M.

O_{1} = ⟨(ι ⊗ ω)Δ(z) | ω ∈ M_{∗}, z ∈ M⟩^{− σ-strong^{∗}}
O_{2} = {(ι ⊗ ω)(W^{∗}) | ω ∈ B(H)_{∗}}^{− σ-strong^{∗}} .

Using [24], Section 4 we get

O_{1} = ⟨(ι ⊗ ω_{Λ(a),Λ(b)})Δ(x) | a, b ∈ T_{φ}, x ∈ N_{φ}⟩^{− σ-strong^{∗}}
= ⟨(ι ⊗ φ)((1 ⊗ σ_{i}(a)b^{∗})Δ(x)) | a, b ∈ T_{φ}, x ∈ N_{φ}⟩^{− σ-strong^{∗}}
= ⟨(ι ⊗ ω_{Λ(x),Λ(bσ_{−i}(a^{∗}))})(W^{∗}) | a, b ∈ T_{φ}, x ∈ N_{φ}⟩^{− σ-strong^{∗}} = O_{2} .

It is clear that O_{1} is self-adjoint. Because W is a multiplicative unitary we get that O_{2} is an algebra. Using the previous computation we get that O_{1} = O_{2} := O is a ∗-algebra closed in σ-strong^{∗} topology. We have

(ι ⊗ ω)Δ(α(x)) = α((ι ⊗ ωα)Δ_{2}(x)) for all x ∈ M_{2} and ω ∈ M_{∗}

and because α is faithful we may conclude that α(M_{2}) ⊂ O. In particular 1 ∈ M and hence O is a von Neumann subalgebra of M. Then the bicommutant theorem gives, for all z ∈ M_{1} ⊗ M:

(ι ⊗ Δ^{op})(z) ∈ M_{1} ⊗ M ⊗ O.

In particular we get that (ι ⊗ Δ^{op})(W̃) belongs to M_{1} ⊗ M ⊗ O. Proposition 2.5 implies

(W̃ ⊗ 1 ⊗ 1) ((ι ⊗ α)β ⊗ ι ⊗ ι)(W̃) (ι ⊗ α ⊗ α)(V) ∈ M_{1} ⊗ M ⊗ O .

Because α(M_{2}) ⊂ O it follows that

((ι ⊗ α)β ⊗ ι ⊗ ι)(W̃) ∈ M_{1} ⊗ M ⊗ O .

Because (ι ⊗ α)β is faithful we may conclude that (ω ⊗ ι ⊗ ι)(W̃) belongs to O for all ω ∈ M_{1 ∗}. Together with α(M_{2}) ⊂ O we get that O = M. But then also O^{∗} = M which concludes the proof.

Then we can finally prove the needed formula.

Proposition 2.12. The map R(z) = Ĵz^{∗}Ĵ (z ∈ M) is a ∗-anti-automorphism of M satisfying

ΔR = σ(R ⊗ R)Δ .

Proof. R is a ∗-anti-homomorphism from M into B(H). Lemma 2.10 gives

R((ι ⊗ ω_{ξ,η})(W)) = (ι ⊗ ω_{Jη,Jξ})(W)

for all ξ, η ∈ H. Because R is continuous in σ-strong^{∗} topology, it follows from Lemma 2.11 that R(M) ⊂ M. But clearly R(R(M)) = M and so R(M) = M. Hence R is a ∗-anti-automorphism of M.

Let (e_{i})_{i∈I} be an orthonormal basis for H. Because W is a multiplicative unitary we get for all ξ, η ∈ H, with σ-strong^{∗} convergence

Δ((ι ⊗ ω_{ξ,η})(W)) = (ι ⊗ ι ⊗ ω_{ξ,η})(W_{13}W_{23}) = ∑_{i∈I}(ι ⊗ ω_{e_{i},η})(W) ⊗ (ι ⊗ ω_{ξ,e_{i}})(W) .

So we also get for all ξ, η ∈ H

σ(R ⊗ R)Δ((ι ⊗ ω_{ξ,η})(W)) = ∑_{i∈I}(ι ⊗ ω_{Je_{i},Jξ})(W) ⊗ (ι ⊗ ω_{Jη,Je_{i}})(W) = (ι ⊗ ι ⊗ ω_{Jη,Jξ})(W_{13}W_{23})

= Δ((ι ⊗ ω_{Jη,Jξ})(W)) = ΔR((ι ⊗ ω_{ξ,η})(W)) .

Because both ΔR and σ(R ⊗ R)Δ are continuous in σ-strong^{∗} topology, it follows from Lemma 2.11 that

ΔR = σ(R ⊗ R)Δ .

We have now gathered enough results to prove the main result of this section.

Theorem 2.13. (M, Δ) is a l.c. quantum group. The weight φ is left invariant and (H, ι, Λ) is a GNSconstruction for φ. The left regular representation of (M, Δ) in this GNS-construction is W. The associated dual l.c. quantum group is (M̂, Δ̂). We call (M, Δ) the cocycle bicrossed product of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}).

Proof. Let R be as above, then φR is a n.s.f. weight on M which is right invariant. So (M, Δ) is a l.c. quantum group, and Proposition 2.8 gives that W is its left regular representation. It follows from [25], Definition 1.5 that the underlying von Neuman algebra of the dual locally compact quantum group is

{(ω ⊗ ι)(W) | ω ∈ M_{∗}}^{− σ-strong^{∗}} .

Applying Lemma 2.11 to the cocycle matching (τ̃, Ũ, Ṽ) of (M_{2}, Δ_{2}) and (M_{1}, Δ_{1}), we get that the dual l.c. quantum group precisely is (M̂, Δ̂).

Now we have at our disposal the l.c. quantum group (M, Δ) and the GNS-construction (H, ι, Λ) for the left invariant weight φ. From [25], Section 1 we get in a canonical way a left invariant weight φ̂_{0} on (M̂, Δ̂) with GNS-construction (H, ι, Λ̂_{0}). In the remaining part of this subsection we prove that φ̂_{0} = φ̂ and Λ̂_{0} = Λ̂.

Whenever K is a Hilbert space and ρ ∈ K, we denote by θ_{ρ} the rank-one operator in B(ℂ, K) given by θ_{ρ}(λ) = λρ for all λ ∈ ℂ.

Lemma 2.14. For all ξ ∈ H_{1} ⊗ H_{2}, ρ ∈ H_{2}, b ∈ T_{φ_{1}} and x ∈ N_{φ_{2}} the element z ∈ M defined by

z := (ω_{ξ,U_{β}(Λ_{1}(b)⊗ρ)} ⊗ ι ⊗ ι)(Ŵ) α(x),

where U_{β} = Ĵ(J_{1} ⊗ Ĵ_{2}) is the canonical implementation of β as in Proposition 1.19, belongs to N_{φ} and

Λ(z) = ((J_{1}σ_{i/2}^{1}(b)J_{1} ⊗ θ_{ρ}^{∗})U_{β}^{∗} ⊗ 1)V(1 ⊗ Ŵ_{2})(ξ ⊗ Λ_{2}(x)) .

Here (σ_{t}^{1}) denotes the modular group of the weight φ_{1} on M_{1}.

Proof. Let (e_{i})_{i∈I} be an orthonormal basis for H_{1} ⊗ H_{2}. Then we have, with σ-strong^{∗} convergence

z = ∑_{i∈I}(ω_{e_{i},U_{β}(Λ_{1}(b)⊗ρ)}β ⊗ ι ⊗ ι)(W̃) α((ω_{ξ,e_{i}} ⊗ ι)(V(1 ⊗ Ŵ_{2})) x) .

Denote by z_{I_{0}} the same expression summed over a finite subset I_{0} ⊂ I. We claim that for all η ∈ H_{1} ⊗ H_{2} the normal functional μ := ω_{η,U_{β}(Λ_{1}(b)⊗ρ)}β belongs to I_{1} and

ξ_{1}(μ) = (J_{1}σ_{i/2}^{1}(b)J_{1} ⊗ θ_{ρ}^{∗})U_{β}^{∗}η .

To prove this claim, we choose y ∈ N_{φ_{1}}. Then

μ(y^{∗}) = ⟨β(y^{∗})η, U_{β}(Λ_{1}(b) ⊗ ξ)⟩ = ⟨(y^{∗} ⊗ 1)U_{β}^{∗}η, Λ_{1}(b) ⊗ ρ⟩ = ⟨(J_{1}σ_{i/2}^{1}(b)J_{1} ⊗ θ_{ρ}^{∗})U_{β}^{∗}η, Λ_{1}(y)⟩ .

But then for all finite subsets I_{0} ⊂ I we have z_{I_{0}} ∈ N_{φ} and

Λ(z_{I_{0}}) = ∑_{i∈I_{0}}(J_{1}σ_{i/2}^{1}(b)J_{1} ⊗ θ_{ρ}^{∗})U_{β}^{∗}e_{i} ⊗ (ω_{ξ,e_{i}} ⊗ ι)(V(1 ⊗ Ŵ_{2}))Λ_{2}(x) .

Because Λ is σ-strong^{∗}– norm closed the claim of the lemma follows.

Lemma 2.15. Let (M, Δ) be a l.c. quantum group with left invariant weight φ and GNS-construction (H, ι, Λ), (M̂, Δ̂) the associated dual l.c. quantum group with canonical left invariant weight φ̂ and GNSconstruction (H, ι, Λ̂), (σ̂_{t}) the modular group of the weight φ̂ on M̂.

Suppose that y ∈ D(σ̂_{i/2}) and that there exist elements (p_{i})_{i∈I} and (q_{i})_{i∈I} in M̂ such that ∑ p_{i}p_{i}^{∗} and ∑ q_{i}^{∗}q_{i} are bounded and

1 ⊗ y = ∑_{i∈I} Δ̂(p_{i})(q_{i} ⊗ 1) .

Then there exists a unique linear map P : M → M continuous in σ-strong^{∗} topology and such that for every x ∈ N_{φ} the element P (x) belongs to N_{φ} and

Λ(P (x)) = Ĵσ̂_{i/2}(y)ĴΛ(x) .

Moreover, whenever ω ∈ M_{∗} and (ω ⊗ ι)(W)y ∈ Nφ̂ we have, for all x ∈ N_{φ}:

⟨Λ̂((ω ⊗ ι)(W)y), Λ(x)⟩ = ω(P (x)^{∗}).

Proof. Denote, for all i ∈ I, a_{i} = R̂(q_{i}) and b_{i} = R̂(p_{i}). Then ∑ a_{i}a_{i}^{∗} and ∑ b_{i}^{∗}b_{i} are bounded and

1 ⊗ R̂(y) = ∑_{i∈I}(a_{i} ⊗ 1)Δ̂^{op}(b_{i}) .

So we can define for every x ∈ M an operator P (x) ∈ B(H) as

P (x) = ∑_{i∈I} Ĵa_{i}ĴxĴb_{i}Ĵ .

Considering A = (Ĵa_{i}Ĵ)_{i∈I} as an operator in B(H ⊗ℓ^{2}(I), H) and B = (Ĵb_{i}Ĵ)_{i∈I} as an operator in B(H, H ⊗ ℓ^{2}(I)), we get P (x) = A(x ⊗ 1)B for all x ∈ M, so P is continuous in σ-strong^{∗} topology. We claim that

P ((ι ⊗ μ)(W^{∗})) = (ι ⊗ μy^{∗})(W^{∗})

for all μ ∈ M̂_{∗}. Indeed, using the formula (Ĵ ⊗ J)W = W^{∗}(Ĵ ⊗ J), proved in [25], Corollary 2.2, we have

∑_{i∈I}(Ĵa_{i}Ĵ⊗1)W^{∗}(Ĵb_{i}Ĵ⊗1) = ∑_{i∈I}(Ĵ⊗J)(a_{i}⊗1)Δ̂^{op}(b_{i})(Ĵ⊗J)W^{∗} = (Ĵ⊗J)(1⊗R̂(y))(Ĵ⊗J)W^{∗} = (1⊗y^{∗})W^{∗},

from where the claim follows. For all μ ∈ Î we have μy^{∗} ∈ Î, hence P ((ι ⊗ μ)(W^{∗})) ∈ N_{φ} and

Λ(P ((ι ⊗ μ)(W^{∗}))) = Λ((ι ⊗ μy^{∗})(W^{∗})) = Ĵσ̂_{i/2}(y)ĴΛ((ι ⊗ μ)(W^{∗})) .

If now x ∈ N_{φ}, there is a net (μ_{α}) in Î such that (ι ⊗ μ_{α})(W^{∗}) → x in σ-strong^{∗} topology and Λ((ι ⊗ μ_{α})(W^{∗})) → Λ(x) in norm. Because P is continuous in σ-strong^{∗} topology and Λ is σ-strong^{∗}– norm closed, we get P (x) ∈ N_{φ} and

Λ(P (x)) = Ĵσ̂_{i/2}(y)ĴΛ(x) .

This concludes the first part of the proof. Now suppose that we also have ω ∈ M_{∗} such that (ω⊗ι)(W)y ∈ Nφ̂. Then we get, for all μ ∈ Î:

⟨Λ̂((ω ⊗ ι)(W)y), Λ((ι ⊗ μ)(W^{∗}))⟩ = μ̄((ω ⊗ ι)(W)y) = ω(P ((ι ⊗ μ)(W^{∗}))^{∗}) .

If x ∈ N_{φ} we take the same net (μ_{α}) in Î as above and obtain that

⟨Λ̂((ω ⊗ ι)(W)y), Λ(x)⟩ = ω(P (x)^{∗}) .

Now we are ready to prove the following nice result.

Proposition 2.16. We have φ̂_{0} = φ̂ and Λ̂_{0} = Λ̂. Hence the left invariant weight with GNS-construction obtained by considering (M̂, Δ̂) as a cocycle bicrossed product, as in Proposition 2.9, is the same as the left invariant weight with GNS-construction obtained by considering (M̂, Δ̂) as the dual of (M, Δ), as in [24], Proposition 8.14.

Proof. Consider the l.c. quantum group (M̂, Δ̂) with left invariant weight φ̂, GNS-construction (H, ι, Λ̂) for φ̂ and multiplicative unitary Ŵ. Applying [24], Propositions 8.14 and 8.15 to the l.c. quantum group (M̂, Δ̂) we obtain a canonical left invariant weight φ̂ ˆ on the dual (M, Δ) of (M̂, Δ̂), with canonical GNS-construction (H, ι, Λ̂ ˆ ). In view of [24], Proposition 8.30 it suffices to prove that φ̂ ˆ = φ and Λ̂ ˆ = Λ.

By the uniqueness of left invariant weights, there exists a positive number ν > 0 such that φ̂ ˆ = νφ. So N_{φ̂}ˆ = N_{φ}. Then we can define a unitary operator u on H such that uΛ̂ ˆ (x) = ν^{1/2}Λ(x) for all x ∈ N_{φ}. Because both GNS-representations involved are the identical representations, we get u ∈ M′. From the dual version of [24], Proposition 8.16 we get, for all x ∈ N_{φ̂}ˆ and ω ∈ M_{∗}:

Λ̂ ˆ ((ω ⊗ ι)Δ(x)) = (ω ⊗ ι)(W^{∗})Λ̂ ˆ (x).

But from Proposition 2.8 it follows, for all x ∈ N_{φ} and ω ∈ M_{∗}:

Λ((ω ⊗ ι)Δ(x)) = (ω ⊗ ι)(W^{∗})Λ(x).

Both formulas together imply u ∈ M̂′. Because we already have u ∈ M′, it follows that u ∈ ℂ1. Hence we can take λ ∈ ℂ \ {0} such that Λ̂ ˆ = λΛ. We have to prove that λ = 1. We can already conclude that the modular group (σ̂_{ ˆ t}) of φ̂ ˆ equals the modular group (σ_{t}) of φ and that Ĵ ˆ = λ/λ̄ J.

Choose now ξ ∈ H_{1} ⊗ H_{2}, ρ ∈ H_{2}, b ∈ T_{φ_{1}} and y ∈ N_{φ_{2}} ∩ D(σ_{i/2}^{2}) such that there exist elements (p_{i})_{i∈I} and (q_{i})_{i∈I} in M_{2} with ∑ p_{i}p_{i}^{∗} and ∑ q_{i}^{∗}q_{i} bounded and

1 ⊗ y = ∑_{i∈I} Δ_{2}(p_{i})(q_{i} ⊗ 1) .  (2.8)

Remark that such elements y form a dense subspace of M_{2} in σ-strong^{∗} topology. Because ψ_{2}σ_{t}^{2} = ν_{2}^{t} ψ_{2} and ψ_{2}τ_{t}^{2} = ν_{2}^{−t}ψ_{2} we can take elements a, b ∈ M_{2} such that a is analytic w.r.t. (σ_{t}^{2}), b is analytic w.r.t. (τ_{t}^{2}) and σ_{z}^{2}(a) ∈ N_{ψ_{2}}^{∗}N_{φ_{2}}, τ_{z}^{2}(b) ∈ N_{ψ_{2}} for all z ∈ ℂ. Now [24], Result 2.6 shows that

y := (ψ_{2} ⊗ ι)(Δ(a)(b ⊗ 1)) ∈ N_{φ_{2}}.

Using the relation (τ_{t}^{2} ⊗ σ_{t}^{2})Δ_{2} = Δ_{2}σ_{t}^{2} we get that y is analytic w.r.t. (σ_{t}^{2}) and

σ_{z}^{2}(y) = ν_{2}^{z} (ψ_{2} ⊗ ι)(Δ(σ_{z}^{2}(a))(τ_{z}^{2}(b) ⊗ 1)) .

Finally observe that y = (ω_{Γ_{2}(b),Γ_{2}(a^{∗})}⊗ι)(V_{2}^{∗}), so defining p_{i} = (ω_{e_{i},Γ_{2}(a^{∗})}⊗ι)(V_{2}^{∗}) and q_{i} = (ω_{Γ_{2}(b),e_{i}}⊗ι)(V_{2}), where (e_{i}) is an orthonormal basis for H, we have (2.8).

Because y ∈ N_{φ_{2}}, it follows from Lemma 2.14 that the element z ∈ M defined by

z := (ω ⊗ ι ⊗ ι)(Ŵ) α(y) , with ω = ω_{ξ,U_{β}(Λ_{1}(b)⊗ρ)}

belongs to N_{φ} and

Λ(z) = ((J_{1}σ_{i/2}^{1}(b)J_{1} ⊗ θ_{ρ}^{∗})U_{β}^{∗} ⊗ 1)V(1 ⊗ Ŵ_{2})(ξ ⊗ Λ_{2}(y)) .

Choose now μ ∈ I_{2} and x ∈ N_{φ_{1}}. Then

⟨Λ(z), Λ̂((ι ⊗ ι ⊗ μ)((1 ⊗ Ŵ_{2}^{∗})V^{∗}) β(x))⟩
= ⟨((J_{1}σ_{i/2}^{1}(b)J_{1} ⊗ θ_{ρ}^{∗})U_{β}^{∗} ⊗ 1)V(1 ⊗ Ŵ_{2})(ξ ⊗ Λ_{2}(y)), Λ_{1}(x) ⊗ ξ_{2}(μ)⟩
= ⟨(ωβ(x^{∗}) ⊗ ι)(V(1 ⊗ Ŵ_{2}))Λ_{2}(y), ξ_{2}(μ)⟩ = μ̄((ω ⊗ ι)((β(x^{∗}) ⊗ 1)V(1 ⊗ Ŵ_{2}))y)
= ω(((ι ⊗ ι ⊗ μy^{∗})((1 ⊗ Ŵ_{2}^{∗})V^{∗})β(x))^{∗}) .  (2.9)

Now apply Lemma 2.15 to the l.c. quantum group (M̂, Δ̂). Because σ̂_{ ˆ t}α = σ_{t}α = ασ_{t}^{2} for all t ∈ ℝ, and because Δα = (α ⊗ α)Δ_{2} we get that α(y) satisfies the conditions of this lemma. Hence there exists a σ-strong^{∗} continuous map P from M̂ to M̂, such that for all v ∈ Nφ̂ we get P (v) ∈ Nφ̂ and

Λ̂(P (v)) = Ĵ ˆσ̂_{ ˆ i/2}(α(y))Ĵ ˆΛ̂(v) .

Because Ĵ ˆ = λ/λ̄J, σ̂_{ ˆ t}α = ασ_{t}^{2} and α is implemented by J(Ĵ_{1} ⊗ J_{2}) we get, for all v ∈ Nφ̂:

Λ̂(P (v)) = (1 ⊗ J_{2}σ_{i/2}^{2}(y)J_{2})Λ̂(v).

Since z = (ω ⊗ ι ⊗ ι)(Ŵ)α(y) ∈ N_{φ} = N_{φ̂}ˆ , it follows from Lemma 2.15 that

⟨Λ̂ ˆ (z), Λ̂(v)⟩ = ω(P (v)^{∗}) for all v ∈ Nφ̂.  (2.10)

If we now take v = (ι ⊗ ι ⊗ μ)((1 ⊗ Ŵ_{2}^{∗})V^{∗}) β(x), then

(1 ⊗ J_{2}σ_{i/2}^{2}(y)J_{2})Λ̂(v) = Λ_{1}(x) ⊗ J_{2}σ_{i/2}^{2}(y)J_{2}ξ_{2}(μ) = Λ_{1}(x) ⊗ ξ_{2}(μy^{∗}) = Λ̂((ι ⊗ ι ⊗ μy^{∗})((1 ⊗ Ŵ_{2}^{∗})V^{∗}) β(x)) .

So P (v) = (ι ⊗ ι ⊗ μy^{∗})((1 ⊗ Ŵ_{2}^{∗})V^{∗}) β(x). Now combine equations (2.9) and (2.10) to get

ω(P (v)^{∗}) = ⟨Λ̂ ˆ (z), Λ̂(v)⟩ = λ⟨Λ(z), Λ̂(v)⟩ = λ ω(P (v)^{∗}) .

Because this is valid for enough elements v and ω we get that λ = 1 which concludes the proof.

We characterize compact and discrete cocycle bicrossed products as follows.

Proposition 2.17. The l.c. quantum group (M, Δ) is compact if and only if (M_{2}, Δ_{2}) is compact and (M_{1}, Δ_{1}) is discrete. (M, Δ) is discrete if and only if (M_{2}, Δ_{2}) is discrete and (M_{1}, Δ_{1}) is compact.

Proof. From Definition 1.13 it follows that φ(1) = φ̂_{1}(1) φ_{2}(1) from where the first statement follows. The second statement follows from the first one and from the observation that (M̂, Δ̂) is a cocycle bicrossed product of (M_{2}, Δ_{2}) and (M_{1}, Δ_{1}).

3 Extensions and cocycle bicrossed products

We define and study extensions of l.c. quantum groups and prove that there is a one-to-one correspondence between cleft extensions and cocycle bicrossed products.

3.1 Extensions and cleft extensions of l.c. quantum groups

Let us start with the following result which can be deduced from [23], but we prefer to give its direct proof.

Proposition 3.1. Let (M_{1}, Δ_{1}) and (M, Δ) be l.c. quantum groups and β : M_{1} → M̂ a normal ∗-homomor-phism satisfying Δ̂β = (β ⊗ β)Δ_{1}. Define the unitaries Z_{1} and Z_{2} on H_{1} ⊗ H by

Z_{1} = (ι ⊗ β)(Ŵ_{1}^{∗})  and  Z_{2} = (J_{1} ⊗ Ĵ) (ι ⊗ β)(Ŵ_{1}^{∗}) (J_{1} ⊗ Ĵ)

and then the maps μ and θ on M by

μ(z) = Z_{1}(1 ⊗ z)Z_{1}^{∗}  and  θ(z) = Z_{2}(1 ⊗ z)Z_{2}^{∗}.

Then μ is an action of (M̂_{1}, Δ̂_{1}) on M and θ is an action of (M̂_{1}, Δ̂_{1}^{op}) on M such that

θ(z) = (R̂_{1} ⊗ R)μ(R(z)) for all z ∈ M
(μ ⊗ ι)(W) = (ι ⊗ β)(Ŵ_{1})_{13}W_{23}
(θ ⊗ ι)(W) = W_{23}(ι ⊗ β)(Ŵ_{1})_{13}
(ι ⊗ Δ)μ = (μ ⊗ ι)Δ and (ι ⊗ Δ^{op})θ = (θ ⊗ ι)Δ^{op} .

Finally, for all z ∈ N_{φ} and ω ∈ M̂_{1 ∗} we have (ω ⊗ ι)μ(z) ∈ N_{φ} and

Λ((ω ⊗ ι)μ(z)) = β((ι ⊗ ω)(W_{1})) Λ(z) .

Proof. Observe that

(ι ⊗ β)(Ŵ_{1}^{∗})_{12} W_{23} (ι ⊗ β)(Ŵ_{1})_{12} = (ι ⊗ β)(Ŵ_{1}^{∗})_{12} (ι ⊗ Δ̂^{op}β)(Ŵ_{1}) W_{23}
= (ι ⊗ β)(Ŵ_{1}^{∗})_{12} (ι ⊗ β ⊗ β)(ι ⊗ Δ_{1}^{op})(Ŵ_{1}) W_{23}
= (ι ⊗ β)(Ŵ_{1})_{13} W_{23} .

Then one can conclude that μ(M) ⊂ M̂_{1} ⊗ M and

(μ ⊗ ι)(W) = (ι ⊗ β)(Ŵ_{1})_{13} W_{23} .

Now one can check that (Δ̂_{1} ⊗ ι)μ = (ι ⊗ μ)μ, i.e., μ is an action of (M̂_{1}, Δ̂_{1}) on M, and (ι ⊗ Δ)μ = (μ ⊗ ι)Δ. We get similar formulas for θ from

θ(z) = (R̂_{1} ⊗ R)μ(R(z)) for all z ∈ M

and R̂β = βR_{1} (this follows from the von Neumann algebraic version of [24], Proposition 5.45). So it remains to prove the final statement. Using [24], choose ω ∈ M̂_{1 ∗} and η ∈ Î. Then

(ω ⊗ ι)μ((ι ⊗ η)(W^{∗})) = (ι ⊗ ρ)(W^{∗}) where ρ(z) = (ω ⊗ η)((1 ⊗ z)(ι ⊗ β)(Ŵ_{1}^{∗})) for all z ∈ M̂ . (3.1)

We claim that ρ ∈ Î. For this we choose z ∈ Nφ̂ and compute

ρ(z^{∗}) = (ω ⊗ η)((1 ⊗ z^{∗})(ι ⊗ β)(Ŵ_{1}^{∗})) = η(z^{∗}β((ι ⊗ ω)(W_{1})))

= ⟨Λ((ι ⊗ η)(W^{∗})), β((ι ⊗ ω)(W_{1})^{∗})Λ̂(z)⟩ = ⟨β((ι ⊗ ω)(W_{1}))Λ((ι ⊗ η)(W^{∗})), Λ̂(z)⟩ .

From this the claim follows, so (ι ⊗ ρ)(W^{∗}) ∈ N_{φ} and

Λ((ι ⊗ ρ)(W^{∗})) = β((ι ⊗ ω)(W_{1})) Λ((ι ⊗ η)(W^{∗})) .

Combining this with equation (3.1) and the fact that the elements (ι ⊗ η)(W^{∗}) give a σ-strong^{∗}– norm core for Λ, we get the final statement.

The starting point of Proposition 3.1 is the von Neumann algebraic morphism β. If all algebras are finite dimensional, we can dualize β to a morphism β̃ : M → M̂_{1} such that

(β̃ ⊗ ι)(W) = (ι ⊗ β)(Ŵ_{1}) .

The link between β̃ and the action μ is now given by the formula

μ(z) = (β̃ ⊗ ι)Δ(z) for all z ∈ M ,

and the action μ also makes sense in general case. The action μ satisfies (ι ⊗ Δ)μ = (μ ⊗ ι)Δ, and in [23] there is established a natural one-to-one correspondence between actions of this special type and morphisms.

Turning now to the definition of extensions of l.c. quantum groups, let us note that we use the notation β first of all for a morphism M_{1} → M̂, but also symbolically on the arrow pointing from (M, Δ) to (M̂_{1}, Δ̂_{1}) for in a sense dual morphism which does not necessarily exist on the von Neumann algebra level.

Definition 3.2. Let (M_{1}, Δ_{1}), (M_{2}, Δ_{2}) and (M, Δ) be l.c. quantum groups. We call

(M_{2}, Δ_{2}) ⟶^{α} (M, Δ) ⟶^{β} (M̂_{1}, Δ̂_{1})

a short exact sequence, if

α : M_{2} → M and β : M_{1} → M̂

are normal, faithful ∗-homomorphisms satisfying

Δα = (α ⊗ α)Δ_{2} and Δ̂β = (β ⊗ β)Δ_{1}

and if α(M_{2}) = M^{θ}, where θ is the action of (M̂_{1}, Δ̂_{1}^{op}) on M defined in Proposition 3.1 starting with the morphism β. In this situation we call (M, Δ) an extension of (M_{2}, Δ_{2}) by (M̂_{1}, Δ̂_{1}).

The faithfulness of the morphisms α and β reflects the exactness of the sequence in the first and third place. The formula α(M_{2}) = M^{θ} reflects its exactness in the second place. In terms of the above morphism β̃ : M → M̂_{1} the exactness of the sequence in the second place is formally expressed by

α(M_{2}) = {z ∈ M | (ι ⊗ β̃)Δ(z) = z ⊗ 1} .

This formula is precisely the one used in Hopf algebra theory, see [2]. If α and β give rise to a short exact sequence as in the previous definition, then the dual sequence

(M_{1}, Δ_{1}) ⟶^{β} (M̂, Δ̂) ⟶^{α} (M̂_{2}, Δ̂_{2})

also is a short exact sequence. This follows from the argument in part 1.a) of the proof of Theorem 3.5.

Definition 3.3. An extension (M, Δ) of l.c. quantum groups is said to be a cleft extension of (M_{1}, Δ_{1}) by (M̂_{2}, Δ̂_{2}) if the extension α(M_{2}) = M^{θ} ⊂ M is a cleft extension of von Neumann algebras.

From part 1.b) of the proof of Theorem 3.5 it follows that the dual extension of a cleft extension is again cleft.

3.2 From cocycle bicrossed products to cleft extensions

Proposition 3.4. Let (M, Δ) be the cocycle bicrossed product of l.c. quantum groups (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}), which is also a l.c. quantum group by Theorem 2.13. Let α and β be as in Definition 2.1. Then

α : M_{2} → M  and  β : M_{1} → M̂

and (M, Δ) is a cleft extension of (M_{2}, Δ_{2}) by (M̂_{1}, Δ̂_{1}).

Proof. Theorem 2.13 shows that (M̂, Δ̂) is the dual l.c. quantum group of (M, Δ), so β maps M_{1} faithfully into M̂. Definition 2.2 implies that α maps M_{2} faithfully into M and Proposition 2.4 shows that Δα = (α⊗α)Δ_{2}. Because the construction of (M̂, Δ̂) corresponds to the cocycle matching (τ̃, Ũ, Ṽ), Proposition 2.4 also shows that Δ̂β = (β⊗β)Δ_{1}. Let J and Ĵ be the modular conjugations of φ and φ̂ in the GNS-constructions (H, ι, Λ) and (H, ι, Λ̂) respectively (see Definition 2.7 and Proposition 2.9). From Proposition 2.16 we know that φ̂ is the canonical left invariant weight on the dual (M̂, Δ̂) with canonical dual GNS-construction (H, ι, Λ̂). So, the action θ of (M̂_{1}, Δ̂_{1}^{op}) on M constructed in Proposition 3.1 verifies θ(z) = Z(1⊗z)Z^{∗} for all z ∈ M where

Z = (J_{1} ⊗ Ĵ) (ι ⊗ β)(Ŵ_{1}^{∗}) (J_{1} ⊗ Ĵ) .

Up to a flip map, M̂ is the cocycle crossed product M_{2} _{σβ,V_{321}}⋉ M_{1} and φ̂ is the dual weight of φ_{1} with accompanying dual weight GNS-construction (H, ι, Λ̂). Hence it follows from Proposition 1.19 that β is implemented by Ĵ(J_{1} ⊗ Ĵ_{2}). Then we get

Z = (J_{1} ⊗ J_{1})ΣW_{1}Σ(J_{1} ⊗ J_{1}) ⊗ 1 .

So it follows from Proposition 1.4 that θ = α̂, therefore Theorem 1.11 gives

M^{θ} = (M_{1} _{α,U}⋉ M_{2})α̂ = α(M_{2}) .

Since θ = α̂, Propositions 1.22 and 1.24 show that M^{θ} ⊂ M is a cleft extension.

3.3 From cleft extensions to cocycle bicrossed products

Theorem 3.5. Let (M_{1}, Δ_{1}), (M_{2}, Δ_{2}) and (N, Γ) be l.c. quantum groups, ρ : M_{2} → N and γ : M_{1} → N̂ normal and faithful ∗-homomorphisms turning (N, Γ) into a cleft extension of (M_{2}, Δ_{2}) by (M̂_{1}, Δ̂_{1}).

Then there exists a cocycle matching (τ, U, V) of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}) and an isomorphism

Φ : (N, Γ) → (M, Δ)

of (N, Γ) onto the cocycle bicrossed product (M, Δ), satisfying

Φρ = α  and  Φ̂γ = β

where α and β are as in Definition 2.1, Φ̂ : (N̂, Γ̂) → (M̂, Δ̂) is the isomorphism canonically associated with Φ and characterized by

(Φ ⊗ Φ̂)(W) = W

(W is the multiplicative unitary of (N, Γ) and W is the one of (M, Δ)).

Proof. 1. The definition of α, U, β, V and Φ.

a) Represent (M_{1}, Δ_{1}), (M_{2}, Δ_{2}) and (N, Γ) on the GNS-space of a left Haar weight and denote by K the Hilbert space on which (N, Γ) acts, by W the multiplicative unitary of (N, Γ) and by J_{N} and Ĵ_{N} the modular conjugations of the invariant weights on N and N̂.

Let θ : N → M̂_{1} ⊗ N be the action of (M̂_{1}, Δ̂_{1}^{op}) on N constructed by γ, as in Proposition 3.1, then

θ(z) = Z(1 ⊗ z)Z^{∗} for all z ∈ N  and  Z = (J_{1} ⊗ Ĵ_{N})(ι ⊗ γ)(Ŵ_{1}^{∗})(J_{1} ⊗ Ĵ_{N}) .

One also has

(ι ⊗ θ)(Ŵ) = (γ ⊗ ι)(W_{1})_{12}Ŵ_{13} .

Applying now Proposition 3.1 to ρ : M_{2} → N we get an action η : N̂ → M̂_{2} ⊗ N̂ of (M̂_{2}, Δ̂_{2}^{op}) on N̂ by

η(z) = Z̃(1 ⊗ z)Z̃^{∗} for all z ∈ N̂  and  Z̃ = (J_{2} ⊗ J_{N})(ι ⊗ ρ)(Ŵ_{2}^{∗})(J_{2} ⊗ J_{N}) .  (3.2)

The action η satisfies

(η ⊗ ι)(Ŵ) = Ŵ_{23}(ι ⊗ ρ)(Ŵ_{2})_{13} .

Because γ and ρ turn (N, Γ) into an extension, one has N^{θ} = ρ(M_{2}). We claim that also N̂^{η} = γ(M_{1}). Indeed, (3.2) gives

N̂^{η} = N̂ ∩ J_{N}ρ(M_{2})′J_{N} .

Similarly we have

N^{θ} = N ∩ Ĵ_{N}γ(M_{1})′Ĵ_{N} .

So

ρ(M_{2}) = N ∩ Ĵ_{N}γ(M_{1})′Ĵ_{N} .

But for all x ∈ M_{2} we have Ĵ_{N}ρ(x)Ĵ_{N} = R(ρ(x)^{∗}) = ρ(R_{2}(x^{∗})) and so Ĵ_{N}ρ(M_{2})Ĵ_{N} = ρ(M_{2}). Therefore,

ρ(M_{2}) = N ∩ γ(M_{1})′ .

Since Γ̂γ = (γ ⊗ γ)Δ_{1} we get that γ(M_{1}) is a two-sided coideal in (N̂, Γ̂) in the sense of [13], Définition 3.1. Then it follows from [13], Théorème 3.3 that

γ(M_{1}) = N̂ ∩ (N ∩ γ(M_{1})′)′ = N̂ ∩ ρ(M_{2})′ .

Because J_{N}γ(M_{1})J_{N} = γ(M_{1}) one has

γ(M_{1}) = N̂ ∩ J_{N}ρ(M_{2})′J_{N} = N̂^{η}

which proves the claim.

Since γ and ρ turn (N, Γ) into a cleft extension, there exists a unitary X ∈ M_{1} ⊗ N satisfying

(ι ⊗ θ)(X) = W_{1,12}X_{13} .

From Proposition 1.22 it follows that (α̃, Ũ) is a cocycle action of (M_{1}, Δ_{1}) on N^{θ}, where

α̃(z) = X^{∗}(1 ⊗ z)X for all z ∈ N^{θ}  and  Ũ = X_{23}^{∗}X_{13}^{∗}(Δ_{1} ⊗ ι)(X).

From N^{θ} = ρ(M_{2}) we can define

α = (ι ⊗ ρ^{−1})α̃ρ  and  U = (ι ⊗ ι ⊗ ρ^{−1})(U)

and then α : M_{2} → M_{1} ⊗ M_{2} is a faithful and normal ∗-homomorphism, U ∈ M_{1} ⊗ M_{1} ⊗ M_{2} is a unitary and

(ι ⊗ α)α(x) = U (Δ_{1} ⊗ ι)α(x) U^{∗} for all x ∈ M_{2}
(ι ⊗ ι ⊗ α)(U) (Δ_{1} ⊗ ι ⊗ ι)(U) = (1 ⊗ U) (ι ⊗ Δ_{1} ⊗ ι)(U) .

Still using Proposition 1.22 we can define a ∗-isomorphism

Φ : N → M_{1} _{α,U}⋉ M_{2}

satisfying

Φ(ρ(x)) = α(x) for all x ∈ M_{2}  and  (ι ⊗ Φ)(X) = W_{1,12}U^{∗} .

From now on we denote M = M_{1} _{α,U}⋉ M_{2}.

b) To show that also N̂ has a cocycle crossed product structure we will find a cocycle action of (M_{2}, Δ_{2}) on M_{1}. Let us define a unitary Ỹ in N̂ ⊗ N by

Ỹ = (γ ⊗ ι)(X^{∗})Ŵ .  (3.3)

Then we have

(ι ⊗ θ)(Ỹ) = (γ ⊗ θ)(X^{∗})(ι ⊗ θ)(Ŵ) = (γ ⊗ ι)(X^{∗})_{13} (γ ⊗ ι)(W_{1}^{∗})_{12} (γ ⊗ ι)(W_{1})_{12} Ŵ_{13} = Ỹ_{13} .

So Ỹ ∈ N̂ ⊗ N^{θ} and we define Y = (ι ⊗ ρ^{−1})(Ỹ) ∈ N̂ ⊗ M_{2}. Observe now that

(η ⊗ ι)(Ỹ) = (γ ⊗ ι)(X^{∗})_{23} (η ⊗ ι)(Ŵ) = (γ ⊗ ι)(X^{∗})_{23} Ŵ_{23} (ι ⊗ ρ)(Ŵ_{2})_{13} = Ỹ_{23}(ι ⊗ ρ)(Ŵ_{2})_{13} .

From this we may conclude that

(ι ⊗ η)(ΣY^{∗}Σ) = W_{2,12}(ΣY^{∗}Σ)_{13} .

Hence Proposition 1.22 shows that N̂ has a cocycle crossed product structure and we can define

β̃ : N̂^{η} → N̂^{η} ⊗ M_{2} : β̃(z) = Y (z ⊗ 1)Y^{∗}
Ṽ = Y_{12}Y_{13}(ι ⊗ Δ_{2}^{op})(Y^{∗}) ∈ N̂^{η} ⊗ M_{2} ⊗ M_{2} .

Then β = (γ^{−1} ⊗ ι)β̃γ : M_{1} → M_{1} ⊗ M_{2} is a faithful and normal ∗-homomorphism, V = (γ^{−1} ⊗ ι ⊗ ι)(Ṽ) is a unitary in M_{1} ⊗ M_{2} ⊗ M_{2} and

(β ⊗ ι)β(x) = V(ι ⊗ Δ_{2}^{op})β(x)V^{∗} for all x ∈ M_{1}
(β ⊗ ι ⊗ ι)(V)(ι ⊗ ι ⊗ Δ_{2}^{op})(V) = (V ⊗ 1)(ι ⊗ Δ_{2}^{op} ⊗ ι)(V) .

If M̃ is the von Neumann algebra acting on H_{1} ⊗ H_{2} generated by β(M_{1}) and the elements (ι ⊗ ι ⊗ ω)(V(1 ⊗ Ŵ_{2})), it also follows from Proposition 1.22 that there is a unique ∗-isomorphism Φ̃ : N̂ → M̃ satisfying

Φ̃(γ(x)) = β(x) for all x ∈ M_{1}  and  (Φ̃ ⊗ ι)(Y) = V(1 ⊗ Ŵ_{2}) .

2. The definition of τ.

a) The comultiplication Δ = (Φ⊗Φ)ΓΦ^{−1} on M makes (M, Δ) a l.c. quantum group and Φ : (N, Γ) → (M, Δ) an isomorphism. We claim that for W̃ = W_{1,12}U^{∗} one has

(ι ⊗ Δ^{op})(W̃) = (W̃ ⊗ 1 ⊗ 1) ((ι ⊗ α)β ⊗ ι ⊗ ι)(W̃) (ι ⊗ α ⊗ α)(V) .

Indeed, observe that, by definition of Y and equation (3.3), we have

Ŵ = (γ ⊗ ι)(X)(ι ⊗ ρ)(Y) .

From this it follows that

(ι ⊗ Γ^{op} )(Ŵ) = (γ ⊗ Γ^{op} )(X) (ι ⊗ ρ ⊗ ρ)(ι ⊗ Δ_{2}^{op})(Y)
= (γ ⊗ Γ^{op} )(X) (ι ⊗ ρ ⊗ ρ)(Ṽ^{∗}Y_{12}Y_{13})
= (γ ⊗ Γ^{op} )(X) (γ ⊗ ρ ⊗ ρ)(V^{∗}) (ι ⊗ ρ)(Y)_{12} (ι ⊗ ρ)(Y)_{13} .

On the other hand,

(ι ⊗ Γ^{op} )(Ŵ) = Ŵ_{12}Ŵ_{13} = (γ ⊗ ι)(X)_{12}(ι ⊗ ρ)(Y)_{12} (γ ⊗ ι)(X)_{13}(ι ⊗ ρ)(Y)_{13} .

Both computations together yield

(γ ⊗ Γ^{op} )(X) = (γ ⊗ ι)(X)_{12}(ι ⊗ ρ)(Y)_{12}(γ ⊗ ι)(X)_{13}(ι ⊗ ρ)(Y^{∗})_{12}(γ ⊗ ρ ⊗ ρ)(V) .  (3.4)

But

(ι ⊗ ρ)(Y)_{12}(γ ⊗ ι)(X)_{13}(ι ⊗ ρ)(Y^{∗})_{12} = (ι ⊗ ρ ⊗ ι)(Y_{12}(γ ⊗ ι)(X)_{13}Y_{12}^{∗})
= (ι ⊗ ρ ⊗ ι)(β̃γ ⊗ ι)(X) = (γ ⊗ ρ ⊗ ι)(β ⊗ ι)(X) .

Then one can conclude, applying γ^{−1} ⊗ ι ⊗ ι to equation (3.4), that

(ι ⊗ Γ^{op} )(X) = X_{12} (ι ⊗ ρ ⊗ ι)(β ⊗ ι)(X) (ι ⊗ ρ ⊗ ρ)(V) .

Apply ι ⊗ Φ ⊗ Φ to this expression, then

(ι ⊗ Δ^{op})(W̃) = (W̃ ⊗ 1 ⊗ 1) ((ι ⊗ α)β ⊗ ι ⊗ ι)(W̃) (ι ⊗ α ⊗ α)(V)  (3.5)

and that is precisely the claim. The formula Γρ = (ρ ⊗ ρ)Δ_{2} implies

Δα = (α ⊗ α)Δ_{2} .  (3.6)

which together with the previous formula determines completely the comultiplication Δ on M.

b) Repeating the proof of Proposition 2.8 and using (3.5), (3.6), we prove that the dual weight φ on M = M_{1} _{α,U}⋉ M_{2} of the weight φ_{2} on M_{2}, with canonical GNS-construction (H_{1} ⊗ H_{2}, ι, Λ) in the sense of Terminology 1.17, is left invariant on the l.c. quantum group (M, Δ), and defining

Ŵ = (β ⊗ ι ⊗ ι)((W_{1} ⊗ 1)U^{∗}) (ι ⊗ ι ⊗ α)(V(1 ⊗ Ŵ_{2}))  and  W = ΣŴ^{∗}Σ,

where Σ flips H ⊗ H when H = H_{1} ⊗ H_{2}, we get that

Λ((ω ⊗ ι)Δ(z)) = (ω ⊗ ι)(W^{∗})Λ(z) for all z ∈ N_{φ}, ω ∈ M_{∗} .

So W is the multiplicative unitary associated with the l.c. quantum group (M, Δ) and the GNS-construction (H_{1} ⊗ H_{2}, ι, Λ) for its left invariant weight φ.

c) The comultiplication Δ̃ = (Φ̃ ⊗ Φ̃)Γ̂Φ̃^{−1} makes (M̃, Δ̃) a l.c. quantum group and Φ̃ : (N̂, Γ̂) → (M̃, Δ̃) is an isomorphism. Exactly as above for Δ, now starting from

Ŵ = (γ ⊗ ι)(X)(ι ⊗ ρ)(Y)  and  (Γ̂ ⊗ ι)(Ŵ) = Ŵ_{13}Ŵ_{23}

one can prove that

(Δ̃ ⊗ ι)(W̌) = (1 ⊗ 1 ⊗ W̌) (ι ⊗ ι ⊗ (β ⊗ ι)α)(W̌) (β ⊗ β ⊗ ι)(U)  and  Δ̃β = (β ⊗ β)Δ_{1},

where W̌ = (1 ⊗ Ŵ_{2}^{∗})V^{∗} and that the dual weight φ̃ on M̃ of the weight φ_{1} on M_{1}, with canonical GNSconstruction (H_{1} ⊗ H_{2}, ι, Λ̃), is left invariant on (M̃, Δ̃), and

Λ̃((ω ⊗ ι)Δ̃(z)) = (ι ⊗ ω)(W)Λ̃(z) for all z ∈ Nφ̃, ω ∈ M̃_{∗} .

Then we may conclude that (M̃, Δ̃) is precisely the dual of (M, Δ).

d) Apply [24], Propositions 8.14 and 8.15 to the l.c. quantum group (M, Δ) to get a canonical left invariant weight φ̂ on (M̂, Δ̂) with canonical GNS-construction (H_{1} ⊗ H_{2}, ι, Λ̂). We already have (M̂, Δ̂) = (M̃, Δ̃) and the left invariant weight φ̃ on (M̃, Δ̃) with GNS-construction (H_{1} ⊗ H_{2}, ι, Λ̃). Just as in the beginning of the proof of Proposition 2.16, we get λ ∈ ℂ \ {0} such that Λ̂ = λΛ̃. Denote by Ĵ the modular conjugation of φ̂ in the GNS-construction (H_{1} ⊗ H_{2}, ι, Λ̂). Putting u = λ̄/λ we get that uĴ is the modular conjugation of φ̃ in the GNS-construction (H_{1} ⊗ H_{2}, ι, Λ̃). Combining this with Proposition 1.19, we have

β(x) = uĴ(J_{1} ⊗ Ĵ_{2})(x ⊗ 1)(J_{1} ⊗ Ĵ_{2})Ĵū = Ĵ(J_{1} ⊗ Ĵ_{2})(x ⊗ 1)(J_{1} ⊗ Ĵ_{2})Ĵ

for all x ∈ M_{1}. Further we get Ĵ(J_{1} ⊗ Ĵ_{2}) ∈ B(H_{1}) ⊗ M_{2}. If J is the modular conjugation of φ in the GNS-construction (H_{1} ⊗ H_{2}, ι, Λ), it follows from Proposition 1.19 that

α(x) = J(Ĵ_{1} ⊗ J_{2})(1 ⊗ x)(Ĵ_{1} ⊗ J_{2})J

for all x ∈ M_{2}. Further we get J(Ĵ_{1} ⊗ J_{2}) ∈ M_{1} ⊗ B(H_{2}). Ddefine a unitary R on H_{1} ⊗ H_{2} by

R = JĴ(J_{1}Ĵ_{1} ⊗ Ĵ_{2}J_{2}) .

For all x ∈ M_{2} one has, using that Ĵ(J_{1} ⊗ Ĵ_{2}) ∈ B(H_{1}) ⊗ M_{2},

R(1 ⊗ x)R^{∗} = J Ĵ(J_{1} ⊗ Ĵ_{2}) (1 ⊗ J_{2}xJ_{2}) (J_{1} ⊗ Ĵ_{2})Ĵ J
= J(1 ⊗ J_{2}xJ_{2})J = α(x)

From [25], Corollary 2.12 we have the existence of a complex number ν such that

R = ν ĴJ(Ĵ_{1}J_{1} ⊗ J_{2}Ĵ_{2}) .

Using this and J(Ĵ_{1} ⊗ J_{2}) ∈ M_{1} ⊗ B(H_{2}) we get, for all x ∈ M_{1},

R(x ⊗ 1)R^{∗} = Ĵ J(Ĵ_{1} ⊗ J_{2}) (J_{1}xJ_{1} ⊗ 1) (Ĵ_{1} ⊗ J_{2})J Ĵ
= Ĵ(J_{1}xJ_{1} ⊗ 1)Ĵ = β(x)

Taking both computations together we can define a faithful ∗-homomorphism τ

τ : M_{1} ⊗ M_{2} → M_{1} ⊗ M_{2} : τ(z) = RzR^{∗}

satisfying α(x) = τ(1 ⊗ x) for all x ∈ M_{2} and β(x) = τ(x ⊗ 1) for all x ∈ M_{1}.

3. The proof of compatibility relations.

First we prove the relation

V_{132} (ι ⊗ Δ_{2})α(x) V_{132}^{∗} = τ_{13}(α ⊗ ι)Δ_{2}(x)

for all x ∈ M_{2}. Recall that

Ŵ = (β ⊗ ι ⊗ ι)(W̃) (ι ⊗ ι ⊗ α)(W̌^{∗}) where W̃ = (W_{1} ⊗ 1)U^{∗} and W̌ = (1 ⊗ Ŵ_{2}^{∗})V^{∗} .

Hence we obtain, for all z ∈ M,

Δ^{op}(z) = Ŵ (z ⊗ 1)Ŵ^{∗} = (β ⊗ ι ⊗ ι)(W̃) (ι ⊗ ι ⊗ α)(V(ι ⊗ Δ_{2}^{op})(z)V^{∗}) (β ⊗ ι ⊗ ι)(W̃^{∗}) .

Using that Δ(α(x)) = (α ⊗ α)Δ_{2}(x) for all x ∈ M_{2}, and using the previous formula, we get for all x ∈ M_{2}

(ι ⊗ ι ⊗ α)(V(ι ⊗ Δ_{2}^{op})(α(x))V^{∗}) = (β ⊗ ι ⊗ ι)(W̃^{∗}) Δ^{op}(α(x)) (β ⊗ ι ⊗ ι)(W̃)
= (β ⊗ ι ⊗ ι)(W̃^{∗}) (α ⊗ α)Δ_{2}^{op}(x) (β ⊗ ι ⊗ ι)(W̃)
= (τ ⊗ ι ⊗ ι)(W̃_{134}^{∗} ((ι ⊗ α)Δ_{2}^{op}(x))_{234} W̃_{134}) .

Because (α, U) is a cocycle action of (M_{1}, Δ_{1}) on M_{2} we have

W̃^{∗}(1 ⊗ α(y))W̃ = (ι ⊗ α)α(y) for all y ∈ M_{2}

and using this we can continue the previous computation as

(ι ⊗ ι ⊗ α)(V(ι ⊗ Δ_{2}^{op})(α(x))V^{∗}) = (τ ⊗ ι ⊗ ι)((ι ⊗ (ι ⊗ α)α)Δ_{2}^{op}(x)_{2134})
= (ι ⊗ ι ⊗ α)τ_{12}σ_{12}(ι ⊗ α)Δ_{2}^{op}(x) .

Because α is faithful we may conclude that

V (ι ⊗ Δ_{2}^{op})α(x) V^{∗} = τ_{12}σ_{12}(ι ⊗ α)Δ_{2}^{op}(x)

for all x ∈ M_{2}, from where the needed relation follows. Using now the formula Δ̂(β(x)) = (β ⊗ β)Δ_{1}(x) for all x ∈ M_{1} and the fact that (σβ, V_{321}) is a cocycle action of (M_{2}, Δ_{2}) on M_{1} we obtain similarly, for all x ∈ M_{1},

U (Δ_{1} ⊗ ι)β(x) U^{∗} = τ_{23}σ_{23}(β ⊗ ι)Δ_{1}(x).

b) Let us prove the relation (Δ_{1} ⊗ ι ⊗ ι)(V)(ι ⊗ ι ⊗ Δ_{2}^{op})(U^{∗}) = (U^{∗} ⊗ 1)(ι ⊗ τσ ⊗ ι)((β ⊗ ι ⊗ ι)(U^{∗})(ι ⊗ ι ⊗ α)(V))(1 ⊗ V) .

From equation (3.5) one has

(W̃ ⊗ 1 ⊗ 1) ((ι ⊗ α)β ⊗ ι ⊗ ι)(W̃) (ι ⊗ α ⊗ α)(V) = (ι ⊗ Δ^{op})(W̃)

= (1 ⊗ (β ⊗ ι ⊗ ι)(W̃)) (ι ⊗ ι ⊗ ι ⊗ α)((1 ⊗ V)(ι ⊗ ι ⊗ Δ_{2}^{op})(W̃)(1 ⊗ V^{∗})) (1 ⊗ (β ⊗ ι ⊗ ι)(W̃^{∗}))

and so

(ι ⊗ ι ⊗ ι ⊗ α)((1 ⊗ V)(ι ⊗ ι ⊗ Δ_{2}^{op})(W̃)(1 ⊗ V^{∗}))

= (1 ⊗ (β ⊗ ι ⊗ ι)(W̃^{∗})) (W̃ ⊗ 1 ⊗ 1) ((ι ⊗ α)β ⊗ ι ⊗ ι)(W̃) (ι ⊗ α ⊗ α)(V) (1 ⊗ (β ⊗ ι ⊗ ι)(W̃)) . Using now that for all x ∈ M_{1} we have

W̃^{∗}(1 ⊗ β(x))W̃ = U (Δ_{1} ⊗ ι)β(x) U^{∗} = τ_{23}σ_{23}(β ⊗ ι)Δ_{1}(x)

we obtain

(ι ⊗ ι ⊗ ι ⊗ α) ((1 ⊗ V)(ι ⊗ ι ⊗ Δ_{2}^{op})(W̃)(1 ⊗ V^{∗}))
= (W̃ ⊗ 1 ⊗ 1) τ_{23}σ_{23}((β ⊗ ι ⊗ ι ⊗ ι)(Δ_{1} ⊗ ι ⊗ ι)(W̃^{∗})) ((ι ⊗ α)β ⊗ ι ⊗ ι)(W̃) (ι ⊗ α ⊗ α)(V)
(1 ⊗ (β ⊗ ι ⊗ ι)(W̃))

= (W̃ ⊗ 1 ⊗ 1) τ_{23}σ_{23}((β ⊗ ι ⊗ ι ⊗ ι)(Δ_{1} ⊗ ι ⊗ ι)(W̃^{∗}) (β ⊗ ι ⊗ ι)(W̃)_{1245} (ι ⊗ ι ⊗ α)(V)_{1245} W̃_{345}) . Using now that

W̃^{∗}(1 ⊗ α(x))W̃ = (ι ⊗ α)α(x)

for all x ∈ M_{1} we get

(ι ⊗ ι ⊗ ι ⊗ α) ((1 ⊗ V)(ι ⊗ ι ⊗ Δ_{2}^{op})(W̃)(1 ⊗ V^{∗}))

= (W̃ ⊗ 1 ⊗ 1) τ_{23}σ_{23}((β ⊗ ι ⊗ ι ⊗ ι)(Δ_{1} ⊗ ι ⊗ ι)(W̃^{∗}) (β ⊗ ι ⊗ ι)(W̃)_{1245} W̃_{345} (ι ⊗ ι ⊗ (ι ⊗ α)α)(V))

= (W̃ ⊗ 1 ⊗ 1) τ_{23}σ_{23}((β ⊗ ι ⊗ ι ⊗ ι)((Δ_{1} ⊗ ι ⊗ ι)(W̃^{∗}) W̃_{134}W̃_{234}) (ι ⊗ ι ⊗ (ι ⊗ α)α)(V)) .

Before we continue this computation, we use the cocycle property of U to obtain

(Δ_{1} ⊗ ι ⊗ ι)(W̃^{∗}) W̃_{134} W̃_{234} = (Δ_{1} ⊗ ι ⊗ ι)(U)W_{1,23}^{∗}W_{1,13}^{∗} W_{1,13}U_{134}^{∗} W_{1,23}U_{234}^{∗}
= (Δ_{1} ⊗ ι ⊗ ι)(U) (ι ⊗ Δ_{1} ⊗ ι)(U^{∗}) U_{234}^{∗}
= (ι ⊗ ι ⊗ α)(U^{∗}) .

Then we can continue the computation above and obtain

(ι ⊗ ι ⊗ ι ⊗ α) ((1 ⊗ V)(ι ⊗ ι ⊗ Δ_{2}^{op})(W̃)(1 ⊗ V^{∗}))
= (W̃ ⊗ 1 ⊗ 1) τ_{23}σ_{23}((β ⊗ ι ⊗ α)(U^{∗}) (ι ⊗ ι ⊗ (ι ⊗ α)α)(V))
= (ι ⊗ ι ⊗ ι ⊗ α)((W̃ ⊗ 1) τ_{23}σ_{23}((β ⊗ ι ⊗ ι)(U^{∗}) (ι ⊗ ι ⊗ α)(V))) .

Because α is faithful it now follows that

(1 ⊗ V)(ι ⊗ ι ⊗ Δ_{2}^{op})(W̃)(1 ⊗ V^{∗}) = (W̃ ⊗ 1) τ_{23}σ_{23}((β ⊗ ι ⊗ ι)(U^{∗}) (ι ⊗ ι ⊗ α)(V)) .

Because W̃ = (W_{1} ⊗ 1)U^{∗} this gives us

(1 ⊗ V)W_{1,12} (ι ⊗ ι ⊗ Δ_{2}^{op})(U^{∗}) (1 ⊗ V^{∗}) = W_{1,12}(U^{∗} ⊗ 1) τ_{23}σ_{23}((β ⊗ ι ⊗ ι)(U^{∗}) (ι ⊗ ι ⊗ α)(V))

so that finally

(Δ_{1} ⊗ ι ⊗ ι)(V) (ι ⊗ ι ⊗ Δ_{2}^{op})(U^{∗}) (1 ⊗ V^{∗}) = (U^{∗} ⊗ 1) (ι ⊗ τσ ⊗ ι)((β ⊗ ι ⊗ ι)(U^{∗}) (ι ⊗ ι ⊗ α)(V)) .

c) It follows from equations (3.5) and (3.6) that (M, Δ) is indeed the cocycle bicrossed product of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}). We also have the isomorphism Φ : (N, Γ) → (M, Δ) satisfying Φρ = α. Let Φ̂ be the canonically associated isomorphism Φ̂ : (N̂, Γ̂) → (M̂, Δ̂) characterized by

(Φ ⊗ Φ̂)(W) = W .

Since Ŵ = (γ ⊗ ι)(X)(ι ⊗ ρ)(Y), one has

(Φ ⊗ Φ̃)(W) = W .

So we obtain Φ̃ = Φ̂ and Φ̂γ = β, which concludes the proof.

3.4 Isomorphisms of cleft extensions

Definition 3.6. Let (M_{1}, Δ_{1}), (M_{2}, Δ_{2}), (M_{a}, Δ_{a}) and (M_{b}, Δ_{b}) be l.c. quantum groups. Extensions

(M_{2}, Δ_{2}) ⟶^{α_{a}} (M_{a}, Δ_{a}) ⟶^{β_{a}} (M̂_{1}, Δ̂_{1})  and  (M_{2}, Δ_{2}) ⟶^{α_{b}} (M_{b}, Δ_{b}) ⟶^{β_{b}} (M̂_{1}, Δ̂_{1})

are said to be isomorphic, if there exists an isomorphism of l.c. quantum groups

π : (M_{a}, Δ_{a}) → (M_{b}, Δ_{b})

satisfying πα_{a} = α_{b} and π̂β_{a} = β_{b}, where π̂ is the canonical isomorphism of (M̂_{a}, Δ̂_{a}) onto (M̂_{b}, Δ̂_{b}) associated with π.

Then we can prove the following result.

Proposition 3.7. Let (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}) be l.c. quantum groups, (τ_{a}, U_{a}, V_{a}) and (τ_{b}, U_{b}, V_{b}) two cocycle matchings of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}) with the corresponding actions α_{a}, α_{b}, β_{a} and β_{b}, and (M_{a}, Δ_{a}) and (M_{b}, Δ_{b}) the respective cocycle bicrossed products. If the extensions

(M_{2}, Δ_{2}) ⟶^{α_{a}} (M_{a}, Δ_{a}) ⟶^{β_{a}} (M̂_{1}, Δ̂_{1})  and  (M_{2}, Δ_{2}) ⟶^{α_{b}} (M_{b}, Δ_{b}) ⟶^{β_{b}} (M̂_{1}, Δ̂_{1})

are isomorphic through the isomorphism π, then there exists a unitary R in M_{1} ⊗ M_{2} satisfying

τ_{b}(z) = R τ_{a}(z) R^{∗} for all z ∈ M_{1} ⊗ M_{2}
U_{b} = (1 ⊗ R) (ι ⊗ α_{a})(R) U_{a} (Δ_{1} ⊗ ι)(R^{∗})
V_{b} = (R ⊗ 1) (β_{a} ⊗ ι)(R) V_{a} (ι ⊗ Δ_{2}^{op})(R^{∗})
π(z) = R z R^{∗} for all z ∈ M_{a}
π̂(z) = R z R^{∗} for all z ∈ M̂_{a} .

If conversely (τ_{a}, U_{a}, V_{a}) is a cocycle matching of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}), and R is a unitary in M_{1} ⊗ M_{2}, then the formulas above define a cocycle matching (τ_{b}, U_{b}, V_{b}) of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}). If (M_{a}, Δ_{a}) and (M_{b}, Δ_{b}) are the corresponding cocycle bicrossed products, one can define an isomorphism

π : (M_{a}, Δ_{a}) → (M_{b}, Δ_{b}) : π(z) = R z R^{∗}

of l.c. quantum groups such that πα_{a} = α_{b} and π̂β_{a} = β_{b}.

Observe that when both M_{1} and M_{2} are commutative, the extensions given by cocycle matchings (τ_{a}, U_{a}, V_{a}) and (τ_{b}, U_{b}, V_{b}) are isomorphic if and only if τ_{a} = τ_{b} and there exists a unitary R in M_{1} ⊗ M_{2} satisfying

U_{b} = (1 ⊗ R) (ι ⊗ α_{a})(R) U_{a} (Δ_{1} ⊗ ι)(R^{∗})
V_{b} = (R ⊗ 1) (β_{a} ⊗ ι)(R) V_{a} (ι ⊗ Δ_{2}^{op})(R^{∗}) .

We will come back to this situation in Subsection 4.3.

Proof. First let π be an isomorphism of (M_{a}, Δ_{a}) onto (M_{b}, Δ_{b}) satisfying πα_{a} = α_{b} and π̂β_{a} = β_{b}. Recall that π̂ satisfies (π ⊗ π̂)(W_{a}) = W_{b} where W_{a} and W_{b} denote the multiplicative unitaries of (M_{a}, Δ_{a}) and (M_{b}, Δ_{b}) respectively. For the dual actions α̂_{a} and α̂_{b} of (M̂_{1}, Δ̂_{1}^{op}) on M_{a} and M_{b} respectively it follows from Proposition 1.4 and Definition 2.2 that

(ι ⊗ ι ⊗ α̂_{a})(Ŵ_{a}) = (β_{a} ⊗ ι)(W_{1})_{123} Ŵ_{a,1245}  and  (ι ⊗ ι ⊗ α̂_{b})(Ŵ_{b}) = (β_{b} ⊗ ι)(W_{1})_{123} Ŵ_{b,1245} .

Hence we get

(ι ⊗ ι ⊗ (ι ⊗ π)α̂_{a})(Ŵ_{a}) = (β_{a} ⊗ ι)(W_{1})_{123} (ι ⊗ ι ⊗ π)(Ŵ_{a})_{1245} = (π̂^{−1} ⊗ ι ⊗ ι ⊗ ι)((β_{b} ⊗ ι)(W_{1})_{123} Ŵ_{b,1245})
= (π̂^{−1} ⊗ α̂_{b})(Ŵ_{b}) = (ι ⊗ ι ⊗ α̂_{b}π)(Ŵ_{a}) .

So α̂_{b}π = (ι ⊗ π)α̂_{a}.

As before we define the unitaries W̃_{a} and W̃_{b} as

W̃_{a} = (W_{1} ⊗ 1)U_{a}^{∗}  and  W̃_{b} = (W_{1} ⊗ 1)U_{b}^{∗} .

Then we define a unitary Y := W̃_{b}^{∗}(ι ⊗ π)(W̃_{a}) ∈ M_{1} ⊗ M_{b}. Proposition 1.4 and the formula above imply

(ι ⊗ α̂_{b})(Y) = W̃_{b,134}^{∗}W_{1,12}^{∗} (ι ⊗ ι ⊗ π)(W_{1,12}W̃_{a,134}) = Y_{134} .

Therefore, Y ∈ M_{1} ⊗ M α̂_{b} ; b. Theorem 1.11 shows that there is a unitary R ∈ M_{1} ⊗ M_{2} such that Y = (ι ⊗ α_{b})(R), hence

(ι ⊗ π)(W̃_{a}) = W̃_{b}(ι ⊗ α_{b})(R) .  (3.7)

Let φ_{a} and φ_{b} be the canonical left invariant weights on (M_{a}, Δ_{a}) and (M_{b}, Δ_{b}) with GNS-construction (H_{1} ⊗ H_{2}, ι, Λ_{a}) and (H_{1} ⊗ H_{2}, ι, Λ_{b}). We claim that for all z ∈ N_{φ_{a}} we have π(z) ∈ N_{φ_{b}} and

Λ_{b}(π(z)) = RΛ_{a}(z) .

To prove this, choose ξ ∈ H_{1}, b ∈ T_{φ_{1}}, x ∈ N_{φ_{2}}, an orthonormal basis (e_{i})_{i∈I} in H_{1} and define the element

z := (ω_{ξ,Λ_{1}(b)} ⊗ ι ⊗ ι)(W̃_{a}) α_{a}(x) ∈ N_{φ_{a}} .

So, with σ-strong^{∗} convergence π(z) is given by

π(z) = ∑_{i∈I}(ω_{e_{i},Λ_{1}(b)} ⊗ ι ⊗ ι)(W̃_{b}) α_{b}((ω_{ξ,e_{i}} ⊗ ι)(R)x) .

If we denote with y_{I_{0}} the sum over a finite subset I_{0} ⊂ I, we get that y_{I_{0}} ∈ N_{φ_{b}} and

Λ_{b}(y_{I_{0}}) = ∑_{i∈I_{0}} J_{1}σ_{i/2}^{1}(b)J_{1}e_{i} ⊗ (ω_{ξ,e_{i}} ⊗ ι)(R)Λ_{2}(x) .

Since Λ_{b} is σ-strong^{∗}– norm closed, we get that π(z) ∈ N_{φ_{b}} and

Λ_{b}(π(z)) = (J_{1}σ_{i/2}^{1}(b)J_{1} ⊗ 1)R(ξ ⊗ Λ_{2}(x)) = RΛ_{a}(z) .

Because such elements z form a σ-strong^{∗}– norm core for Λ_{a}, the claim is proved. Similarly z ∈ N_{φ_{a}} if and only if π(z) ∈ N_{φ_{b}}. From the formula Λ_{b}(π(z)) = RΛ_{a}(z) for all z ∈ N_{φ_{a}} one can conclude that

π(z) = R z R^{∗} for all z ∈ M_{a}  and  π̂(z) = R z R^{∗} for all z ∈ M̂_{a} .

Then also α_{b}(x) = π(α_{a}(x)) = R α_{a}(x) R^{∗} for all x ∈ M_{2} and β_{b}(y) = π̂(β_{a}(x)) = R β_{a}(x) R^{∗} for all x ∈ M_{1}. Therefore, τ_{b}(z) = R τ_{a}(z) R^{∗} for all z ∈ M_{1} ⊗ M_{2}. So it follows from equation (3.7) that

U_{b}(W_{1}^{∗} ⊗ 1) = W̃_{b}^{∗} = (ι ⊗ α_{b})(R) (ι ⊗ π)(W̃_{a}^{∗}) = (1 ⊗ R) (ι ⊗ α_{a})(R) U_{a} (W_{1}^{∗} ⊗ 1) (1 ⊗ R^{∗})

and hence

U_{b} = (1 ⊗ R) (ι ⊗ α_{a})(R) U_{a} (Δ_{1} ⊗ ι)(R^{∗}) .

It remains to prove the formula for V_{b}. We have that

Ŵ_{a} = (β_{a} ⊗ ι ⊗ ι)(W̃_{a}) (ι ⊗ ι ⊗ α_{a})(V_{a}(1 ⊗ Ŵ_{2}))  and  Ŵ_{b} = (β_{b} ⊗ ι ⊗ ι)(W̃_{b}) (ι ⊗ ι ⊗ α_{b})(V_{b}(1 ⊗ Ŵ_{2})) .

Because Ŵ_{b} = (π̂ ⊗ π)(Ŵ_{a}) and because of equation (3.7), we get that

(β_{b} ⊗ ι ⊗ ι)(W̃_{b}) (ι ⊗ ι ⊗ α_{b})(V_{b}(1 ⊗ Ŵ_{2})) = (π̂ ⊗ π)(Ŵ_{a})

= (β_{b} ⊗ ι ⊗ ι)(W̃_{b}(ι ⊗ α_{b})(R)) (ι ⊗ ι ⊗ α_{b})(π̂ ⊗ ι)(V_{a}(1 ⊗ Ŵ_{2})) .

From the faithfulness of α_{b} it follows that

V_{b}(1 ⊗ Ŵ_{2}) = (β_{b} ⊗ ι)(R) (π̂ ⊗ ι)(V_{a}(1 ⊗ Ŵ_{2}))

and so

V_{b} = (R ⊗ 1) (β_{a} ⊗ ι)(R) V_{a} (1 ⊗ Ŵ_{2})(R^{∗} ⊗ 1)(1 ⊗ Ŵ_{2}^{∗}) = (R ⊗ 1) (β_{a} ⊗ ι)(R) V_{a} (ι ⊗ Δ_{2}^{op})(R^{∗}) .

This concludes the proof of the first statement. Vice versa, let (τ_{a}, U_{a}, V_{a}) be a cocycle matching of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}), R a unitary in M_{1} ⊗ M_{2} and define (τ_{b}, U_{b}, V_{b}) by the formulas mentioned in the proposition. Then it is straightforward to check that (τ_{b}, U_{b}, V_{b}) is a cocycle matching of (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}). Let (M_{a}, Δ_{a}) and (M_{b}, Δ_{b}) the corresponding cocycle bicrossed products and Ŵ_{a} and Ŵ_{b} be the corresponding multiplicative unitaries. Then one can check that

(R ⊗ R)Ŵ_{a}(R^{∗} ⊗ R^{∗}) = Ŵ_{b} .

So π(z) = R z R^{∗} (z ∈ M_{a}), gives an isomorphism of l.c. quantum groups of (M_{a}, Δ_{a}) and (M_{b}, Δ_{b}) such that π̂(z) = R z R^{∗} (z ∈ M̂_{a}). Hence we get that πα_{a} = α_{b} and π̂β_{a} = β_{b}, which concludes the proof.

4 Extensions of l.c. groups

In this section we study extensions of l.c. groups, i.e., short exact sequences of the form

(L^{∞}(G_{2}), Δ_{2}) ⟶^{α} (M, Δ) ⟶^{β} (L(G_{1}), Δ̂_{1}),

where G_{1} and G_{2} are usual l.c. groups.

4.1 Extensions of discrete groups

If both G_{1} and G_{2} are finite, G.I. Kac [18], Theorem 3 showed that every extension has a cocycle bicrossed product structure (later it was shown in [1] that it follows from H.-J. Schneider’s work [40] that every extension of finite-dimensional Hopf algebras is cleft). We generalize the result of G.I. Kac as follows.

Theorem 4.1. Any extension

(L^{∞}(G_{2}), Δ_{2}) ⟶^{α} (M, Δ) ⟶^{β} (L(G_{1}), Δ̂_{1}),

where G_{1} and G_{2} are discrete groups is automatically cleft; applying then Theorem 3.5, we have that (M, Δ) is a cocycle bicrossed product of (L^{∞}(G_{1}), Δ_{1}) and (L^{∞}(G_{2}), Δ_{2}).

The proof of this theorem will consist of several lemmas, very much inspired by [18].

Define, as in Preliminaries, quantum groups (M_{1}, Δ_{1}) and (M_{2}, Δ_{2}), their duals and all their ingredients related to G_{1} and G_{2}. For g ∈ G_{i} denote by δ_{g} the element in M_{i} (i = 1, 2) determined by

δ_{g}(h) = δ_{g,h},

where δ_{g,h} is the Kronecker symbol; the elements δ_{g} form an orthonormal basis in H_{i} (i = 1, 2). Next define for all g ∈ G_{1} the functional ω_{g} ∈ M̂_{1 ∗} by

ω_{g}(x) = ⟨xδ_{e}, δ_{g}⟩ for all x ∈ M̂_{1}

and observe that

(ι ⊗ ω_{g})(W_{1}) = δ_{g}  and  ω_{g}(λ_{h}) = δ_{g,h}

for all g, h ∈ G_{1}, \overline{ω_{g}} = ω_{g^{−1}} and

(ι ⊗ ω_{g})Δ̂_{1}(x) = ω_{g}(x)λ_{g} for all x ∈ M̂_{1} .

Let α : M_{2} → M and β : M_{1} → M̂ be normal and faithful ∗-homomorphisms satisfying Δα = (α ⊗α)Δ_{2} and Δ̂β = (β ⊗β)Δ_{1}. Denote by μ the action defined in Proposition 3.1 such that μ = (R̂_{1} ⊗R)θR, M^{μ} = α(M_{2}) and (ι ⊗ Δ)μ = (μ ⊗ ι)Δ. We fix a left invariant weight φ on (M, Δ) with GNS-construction (H, ι, Λ).

Lemma 4.2. For all g ∈ G_{1} and s ∈ G_{2} there exists z ∈ M such that

(ω_{g} ⊗ ι)μ(z) α(δ_{s}) ≠ 0 .

Proof. We first claim that there exist z ∈ N_{φ} and k ∈ G_{2} such that α(δ_{k}) (ω_{g} ⊗ ι)μ(z) ≠ 0. If this is not the case, then for every z ∈ N_{φ} and k ∈ G_{2} we have α(δ_{k}) (ω_{g} ⊗ ι)μ(z) = 0, hence, using Proposition 3.1

0 = Λ(α(δ_{k}) (ω_{g} ⊗ ι)μ(z)) = α(δ_{k}) β((ι ⊗ ω_{g})(W_{1})) Λ(z) = α(δ_{k})β(δ_{g})Λ(z) .

Summing over k ∈ G_{2}, it then follows that β(δ_{g}) = 0 which contradicts the faithfulness of β. So one can take k ∈ G_{2} and z ∈ N_{φ} such that Z := α(δ_{k}) (ω_{g} ⊗ ι)μ(z) ≠ 0. Since Z ∈ N_{φ}, we have 0 < φ(Z^{∗}Z) < ∞. Fix g ∈ G_{1} and define

I_{g} = {s ∈ G_{2} | There exists a z ∈ M with (ω_{g} ⊗ ι)μ(z) α(δ_{s}) ≠ 0} .

Suppose s ∈ G_{2} \ I_{g}. One has (ω_{g} ⊗ ι)μ(y) α(δ_{s}) = 0 for all y ∈ M. We claim that Δ(Z)(α(δ_{s}) ⊗ 1) = 0. Indeed,

Δ(Z) = (α ⊗ α)Δ_{2}(δ_{k}) Δ((ω_{g} ⊗ ι)μ(z)) = (α ⊗ α)Δ_{2}(δ_{k}) ((ω_{g} ⊗ ι)μ ⊗ ι)Δ(z) .

For all η ∈ M_{∗}

(ι ⊗ η)(((ω_{g} ⊗ ι)μ ⊗ ι)Δ(z)) α(δ_{s}) = (ω_{g} ⊗ ι)μ((ι ⊗ η)Δ(z)) α(δ_{s}) = 0

and so

((ω_{g} ⊗ ι)μ ⊗ ι)Δ(z) (α(δ_{s}) ⊗ 1) = 0 .

From this the claim follows. But then

(α(δ_{s}) ⊗ 1)Δ(Z^{∗}Z)(α(δ_{s}) ⊗ 1) = 0 .

Applying ι ⊗ φ we obtain

φ(Z^{∗}Z) α(δ_{s}) = 0

i.e., a contradiction with the faithfulness of α. Therefore, G_{2} = I_{g}, which concludes the proof. Lemma 4.3. Let g ∈ G_{1}. Then

{(ω_{g} ⊗ ι)μ(z) | z ∈ M} = {z ∈ M | μ(z) = λ_{g} ⊗ z} .

Proof. Let z ∈ M. Then

μ((ω_{g} ⊗ ι)μ(z)) = ((ω_{g} ⊗ ι)Δ̂_{1} ⊗ ι)μ(z) = λ_{g} ⊗ (ω_{g} ⊗ ι)μ(z) .

This proves the inclusion ⊂. Vice versa, let z ∈ M and suppose μ(z) = λ_{g} ⊗ z. Then z = (ω_{g} ⊗ ι)μ(z), which gives the converse inclusion. Lemma 4.4. Let g ∈ G_{1}, s, k ∈ G_{2} and z ∈ M. If the element u ∈ M defined by

u := α(δ_{k}) (ω_{g} ⊗ ι)μ(z) α(δ_{s})

is different from 0, then \frac{u}{∥u∥} is a partial isometry with initial projection α(δ_{s}) and final projection α(δ_{k}). Proof. Define

v := (ω_{g} ⊗ ι)μ(z)^{∗} α(δ_{k}) (ω_{g} ⊗ ι)μ(z) = (ω_{g^{−1}} ⊗ ι)μ(z^{∗}) α(δ_{k}) (ω_{g} ⊗ ι)μ(z) .

Then u^{∗}u = α(δ_{s}) v α(δ_{s}) and further, using the previous lemma,

μ(v) = (λ_{g^{−1}} ⊗ (ω_{g^{−1}} ⊗ ι)μ(z^{∗})) (1 ⊗ α(δ_{k})) (λ_{g} ⊗ (ω_{g} ⊗ ι)μ(z)) = 1 ⊗ v .

Hence v ∈ M^{μ} = α(M_{2}) and one can take x ∈ M_{2} such that v = α(x). Then

u^{∗}u = α(δ_{s}xδ_{s}) = x(s)α(δ_{s}) .

From this we have u^{∗}u = ∥u∥^{2}α(δ_{s}) and similarly uu^{∗} = ∥u∥^{2}α(δ_{k}), which concludes the proof. Lemma 4.5. Let g ∈ G_{1} and s ∈ G_{2}. Then there exists at most one k ∈ G_{2} such that

{α(δ_{k}) (ω_{g} ⊗ ι)μ(z) α(δ_{s}) | z ∈ M} ≠ {0} .

Proof. If there are k, l ∈ G_{2}, both satisfying the condition above, one can take y, z ∈ M such that

u := α(δ_{k}) (ω_{g} ⊗ ι)μ(y) α(δ_{s}) ≠ 0  and  v := α(δ_{l}) (ω_{g} ⊗ ι)μ(z) α(δ_{s}) ≠ 0 .

We may suppose that ∥u∥ = ∥v∥ = 1. By Lemma 4.4 u (resp., v) is a partial isometry with initial projection α(δ_{s}) and final projection α(δ_{k}) (resp., α(δ_{l})). Hence uv^{∗} is a partial isometry with initial projection α(δ_{l}) and final projection α(δ_{k}). On the other hand, Lemma 4.3 gives

μ(u) = λ_{g} ⊗ u  and  μ(v) = λ_{g} ⊗ v,

so uv^{∗} ∈ M^{μ} = α(M_{2}). Choose x ∈ M_{2} with α(x) = uv^{∗}. Then

α(δ_{l}) = (uv^{∗})^{∗}uv^{∗} = α(x^{∗}x) = α(xx^{∗}) = uv^{∗}(uv^{∗})^{∗} = α(δ_{k}),

so l = k.

The following proposition is the main ingredient to prove Theorem 4.1.

Proposition 4.6. Let g ∈ G_{1} and s ∈ G_{2}. Then the subspace of M

E_{g,s} = {(ω_{g} ⊗ ι)μ(z) α(δ_{s}) | z ∈ M}

is one-dimensional and spanned by a partial isometry u_{g,s} with initial projection α(δ_{s}) and final projection α(δ_{P_{g}(s)}), where P_{g}(s) ∈ G_{2}. For every g ∈ G_{1}, the map s ↦ P_{g}(s) is a bijection of G_{2}. Further

μ(u_{g,s}) = λ_{g} ⊗ u_{g,s} .

Proof. By the previous lemma there is at most one k ∈ G_{2} such that α(δ_{k})E_{g,s} ≠ {0}. On the other hand, if such k does not exist, then α(δ_{k})E_{g,s} = {0} for all k ∈ G_{2} and summing over k we get E_{g,s} = {0}, which contradicts Lemma 4.2. So, let P_{g}(s) be the unique element in G_{2} satisfying

α(δ_{P_{g}(s)})E_{g,s} ≠ {0} .

Then y = α(δ_{P_{g}(s)})y for all y ∈ E_{g,s}. Suppose now that u, v ∈ E_{g,s} and ∥u∥ = ∥v∥ = 1. By Lemma 4.4, both u and v are partial isometries with initial projection α(δ_{s}) and final projection α(δ_{P_{g}(s)}). Lemma 4.3 shows that all elements y ∈ E_{g,s} satisfy μ(y) = λ_{g} ⊗ y, and hence v^{∗}u ∈ M^{μ} = α(M_{2}). But v^{∗}u is a partial isometry with initial and final projection equal to α(δ_{s}). Hence v^{∗}u = ±α(δ_{s}) and so u = ±v. Hence E_{g,s} is one-dimensional and generated by a partial isometry u_{g,s} with initial projection α(δ_{s}) and final projection α(δ_{P_{g}(s)}). Also we get μ(u_{g,s}) = λ_{g} ⊗ u_{g,s}.

Fix now g ∈ G_{1} and consider the map s ↦ P_{g}(s). First of all this map is injective. Take s ≠ t and suppose that P_{g}(s) = P_{g}(t) and denote P_{g}(s) = k. Then we can take y, z ∈ M such that

α(δ_{k}) (ω_{g} ⊗ ι)μ(y) α(δ_{s}) ≠ 0  and  α(δ_{k}) (ω_{g} ⊗ ι)μ(z) α(δ_{t}) ≠ 0 .

Taking the adjoint we get

α(δ_{s}) (ω_{g^{−1}} ⊗ ι)μ(y^{∗}) α(δ_{k}) ≠ 0  and  α(δ_{t}) (ω_{g^{−1}} ⊗ ι)μ(z^{∗}) α(δ_{k}) ≠ 0

and this contradicts Lemma 4.5. Therefore, the map s ↦ P_{g}(s) is injective. To prove its surjectivity, choose t ∈ G_{2}. By the above reasoning we can take s ∈ G_{2} and y ∈ N such that

α(δ_{s}) (ω_{g^{−1}} ⊗ ι)μ(y) α(δ_{t}) ≠ 0 .

Taking the adjoint we get

α(δ_{t}) (ω_{g} ⊗ ι)μ(y^{∗}) α(δ_{s}) ≠ 0

and this implies that P_{g}(s) = t.

Proof of Theorem 4.1. Recall that the action θ introduced in Proposition 3.1 satisfies θ(z) = (R̂_{1}⊗R)μ(R(z)) for all z ∈ M. In order to prove that (M, Δ) is a cleft extension, it suffices, by Proposition 1.24, to show the existence of a unitary X ∈ M_{1} ⊗ M satisfying (ι ⊗ θ)(X) = W_{1,12}X_{13}. Applying R_{1} ⊗ R̂_{1} ⊗ R we have to prove the existence of a unitary Y in M_{1} ⊗ M satisfying (ι ⊗ μ)(Y) = Y_{13}W_{1,12}.

Choose partial isometries u_{g,s} as in the previous proposition. Fix g ∈ G_{1}. Then (u_{g,s})_{s∈G_{2}} is a family of partial isometries in M with initial projections (α(δ_{s}))_{s∈G_{2}} and final projections (α(δ_{P_{g}(s)}))_{s∈G_{2}}. Because the map s ↦ P_{g}(s) is a bijection of G_{2} we can define a unitary Y_{g} in M by the sum

Y_{g} = ∑_{s∈G_{2}} u_{g,s}

converging in the σ-strong^{∗} topology. Since μ(u_{g,s}) = λ_{g} ⊗ u_{g,s} for all s ∈ G_{2}, we get μ(Y_{g}) = λ_{g} ⊗ Y_{g}. Identifying M_{1} ⊗ M with ℓ^{∞}(G_{1}, M) we can define Y ∈ M_{1} ⊗ M by Y (g) = Y_{g}. Then Y is unitary and

(ι ⊗ μ)(Y) = Y_{13}W_{1,12} .

This concludes the proof.

4.2 Extensions of l.c. groups

Here we consider regular Borel measures on l.c. spaces. A product measure means regular Borel product, see e.g. [9], Section 7.6. We say that a statement is valid almost everywhere if it is valid everywhere except on a set whose intersection with an arbitrary compact set is a Borel set of measure zero, see e.g. [9], Section 3.3 where they use the terminology locally almost everywhere. A function is called measurable when it is Borel measurable on every compact set. Given a l.c. space X with a regular Borel measure μ, we denote by K(X) the space of continuous compactly supported ℂ-valued functions on X and by L^{∞}(X) the space of measurable functions from X to ℂ such that there exist a number M > 0 satisfying |f(x)| ≤ M for almost all x ∈ X. Denote by L^{2}(X) the usual Hilbert space of (equivalence classes of) square integrable functions. For every f ∈ L^{∞}(X) one can define a multiplication operator M_{f} on L^{2}(X) in the usual way, and then we have a von Neumann algebra (this follows e.g. from [9], Theorem 7.5.3)

L^{∞}(X) = {M_{f} | f ∈ L^{∞}(X)}.

Definition 4.7. Let G, G_{1} and G_{2} be l.c. groups with fixed left invariant Haar measures and let a homomorphism i : G_{1} → G and an anti-homomorphism j : G_{2} → G have closed images and be homeomorphisms onto these images. Let finally

θ : G_{1} × G_{2} → Ω ⊂ G : (g, s) ↦ i(g)j(s)

be a homeomorphism of G_{1} × G_{2} onto an open subset Ω of G having a complement of measure zero. Then we call G_{1} and G_{2} a matched pair of l.c. groups.

From this data we will construct actions α and β and a map τ to obtain a cocycle matching of (L^{∞}(G_{1}), Δ_{1}) and (L^{∞}(G_{2}), Δ_{2}), with trivial cocycles, in the sense of Definition 2.1.

Denote by δ, δ_{1} and δ_{2} the modular functions of G, G_{1} and G_{2} and define the homeomorphism

ρ : G_{1} × G_{2} → Ω^{−1} : (g, s) ↦ j(s)i(g) .

We put

O = θ^{−1}(Ω ∩ Ω^{−1})  and  O′ = ρ^{−1}(Ω ∩ Ω^{−1}) .

So O and O′ are open subsets of G_{1} × G_{2} and ρ^{−1}θ is a homeomorphism of O onto O′. For all (g, s) ∈ O we define β_{s}(g) ∈ G_{1} and α_{g}(s) ∈ G_{2} such that

ρ^{−1}(θ(g, s)) = (β_{s}(g), α_{g}(s)) .

Hence we get j(α_{g}(s)) i(β_{s}(g)) = i(g)j(s) for all (g, s) ∈ O.

Let us first prove an easy lemma.

Lemma 4.8. Let (g, s) ∈ O and h ∈ G_{1}. Then (hg, s) ∈ O if and only if (h, α_{g}(s)) ∈ O, and in that case

α_{hg}(s) = α_{h}(α_{g}(s))  and  β_{s}(hg) = β_{α_{g}(s)}(h) β_{s}(g) .

Let (g, s) ∈ O and t ∈ G_{2}. Then (g, ts) ∈ O if and only if (β_{s}(g), t) ∈ O and in that case

β_{ts}(g) = β_{t}(β_{s}(g))  and  α_{g}(ts) = α_{β_{s}(g)}(t) α_{g}(s) .

Finally, for all g ∈ G_{1} and s ∈ G_{2} we have (g, e) ∈ O, (e, s) ∈ O, and

α_{g}(e) = e, α_{e}(s) = s, β_{s}(e) = e and β_{e}(g) = g .

Proof. Let (g, s) ∈ O and (h, α_{g}(s)) ∈ O. Then we know that

j(α_{h}(α_{g}(s))) i(β_{α_{g}(s)}(h)) = i(h) j(α_{g}(s)) .

So we get that

j(α_{h}(α_{g}(s))) i(β_{α_{g}(s)}(h)) i(β_{s}(g)) = i(h) j(α_{g}(s)) i(β_{s}(g)) = i(hg) j(s) .

From this we may conclude that (hg, s) ∈ O and β_{s}(hg) = β_{α_{g}(s)}(h) β_{s}(g), α_{hg}(s) = α_{h}(α_{g}(s)). The other statements of the lemma are proved analogously.

Lemma 4.9. The Haar measures on G, G_{1} and G_{2} can be normalized in such a way that for all positive Borel functions f on G we have

∫ f(i(g)j(s)) δ(j(s)) dg × ds = ∫ f(x) dx = ∫ f(j(s)i(g)) δ(i(g)) δ_{1}(g^{−1}) δ_{2}(s^{−1}) dg × ds .

Proof. For any f ∈ K(G_{1} × G_{2}) we define f̃ ∈ K(G_{1} × G_{2}) by f̃(g, s) = f(g, s)δ(j(s)^{−1}) and

I(f) = ∫_{Ω} f̃(θ^{−1}(x)) dx .

We claim that I is a left invariant integral on K(G_{1} × G_{2}). To prove this choose (h, t) ∈ G_{1} × G_{2}. Choose f ∈ K(G_{1} × G_{2}) and define F ∈ K(G_{1} × G_{2}) by F (g, s) = f(hg, ts). We have to prove that I(F) = I(f).

Choose x ∈ Ω. Let x = i(g)j(s). Then

F̃(θ^{−1}(x)) = F̃(g, s) = f(hg, ts)δ(j(s)^{−1}) = f̃(hg, ts)δ(j(t)) .

Because θ(hg, ts) = i(h)xj(t) we get that F̃(θ^{−1}(x)) = f̃(θ^{−1}(i(h)xj(t)))δ(j(t)) for all x ∈ Ω. Hence

I(F) = ∫_{Ω} F̃(θ^{−1}(x)) dx = ∫_{Ω} f̃(θ^{−1}(i(h)xj(t)))δ(j(t)) dx = ∫_{Ω} f̃(θ^{−1}(x)) dx = I(f) .

Hence it follows that I is a left invariant integral on K(G_{1} × G_{2}). Then we can normalize the Haar measure on G in such a way that the first equality of the lemma holds for all continuous functions on G with compact support in Ω. The same equality then follows for all positive Borel functions as usual. The second equality follows from the first and the formula

∫ f(x) dx = ∫ f(x^{−1})δ(x^{−1}) dx .

This lemma implies in particular that a Borel set A ⊂ G_{1} × G_{2} has measure zero if and only if θ(A) has measure zero in G. Because Ω ∩ Ω^{−1} has a complement of measure zero in G, O and O′ have complements of measure zero in G_{1} × G_{2}. Hence we can define a ∗-isomorphism τ by

τ : L^{∞}(G_{1}) ⊗ L^{∞}(G_{2}) → L^{∞}(G_{1}) ⊗ L^{∞}(G_{2}) : τ(f)(g, s) = f(β_{s}(g), α_{g}(s)) .  (4.1)

Lemma 4.8 guarantees that, defining

α : L^{∞}(G_{2}) → L^{∞}(G_{1}) ⊗ L^{∞}(G_{2}) : α(f) = τ(1 ⊗ f) and
β : L^{∞}(G_{1}) → L^{∞}(G_{1}) ⊗ L^{∞}(G_{2}) : β(f) = τ(f ⊗ 1)

we have

(ι ⊗ α)α(f) = (Δ_{1} ⊗ ι)α(f)  (β ⊗ ι)β(f) = (ι ⊗ Δ_{2}^{op})β(f)
τ_{13}(α ⊗ ι)Δ_{2}(f) = (ι ⊗ Δ_{2})α(f)  τ_{23}σ_{23}(β ⊗ ι)Δ_{1}(f) = (Δ_{1} ⊗ ι)β(f) .

Therefore, τ gives a cocycle matching with trivial cocycles of (L^{∞}(G_{1}), Δ_{1}) and (L^{∞}(G_{2}), Δ_{2}) in the sense of Definition 2.1. Then we can introduce cocycles by the following obvious lemma.

Lemma 4.10. Suppose that

U : G_{1} × G_{1} × G_{2} → ℂ  and  V : G_{1} × G_{2} × G_{2} → ℂ

are measurable maps with values in the unit circle U(1) ⊂ ℂ, satisfying

U(g, h, α_{k}(s)) U(gh, k, s) = U(h, k, s) U(g, hk, s),  (4.2)
V(β_{s}(g), t, r) V(g, s, rt) = V(g, s, t) V(g, ts, r),
V(gh, s, t) Ū(g, h, ts) = Ū(g, h, s) Ū(β_{α_{h}(s)}(g), β_{s}(h), t) V(g, α_{h}(s), α_{β_{s}(h)}(t)) V(h, s, t)

nearly everywhere. Then (τ, U, V) is a cocycle matching of (L^{∞}(G_{1}), Δ_{1}) and (L^{∞}(G_{2}), Δ_{2}).

Fixing cocycles U and V as above and using Definition 2.2 and Theorem 2.13 we get the cocycle bicrossed product (M, Δ), which is a l.c. quantum group.

Let φ_{2} be the n.s.f. weight on L^{∞}(G_{2}) coming from the Haar measure on G_{2} and φ the left invariant weight on (M, Δ) according to Definition 2.7. The obvious GNS-construction for φ_{2} gives also the canonical GNSconstruction (L^{2}(G_{1} × G_{2}), ι, Λ) for φ. Let W_{1} be the multiplicative unitary of G_{1} and W̃ = (W_{1} ⊗ 1)U^{∗}. Recall that M is generated by α(L^{∞}(G_{2})) and the elements (ω ⊗ ι ⊗ ι)(W̃), ω ∈ L^{∞}(G_{1})_{∗}.

For every function F ∈ K(G_{1}) we define the normal functional ω_{F} ∈ L^{∞}(G_{1})_{∗} by ω_{F} (g) = ∫ F (x)g(x) dx for all g ∈ L^{∞}(G_{1}). It follows from Definition 2.7 that

⟨(ω_{F} ⊗ ι ⊗ ι)(W̃)α(G) | F ∈ K(G_{1}), G ∈ K(G_{2})⟩  (4.3)

is a σ-strong^{∗}– norm core for Λ and

Λ((ω_{F} ⊗ ι ⊗ ι)(W̃)α(G)) = F ⊗ G .

Lemma 4.11. Let k ∈ G_{1}. Define

O_{k} = {s ∈ G_{2} | (k, s) ∈ O} .

Then O_{k} is an open subset of G_{2} with complement of measure zero and α_{k} is a homeomorphism of O_{k} onto O_{k^{−1}} satisfying, for all positive Borel functions h on G_{2},

∫_{O_{k}} h(α_{k}(s)) ds = ∫_{O_{k^{−1}}} h(s)F (k^{−1}, s) ds

Here F is the strictly positive continuous function on O defined by

F (k, s) = δ(i(β_{s}(k)^{−1})) δ_{1}(β_{s}(k)) δ_{2}(α_{k}(s)s^{−1}) .

Proof. Lemma 4.9 shows that ρ^{−1} preserves Borel sets of measure zero. Now i(k^{−1})Ω^{−1}∩Ω^{−1} has complement of measure zero in G and its image under ρ^{−1} is G_{1} ×O_{k}. Hence O_{k} is an open subset of G_{2} with complement of measure zero. Using twice Lemma 4.9, we get for every continuous function h on G with compact support in Ω^{−1} ∩ i(k)Ω^{−1}

∫ h(j(s)i(g)) δ(i(g)) δ_{1}(g^{−1}) δ_{2}(s^{−1}) dg × ds = ∫ h(x) dx = ∫ h(i(k)x) dx
= ∫_{G_{1}×O_{k}} h(i(k)j(s)i(g)) δ(i(g)) δ_{1}(g^{−1}) δ_{2}(s^{−1}) dg × ds
= ∫_{G_{1}×O_{k}} h(j(α_{k}(s))i(β_{s}(k)g)) δ(i(g)) δ_{1}(g^{−1}) δ_{2}(s^{−1}) dg × ds
= ∫_{G_{1}×O_{k}} h(j(α_{k}(s))i(g)) δ(i(β_{s}(k)^{−1}g)) δ_{1}(g^{−1}β_{s}(k)) δ_{2}(s^{−1}) dg × ds ,

where we have used twice the Fubini theorem and the invariance of the Haar measure on G_{2} to obtain the last equality. The Fubini theorem can be applied because the integrand has compact support in G_{1} × O_{k}.

Then the same equality is valid for all positive Borel functions h on G. Taking h of the form (h_{1} ⊗ h_{2})ρ^{−1} we get the result.

Lemma 4.12. Suppose that f is a bounded Borel function on G_{1} × G_{2} whose support has compact closure. Suppose that ξ ∈ K(G_{1} × G_{2}). Define, for nearly all (k, s) ∈ G_{1} × G_{2}

(π(f)ξ)(k, s) = ∫ Ū(g, g^{−1}k, s) f(g, α_{g^{−1}k}(s)) ξ(g^{−1}k, s) dg .

Then π(f) defines a bounded operator on L^{2}(G_{1} × G_{2}) belonging to N_{φ} ⊂ M and such that Λ(π(f)) = f.

Observe that π(f)ξ is a bounded Borel function having a support with compact closure.

Proof. If |f(g, s)| ≤ M for all g and s and f has a support in L_{1} × L_{2} for certain compact sets L_{1} and L_{2}, then π(f)ξ ∈ L^{2}(G_{1} × G_{2}) and ∥π(f)ξ∥ ≤ Mμ_{1}(L_{1})∥ξ∥, where μ_{1} is the Haar measure on G_{1}. Hence π(f) defines a bounded operator. If now f = F ⊗ G with F ∈ K(G_{1}) and G ∈ K(G_{2}), then

π(f) = (ω_{F} ⊗ ι ⊗ ι)(W̃)α(G)

and the result follows in this special case. When f is continuous on L_{1} × L_{2} and equals 0 elsewhere we can find a sequence f_{n} of linear combinations of functions F ⊗ G where F is continuous on L_{1} and G is continuous on L_{2}, converging uniformly to f. It is clear from the inequality above that π(f_{n}) converges in norm to π(f) and that f_{n} converges in L^{2}(G_{1} × G_{2}) to f. Hence π(f) ∈ N_{φ} and Λ(π(f)) = f, because Λ is closed. Finally, if f is Borel, its support is contained in L_{1} × L_{2} and if |f(g, s)| ≤ M for all g and s, we can use the Lusin theorem to obtain a sequence f_{n} of functions continuous on L_{1} × L_{2} and equal to 0 elsewhere, satisfying |f_{n}(g, s)| ≤ M for all g, s and n, and such that

(μ_{1} × μ_{2})({(g, s) | f_{n}(g, s) ≠ f(g, s)}) → 0,

where μ_{i} is the Haar measure on G_{i} (i = 1, 2). Using Lemma 4.11 one can check that π(f_{n}) → π(f) in σ-strong^{∗} topology and f_{n} → f in L^{2}(G_{1} × G_{2}). Because Λ is σ-strong^{∗}– norm closed, the result follows.

Lemma 4.13. There exist unique elements Ũ ∈ L^{∞}(G_{1}) ⊗ L^{∞}(G_{2}) such that, for almost all (g, k, s) ∈ G_{1} × G_{1} × G_{2},

(ι ⊗ α)(Ũ)(g, k, s) = U(g, g^{−1}k, s) U(g^{−1}, k, s)

and Ṽ ∈ L^{∞}(G_{1}) ⊗ L^{∞}(G_{2}) such that, for almost all (g, s, t) ∈ G_{1} × G_{2} × G_{2},

(β ⊗ ι)(Ṽ)(g, s, t) = V(g, s, t^{−1}) V(g, t^{−1}s, t).

Also we have Ũ(g, α_{g}(s)) = Ũ(g^{−1}, s) and Ṽ(β_{s}(g), s) = Ṽ(g, s^{−1}) for almost all (g, s) ∈ G_{1} × G_{2}.

Proof. Define a unitary Z ∈ L^{∞}(G_{1} × G_{1} × G_{2}), for almost all (g, k, s) ∈ G_{1} × G_{1} × G_{2}, by

Z(g, k, s) = U(g, g^{−1}k, s) U(g^{−1}, k, s).

We claim that Z ∈ L^{∞}(G_{1}) ⊗ α(L^{∞}(G_{2})). In view of Corollary 1.21 and Theorem 1.11 it suffices to prove that

(1 ⊗ V^{∗} ⊗ 1) (ι ⊗ ι ⊗ α)(Z) (1 ⊗ V ⊗ 1) = Z_{124}

which amounts to, for almost all (g, h, k, s),

Z(g, h, α_{k}(s)) = Z(g, hk, s).

But applying (4.2) to (g, g^{−1}h, k, s) and to (g^{−1}, h, k, s) we obtain this last formula.

Now we can take Ũ = (ι ⊗ α^{−1})(Z). The second statement can be proved similarly. Finally, to prove that Ũ(g, α_{g}(s)) = Ũ(g^{−1}, s) for almost all (g, s) it suffices to check that Ũ(g, α_{gh}(s)) = Ũ(g^{−1}, α_{h}(s)) for almost all (g, h, s), which is clear. Similarly we get Ṽ(β_{s}(g), s) = Ṽ(g, s^{−1}) for almost all (g, s).

Then we can finally prove the main result.

Proposition 4.14. Denote by X = J∇^{1/2} the modular operator of the left invariant weight φ on the l.c. quantum group (M, Δ) with the GNS-construction (L^{2}(G_{1} × G_{2}), ι, Λ) and by X̂ = Ĵˆ∇^{1/2} the modular operator of the left invariant weight φ̂ on the the dual l.c. quantum group (M̂, Δ̂) with the GNS-construction (L^{2}(G_{1} × G_{2}), ι, Λ̂). Then, for every bounded measurable function ξ on G whose support is contained in a compact subset of O, we get that ξ ∈ D(X), ξ ∈ D(X̂), and

(Xξ)(g, s) = Ũ(g^{−1}, s) δ_{1}(g^{−1}) ξ̄(g^{−1}, α_{g}(s))
(X̂ξ)(g, s) = Ṽ(g, s^{−1}) δ_{2}(s^{−1}) ξ̄(β_{s}(g), s^{−1})
(Jξ)(g, s) = δ(i(β_{s}(g)))^{−1/2} δ_{1}(β_{s}(g)g^{−1})^{1/2} δ_{2}(α_{g}(s)s^{−1})^{1/2} Ũ(g^{−1}, s) ξ̄(g^{−1}, α_{g}(s))
(Ĵξ)(g, s) = δ(j(α_{g}(s)))^{1/2} δ_{1}(β_{s}(g)g^{−1})^{1/2} δ_{2}(α_{g}(s)s^{−1})^{1/2} Ṽ(g, s^{−1}) ξ̄(β_{s}(g), s^{−1}) .

Moreover these elements ξ form a core for both X and X̂.

The operators ∇ and ˆ∇ are strictly positive multiplication operators with the functions

∇(g, s) = δ(i(β_{s}(g)))^{−1} δ_{1}(β_{s}(g)g) δ_{2}(α_{g}(s)s^{−1})
ˆ∇(g, s) = δ(j(α_{g}(s))) δ_{1}(β_{s}(g)g^{−1}) δ_{2}(α_{g}(s)s) .

Proof. Let F be the set of bounded Borel functions on G_{1} × G_{2} whose support is contained in a compact subset of O and let f ∈ F. Using Lemmas 4.12 and 4.13 one can check that for all ξ ∈ K(G_{1} × G_{2})

(π(f)^{∗}ξ)(k, s) = ∫ U(g, k, s) f̄(g, α_{k}(s)) ξ(gk, s) dg
= ∫ Ū(g, g^{−1}k, s) δ_{1}(g^{−1}) U(g, g^{−1}k, s) U(g^{−1}, k, s) f̄(g^{−1}, α_{k}(s)) ξ(g^{−1}k, s) dg
= (π(h)ξ)(k, s)

where h(g, s) = δ_{1}(g^{−1}) Ũ(g^{−1}, s) f̄(g^{−1}, α_{g}(s)). Because (g, s) ↦ (g^{−1}, α_{g}(s)) is a homeomorphism of O we get h ∈ F. Then it follows that f ∈ D(X) and Xf = h. When both f_{1} and f_{2} belong to F, one can check that π(f_{1})π(f_{2}) = π(f_{3}) where

f_{3}(h, s) = ∫ Ū(g, g^{−1}h, s) f_{1}(g, α_{g^{−1}h}(s)) f_{2}(g^{−1}h, s) dg .

We see that f_{3} ∈ F. From Definition 2.7 and Lemma 4.12 it follows that {π(f) | f ∈ F} is a σ-strong^{∗}– norm core for Λ. The computations above show that it is also a ∗-algebra, so F ⊂ L^{2}(G_{1} × G_{2}) is a core for X. Let now f and h be in F. Then we get, using Lemma 4.11 and the notation F introduced there,

⟨Xf, h⟩ = ∫ δ_{1}(g^{−1}) Ũ(g^{−1}, s) f̄(g^{−1}, α_{g}(s)) h̄(g, s) dg × ds
= ∫ (∫ δ_{1}(g^{−1}) Ũ(g^{−1}, s) f̄(g^{−1}, α_{g}(s)) h̄(g, s) ds) dg
= ∫ (∫ F (g^{−1}, s) δ_{1}(g^{−1}) Ũ(g^{−1}, α_{g^{−1}}(s)) f̄(g^{−1}, s) h̄(g, α_{g^{−1}}(s)) ds) dg
= ∫ F (g, s) Ũ(g^{−1}, s) f̄(g, s) h̄(g^{−1}, α_{g}(s)) dg × ds .

In this computation we used twice the Fubini theorem on a bounded Borel function whose support has compact closure. Since the functions f give a core for X, the computation above shows that h ∈ D(X^{∗}) and

(X^{∗}h)(g, s) = Ũ(g^{−1}, s) F (g, s) h̄(g^{−1}, α_{g}(s)) .

Then for all f ∈ F we get f ∈ D(X^{∗}X) and, with ∇(g, s) from the statement of the proposition,

(X^{∗}Xf)(g, s) = ∇(g, s) f(g, s).

Denote by A the strictly positive multiplication operator with the function (g, s) ↦ ∇(g, s). Because X^{∗}X = ∇ we see that Af = ∇f for all f ∈ F. Because the subspace F of L^{2}(G_{1} × G_{2}) is invariant under A^{it} for all t ∈ ℝ, it is a core for A. Hence it follows that A ⊂ ∇, and so A = ∇ because both are self-adjoint. From the formulas for X and ∇ we get the formula for J.

Next Proposition 2.9 and Proposition 2.16 give that (M̂, Δ̂) also is a cocycle bicrossed product. If we define

ĩ: G_{2} → G : ĩ(s) = j(s)^{−1}  and j̃ : G_{1} → G : j̃(g) = i(g)^{−1}

then G_{2} and G_{1} become a matched pair of groups and the maps α̃ and β̃ are given by

α̃_{s}(g) = β_{s}(g)  and  β̃_{g}(s) = α_{g}(s) .

The maps (s, t, g) ↦ V(g, t, s) and (s, g, h) ↦ U(h, g, s) satisfy (4.2). Hence we can make the cocycle bicrossed product, and Proposition 2.9 shows that this equals, up to a flip map, to (M̂, Δ̂). Then the formulas for X̂, Ĵ and ˆ∇ follow from the formulas for X, J and ∇.

Now one could write explicit formulas for τ_{t}(π(f)) and R(π(f)) with a bounded Borel function f whose support has compact closure. Let us give concrete formulas for the remaining operators related to (M, Δ) and its dual.

Proposition 4.15. Let P be the unitary implementation of τ [24], Definition 6.9, δ_{M} the modular element of (M, Δ), δM̂ the modular element of (M̂, Δ̂). Then P is the multiplication operator with the function

P (g, s) = δ(i(gβ_{s}(g)^{−1})) δ_{1}(g^{−1}β_{s}(g)) δ_{2}(s^{−1}α_{g}(s)) ,

δ_{M} is the multiplication operator with the function

δ_{M}(g, s) = δ(j(α_{g}(s)))^{−1} δ_{2}(α_{g}(s))^{−2}

and δM̂ is the multiplication operator with the function

δM̂(g, s) = δ(i(β_{s}(g))) δ_{1}(β_{s}(g))^{−2} .

Proof. Using [24], Definition 6.9 and Proposition 8.23 we get that τ_{t}(x) = P^{it}xP^{−it} and τ̂_{t}(y) = P^{it}yP^{−it} for all t ∈ ℝ, x ∈ M and y ∈ M̂. From [25], Proposition 2.13 it follows that JP J = P^{−1}. If A is the multiplication operator with the function (g, s) ↦ P (g, s) defined in the proposition, one can compute that the closure of A∇^{−1} is the multiplication operator with the function

(g, s) ↦ δ(i(g)) δ_{1}(g)^{−2}

which commutes with M̂. Hence A^{it}yA^{−it} = τ̂_{t}(y) for all t ∈ ℝ and y ∈ M̂. Similarly we prove that A^{it} implements τ_{t} for all t ∈ ℝ. Because M and M̂ generate B(L^{2}(G_{1} × G_{2})), the operators A and P are proportional. Because JAJ = A^{−1}, it follows that A = P.

From [24], Lemma 8.8 it follows that ˆ∇ is the closure of P Jδ_{M}^{−1}J, from where the formula for δ_{M} follows. The formula for δM̂ can be obtained in the same way.

Lemma 4.11 shows that the transformation α_{g} of G_{2} preserves Borel sets of measure zero, and we get the following concrete formula for the Radon-Nikodym derivative.

\frac{dα_{g}(s)}{ds} = δ(i(β_{s}(g)^{−1})) δ_{1}(β_{s}(g)) δ_{2}(α_{g}(s)s^{−1}) .

Following [28] we use the notation

χ(g, s) := \frac{dα_{g}(s)}{ds}  and  Ψ(s, g) := \frac{dβ_{s}(g)}{dg} .

Then we see that

\frac{χ(g, s)}{χ(g, e)} = δ(i(gβ_{s}(g)^{−1})) δ_{1}(g^{−1}β_{s}(g)) δ_{2}(α_{g}(s)s^{−1}) .  (4.4)

By symmetry, we obtain that

\frac{Ψ(s, g)}{Ψ(s, e)} = δ(j(s^{−1}α_{g}(s))) δ_{1}(g^{−1}β_{s}(g)) δ_{2}(α_{g}(s)s^{−1}) .  (4.5)

Because i(g)j(s) = j(α_{g}(s)) i(β_{s}(g)) we get that

δ(i(gβ_{s}(g)^{−1})) = δ(j(s^{−1}α_{g}(s))) .

From this it follows that

\frac{χ(g, s)}{χ(g, e)} = \frac{Ψ(s, g)}{Ψ(s, e)}

and using the S. Majid’s notation, we put

ξ(g, s) := \frac{χ(g, s)}{χ(g, e)} .

The following proposition characterizes all cocycle bicrossed products of l.c. groups which are Kac algebras. Its final statement generalizes [28], Theorem 2.12.

Proposition 4.16. The scaling group τ of the l.c. quantum group (M, Δ) is trivial if and only if

ξ(g, s) = 1 for all (g, s) ∈ O .

The modular element δ_{M} is affiliated to the centre of M if and only if

δ(j(s^{−1}α_{g}(s))) δ_{2}(s^{−1}α_{g}(s))^{2} = 1 for all (g, s) ∈ O .

The l.c. quantum group (M, Δ) is a Kac algebra if and only if

ξ(g, s) = 1 and \frac{δ_{1}(β_{s}(g))}{δ_{1}(g)} = \frac{δ_{2}(α_{g}(s))}{δ_{2}(s)} .

Proof. τ is trivial if and only if P = 1 or if and only if ξ(g, s) = 1 almost everywhere. The continuity of ξ on O gives the first statement.

Proposition 4.15 shows that δ_{M} = α(H) for the function H on G_{2} defined by H(s) = δ(j(s))^{−1} δ_{2}(s)^{−2}. Then δ_{M} is affiliated to the center of M if and only if α(H) = 1 ⊗ H (Proposition 1.6), and the second statement follows.

To prove the final statement, observe that (M, Δ) is a Kac algebra if and only if τ is trivial and δ_{M} is affiliated to the center of M. If (M, Δ) is a Kac algebra, then by the previous statements, ξ(g, s) = 1 for all (g, s) ∈ O and

δ(j(s^{−1}α_{g}(s))) δ_{2}(s^{−1}α_{g}(s))^{2} = 1 for all (g, s) ∈ O .

Dividing this equation by the formula for ξ = 1 (4.5) we get

δ_{1}(gβ_{s}(g)^{−1}) δ_{2}(s^{−1}α_{g}(s)) = 1 for all (g, s) ∈ O .

Rewriting this we get the equality

\frac{δ_{1}(β_{s}(g))}{δ_{1}(g)} = \frac{δ_{2}(α_{g}(s))}{δ_{2}(s)} .

Clearly we can also go the other way around to complete the proof.

Corollary 4.17. If α or β is trivial, (M, Δ) and (M̂, Δ̂) are Kac algebras.

Proof. Let us suppose that α is trivial. Then for all (g, s) ∈ O we have j(s)i(β_{s}(g)) = i(g)j(s). Hence we get that δ(i(g^{−1}β_{s}(g))) = 1 for all (g, s) ∈ O. Because α is trivial, we also have

\frac{dα_{g}(s)}{ds} = 1 for all (g, s) ∈ O .

Applying this equation to (g, s) and (g, e) we get ξ(g, s) = 1 for all (g, s) ∈ O. Combining (4.4) with the equality δ(i(g^{−1}β_{s}(g))) = 1 we get that δ_{1}(g^{−1}β_{s}(g)) = 1 for all (g, s) ∈ O. Because α is trivial this gives the equation

\frac{δ_{1}(β_{s}(g))}{δ_{1}(g)} = \frac{δ_{2}(α_{g}(s))}{δ_{2}(s)} .

Using the previous proposition we obtain that (M, Δ) is a Kac algebra.

Corollary 4.18. If both α and β preserve modular functions and Haar measures, then (M, Δ) and (M̂, Δ̂) are Kac algebras.

Proof. Because α preserves the Haar measure of G_{2} we get ξ(g, s) = 1 for all (g, s) ∈ O. Because α and β preserve the modular functions, we get

\frac{δ_{1}(β_{s}(g))}{δ_{1}(g)} = 1 = \frac{δ_{2}(α_{g}(s))}{δ_{2}(s)}

for all (g, s) ∈ O. Then we can apply Proposition 4.16.

Remark that the conditions of this corollary are fulfilled if both groups are discrete Indeed, any discrete group is unimodular and the Haar measure is constant at arbitrary point of such a group.

Corollary 4.19. If (G_{1}, G_{2}) is a fixed matched pair of groups and cocycles U and V satisfy (4.2), we get a cocycle bicrossed product (M, Δ). If one of these cocycle bicrossed products is a Kac algebra, then all of them are Kac algebras.

Proof. The necessary and sufficient conditions for (M, Δ) to be a Kac algebra of Proposition 4.16 are independent of U and V.

4.3 The group of extensions

If (G_{1}, G_{2}) is a fixed matched pair of l.c. groups, U and V are measurable functions satisfying (4.2), we get by the previous subsection a cocycle matching (τ, U, V) of (L^{∞}(G_{1}), Δ_{1}) and (L^{∞}(G_{2}), Δ_{2}), and hence, by Proposition 3.4, an extension

(L^{∞}(G_{2}), Δ_{2}) ⟶^{α} (M, Δ) ⟶^{β} (L(G_{1}), Δ̂_{1}).

We will say that such an extension (M, Δ) is associated with the matched pair (G_{1}, G_{2}). Proposition 3.7 and the remark following its statement show that the extensions given by two cocycle matchings (τ, U_{a}, V_{a}) and (τ, U_{b}, V_{b}), are isomorphic if and only if there exists a measurable map R from G_{1} × G_{2} to the unit circle in ℂ, satisfying

U_{b}(g, h, s) = U_{a}(g, h, s) R(h, s) R(g, α_{h}(s)) R̄(gh, s)
V_{b}(g, s, t) = V_{a}(g, s, t) R(g, s) R(β_{s}(g), t) R̄(g, ts)

almost everywhere. If this is the case, the pairs (U_{a}, V_{a}) and (U_{b}, V_{b}) will be called cohomologous. Then the set of equivalence classes of cohomologous pairs of cocycles (U, V) satisfying (4.2), exactly corresponds to the set Γ of classes of isomorphic extensions associated with (G_{1}, G_{2}).

The set Γ can be given the structure of an abelian group by defining

π(U_{a}, V_{a}) · π(U_{b}, V_{b}) = π(U_{a}U_{b}, V_{a}V_{b})

where π(U, V) denotes the equivalence class containing the pair (U, V).

Terminology 4.20. The group Γ is called the group of extensions of (L^{∞}(G_{2}), Δ_{2}) by (L(G_{1}), Δ̂_{1}) associated with the matched pair of l.c. groups (G_{1}, G_{2}). The unit of this group corresponds to the equivalence class of cocycles cohomologous to trivial. The corresponding extension is called split extension; all other extensions are called non-trivial extensions.

4.4 Cocycles in the sense of S. Baaj and G. Skandalis

Consider the split extension (M_{0}, Δ_{0}) of (L^{∞}(G_{2}), Δ_{2}) by (L(G_{1}), Δ̂_{1}) associated with the matched pair of l.c. groups (G_{1}, G_{2}). The associated (dual) multiplicative unitary, acting on L^{2}(G_{1} × G_{2} × G_{1} × G_{2}), is given by

Ŵ_{0} = ((α ⊗ ι)(W_{1}) ⊗ 1) (1 ⊗ (ι ⊗ β)(Ŵ_{2})) .

Similarly to [3], Définition 8.24 we call a function Θ : G_{1} × G_{2} × G_{1} × G_{2} → U(1) a Ŵ_{0}-cocycle, if Θ is measurable and the unitary ΘŴ_{0} is multiplicative. Here we consider Θ as a unitary in L^{∞}(G_{1}×G_{2}×G_{1}×G_{2}).

Let us discuss the link between Ŵ_{0}-cocycles Θ and cocycles U and V. Let first U and V be cocycles in the sense of Lemma 4.10. Then (τ, U, V) is a cocycle matching of (L^{∞}(G_{1}), Δ_{1}) and (L^{∞}(G_{2}), Δ_{2}) and hence we obtain the multiplicative unitary

Ŵ = (α ⊗ ι ⊗ ι)((W_{1} ⊗ 1)U^{∗}) (ι ⊗ ι ⊗ β)(V(1 ⊗ Ŵ_{2})) .

One can check that for ξ ∈ L^{2}(G_{1} × G_{2} × G_{1} × G_{2}) we have

(Ŵ_{0}ξ)(g, s, h, t) = ξ(g, α_{β_{s}(g)^{−1}h}(t)s, β_{s}(g)^{−1}h, t)
(Ŵ ξ)(g, s, h, t) = Ū(β_{s}(g), β_{s}(g)^{−1}h, t) V(g, s, α_{β_{s}(g)^{−1}h}(t)) ξ(g, α_{β_{s}(g)^{−1}h}(t)s, β_{s}(g)^{−1}h, t)

nearly everywhere. Hence, defining

Θ(g, s, h, t) = Ū(β_{s}(g), β_{s}(g)^{−1}h, t) V(g, s, α_{β_{s}(g)^{−1}h}(t))  (4.6)

we get that Θ is a Ŵ_{0}-cocycle.

Vice versa, if Θ is a Ŵ_{0}-cocycle, the multiplicativity of ΘŴ_{0} amounts, almost everywhere, to the equation

Θ(g, s, h, t) Θ(g, α_{g^{−1}}(α_{h}(t)α_{g}(s)), k, r) Θ(β_{s}(g)^{−1}h, t, β_{α_{h}(t)α_{g}(s)}(g^{−1})k, r)

= Θ(h, t, k, r) Θ(g, s, h, α_{β_{t}(h)^{−1}k}(r)t). (4.7)

If we know that the previous equality holds whenever its arguments make sense (by this we mean whenever (h, t), (g, s), (g^{−1}, α_{h}(t)α_{g}(s)), … ∈ O), then we can define

U(g, h, t) = Θ̄(g, e, gh, t) Θ(g, e, g, e)  and  V(g, s, t) = Θ(g, s, β_{s}(g), t) Θ̄(e, e, e, t) .

Some computations reveal that U and V are cocycles, i.e. they satisfy (4.2), and Θ is given by equation (4.6). We cannot prove the same result when we only know that (4.7) holds almost everywhere. Nevertheless the formulas for the cocycles U and V given above are sufficient to find these cocycles starting from a Ŵ_{0}-cocycle Θ. If both G_{1} and G_{2} are discrete, the extra assumption is redundant because then ’nearly everywhere’ means everywhere.

5 Examples of cocycle bicrossed products

5.1 A brief survey of known examples

1. The Kac algebra case: application of Corollary 4.17.

Let l.c. quantum groups G_{1}, G_{2} form a matched pair and let, say, the action β be trivial. Then G_{2} is a normal subgroup of G (see the proof of Corollary 4.17) and G_{1} acts on G_{2} by (inner in G) automorphisms (this is seen from Lemma 4.8). Now, as it was explained in [28], the corresponding split extension is nothing but a semidirect product of the commutative Kac algebra L^{∞}(G_{2}) with the group G_{1} acting on it (the construction of a semidirect product of a Kac algebra with a l.c. group is due to J. De Canniere [10]). In the above situation, there is also the induced action of G_{1} on L(G_{2}), and the dual Kac algebra to the above split extension can be again computed following J. De Canniere (for the detailed explanation see [28]).

Turning now to cocycles, the second equation (4.2) shows that V(g, t, s) can be viewed as a function η^{g}(t, s) on G_{1} with values in the group Z_{2}(G_{2}; U(1)) of U(1)-valued 2-cocycles on G_{2}. Let us try to find a nontrivial solution for (4.2) under the hypothesis that U(·) = 1. Then the third equation (4.2) gives

η^{gh}(t, s) = h · η^{g}(t, s) η^{h}(t, s),

where h · η^{g}(t, s) := η^{g}(α_{h}(t), α_{h}(s)), which means, by definition, that g ↦ η^{g} is an equivariant map from G_{1}^{op} to Z_{2}(G_{2}; U(1)).

Such an equivariant map can be found at least in some special cases when both G_{1} and G_{2} are either finite abelian groups or finite-dimensional real vector spaces. Indeed, in [3], 8.26 the Kac-Paljutkin example [20], where G_{1} = ℤ/2ℤ acts on G_{2} = ℤ/2ℤ × ℤ/2ℤ by permutations, has been treated this way. It turns out that this way we find the unique nontrivial extension, so the group of extensions here is ℤ/2ℤ ([18],[32]). More generally, if G_{1} = ℤ/2ℤ acts on G_{2} = ℤ/mℤ × ℤ/mℤ (m ≥ 2 is natural) by permutations, the group of extensions is ℤ/mℤ ([18], [32]), concrete examples of nontrivial extensions can be found in [32], [41], [48]. Let us mention some other matched pairs of finite groups that fit to this framework and for which the group of extensions is nontrivial: a) G_{1} = ℤ/mℤ (m ≥ 2 is natural) acts on G_{2} = ℤ/mℤ × ℤ/mℤ as follows: x · a^{i}b^{j} = a^{i+j}b^{j} (x, a, b are corresponding generators, i, j = 0, 1, ..., m − 1) (see [32]); b) G_{1} = ℤ/2ℤ acts on G_{2} = ℤ/2mℤ (m ≥ 2 is natural) by inversion, then G is the dihedral group (see [36], [48]).

In [3], 8.26 for the other Kac-Paljutkin example [21] with G_{1} = ℝ acting on ℝ^{2} by α_{g}(x) = exp(gK)(x) (where K is a real 2 × 2 matrix) the above equivariant map g ↦ η^{g} was computed as well as for the Rieffel’s example [39], where G_{1} = ℝ^{2} acts on G_{2} = ℝ^{3} by (a, b) · (x, y, z) = (x + ay + bz, y, z). So, in both cases the group of extensions is nontrivial. The same is true for some other matched pairs where both G_{1} and G_{2} are finite-dimensional real vector spaces: a) G_{1} = ℝ^{n} acting on G_{2} = ℝ^{n+1} by α_{a}(b, t) = (b, t + ⟨a, b⟩), where a, b ∈ ℝ^{n}, t ∈ ℝ, n ≥ 2 is natural (see[15],[48]); b) G_{1} = ℝ^{n} acting on G_{2} = ℝ^{m} by certain family of linear operators [49].

Another situation when nontrivial solutions of (4.2) can be easily found, is the one with both trivial actions. Then U(·) can be choosen as a multiplicative function ζ : G_{2} → Z_{2}(G_{1}; U(1)) (i.e., ζ^{ts} = ζ^{t}ζ^{s}) and V(·) as a multiplicative function η : G_{1} → Z_{2}(G_{2}; U(1)) (i.e., η^{gh} = η^{g}η^{h}). If both G_{1} and G_{2} are finite, the whole group of extensions has been computed in [18], 3.5.

2. The Kac algebra case: application of Corollary 4.18.

a) Let G = S_{n} be the symmetric group of order n ≥ 4, G_{1} = S_{n−1} its subgroup which fixes the point n and G_{2} = ℤ/nℤ the subgroup generated by the cyclic permutation. Then it is shown in [3], 8.23b that G_{1}, G_{2} form a matched pair with two nontrivial actions (for instance, the action of s ∈ G_{1} on a^{i} ∈ G_{2} is given by α_{s}(a^{i}) = a^{s(i)}, i = 0, 1, ..., n − 1). The corresponding group of extensions is trivial [32], 4.1, and the split extension is clearly a Kac algebra.

b) Following [30], 6.2.16, consider the pair G_{1} = T_{+}(n, ℝ), G_{2} = T_{−}(n, ℝ), formed by groups of upper (resp., lower) triangular matrices over ℝ with 1 on the diagonal and equipped with the actions

α_{g}(s) = 1 + (s − 1)g^{T}, β_{s}(g) = (1 + θ(s)(g^{−1} − 1))^{−1},

where θ is the composition of inversion and matrix transposition and ^{T} is matrix transposition. These formulas slightly differ from [30], 6.2.16 because there another convention on a matched pair is adopted. Being connected nilpotent groups, both G_{1}, G_{2} are unimodular [16], I.1.6 (morover, one can check that their Haar measures are in fact the product measures of Lebesgue measures of all non diagonal entries g_{ij}), so the actions preserve modular functions. Then one can check that for i > j we have

(α_{g}(s))_{ij} = s_{ij} + ∑_{j<k<i} s_{ik}θ(g)_{kj},

from where it is clear that the transformation α_{g} preserves the Haar measure of G_{2}. So the corresponding split extension is a Kac algebra because of Corollary 4.18. We do not know if other extensions exist in this situation, but if they exist, they all are Kac algebras (see Corollary 4.19).

3. The S. Majid’s example [27]

Using the Iwasawa decomposition of a complex semisimple Lie group G = KAN [16], I.5.1, one can take G_{1} = K, G_{2} = AN, where G_{1} and G_{2} are respectively the maximal compact and the maximal solvable subgroups of G. For instance, for SL(2, ℂ) viewed as a real Lie group we have G_{1} = SU(2, ℂ) (g ∈ G_{1} is defined by a pair x, y ∈ ℂ such that |x|^{2} + |y|^{2} = 1) and G_{2} = {s ∈ ℝ^{3}|s_{3} > −1} with the operation st = s + (s_{3} + 1)t (see [30], 8.3.13). Here and in what follows s_{i} and e_{i} denote the i − th coordinate of the vector s and i-th coordinate vector in ℝ^{3}, i = 1, 2, 3. Then the mutual actions of G_{1}, G_{2} are

α_{g}(s) = Rot_{g}(s − \frac{|s|^{2}}{2(s_{3} + 1)}e_{3}) + \frac{|s|^{2}}{2(s_{3} + 1)}e_{3},
β_{s}(x, y) = \frac{(x + (s_{1} − is_{2})y, y)}{\sqrt{|x + (s_{1} + is_{2})y|^{2} + |y|^{2}}}

(see [30], 8.3.18), where Rot is the usual 3-vector rotation associated to an element of SU(2, ℂ). Now we are able to construct the corresponding split extension, and to decide if it is a Kac algebra, using Proposition 4.16. One can compute (see [27]) that the function ξ(g, s) from this Proposition differs from 1, so the split extension is not a Kac algebra. Again, we do not know if there exist other extensions, but if they exist, they all are not Kac algebras (see Corollary 4.19).

5.2 On infinitesimal objects of l.c. quantum groups

In order to give concrete examples of cocycle bicrossed products we have to take concrete matched pairs of groups (G_{1}, G_{2}) and to solve the equations (4.2) to find cocycles U and V. Then one can perform the cocycle bicrossed product construction to get a l.c. quantum group.

If G_{1} and G_{2} are Lie groups, we also want to describe, at least formally, a Hopf algebra consisting of generators of the l.c. quantum group coming from a matched pair (G_{1}, G_{2}) with cocycles U and V. This infinitesimal Hopf algebra is, in fact, an algebraic cocycle bicrossed product Hopf algebra.

We assume that the cocycles U and V satisfy (4.2) and we require additionally U(g, e, s), U(e, g, s) and V(e, s, r) to be well defined measurable functions in s ∈ G_{2} for all g ∈ G_{1} (resp., in s, r ∈ G_{2}). These extra requirements will always be fulfilled in the concrete examples that we study below. Then one can choose in the class of cohomologous cocycles such representatives that U(e, e, s) = 1 and, taking in the first equation (4.2) subsequently h = k = e and g = h = e, we get U(g, e, s) = U(e, g, s) = 1. Now taking in the third equation (4.2) g = h = e, we get V(e, s, r) = 1.

Let Δ_{1} be the coproduct on L^{∞}(G_{1}), Δ_{2} the coproduct on L^{∞}(G_{2}) and (M, Δ) the cocycle bicrossed product. Let 𝔤_{1} be the Lie algebra of G_{1} and H its universal enveloping algebra which is a Hopf algebra with the usual symmetric coproduct

Δ_{H}(X) = X ⊗ 1 + 1 ⊗ X (∀X ∈ 𝔤_{1}).

We consider H as acting on smooth functions on G_{1} by left invariant differential operators (see [16], II.4), i.e.,

H_{g}[A] = H_{e}[A(g·)]

for all g ∈ G_{1}, H ∈ H and all smooth functions A on G_{1}. In the Sweedler notation, it is clear that, for H, G ∈ H and A a smooth function on G_{1}, we have

∑ H_{(1) e}[A] H_{(2) e}[B] = H_{e}[AB]

and

(HG)_{e}[A] = H_{e}[g ↦ G_{g}[A]] = H_{e}[g ↦ G_{e}[Δ_{1}(A)(g, ·)]] = (H_{e} ⊗ G_{e})[Δ_{1}(A)] .

To make the link with the von Neumann algebra setting, we define formally, for H ∈ H, the unbounded functional ω_{H} on L^{∞}(G_{1}) by

ω_{H}(A) = H_{e}[A] .

By the previous computation we get, for all H, G ∈ H and a, b ∈ L^{∞}(G_{1}),

(ω_{H} ⊗ ω_{G})Δ_{1} = ω_{HG} and ω_{H}(ab) = ∑ ω_{H_{(1)}}(a) ω_{H_{(2)}}(b).

Next suppose that we have some natural algebra A of functions on G_{2} such that (A, Δ_{2}) is a Hopf algebra and compute how to match these Hopf algebras H and A, in the sense of [30, Section 6.3]. Then we construct their algebraic cocycle bicrossed product in the sense of S. Majid.

An arbitrary element H⊗A of the algebraic tensor product H⊗A can be considered formally as an unbounded operator affiliated to the von Neumann algebra M by the identification

H ⊗ A ⟷ (ω_{H} ⊗ ι ⊗ ι)(W̃) α(A)

where, as before, W̃ = (W_{1} ⊗ 1)U^{∗}.

In order to compute the algebra structure that M brings into the vector space H ⊗ A, let us observe that

(Δ_{1} ⊗ ι ⊗ ι)(W̃) (ι ⊗ ι ⊗ α)(U^{∗}) = W̃_{134} W̃_{234} .  (5.1)

This can be verified by the following computation.

(Δ_{1} ⊗ ι ⊗ ι)(W̃) (ι ⊗ ι ⊗ α)(U^{∗}) = (Δ_{1} ⊗ ι ⊗ ι)(W_{1,12}U^{∗}) (ι ⊗ ι ⊗ α)(U^{∗})

= W_{1,13}W_{1,23} (ι ⊗ Δ_{1} ⊗ ι)(U^{∗}) U_{234}^{∗}

= W_{1,13}U_{134}^{∗} W_{1,23}U_{234}^{∗} = W̃_{134} W̃_{234} where we used the cocycle equation for U. Taking H, G ∈ H and A, B ∈ A we get

(ω_{H} ⊗ ι ⊗ ι)(W̃) α(A) (ω_{G} ⊗ ι ⊗ ι)(W̃) α(B) = (ω_{H} ⊗ ω_{G} ⊗ ι ⊗ ι)(W̃_{134} α(A)_{34} W̃_{234}) α(B)
= (ω_{H} ⊗ ω_{G} ⊗ ι ⊗ ι)(W̃_{134} W̃_{234} (ι ⊗ α)α(A)_{234}) α(B)
= (ω_{H} ⊗ ω_{G} ⊗ ι ⊗ ι)((Δ_{1} ⊗ ι ⊗ ι)(W̃) (ι ⊗ ι ⊗ α)(U^{∗}α(A)_{23})) α(B)
= ∑((ω_{H_{(1)}} ⊗ ω_{G_{(1)}})Δ_{1} ⊗ ι ⊗ ι)(W̃) α((ω_{H_{(2)}} ⊗ ω_{G_{(2)}} ⊗ ι)(U^{∗}) (ω_{G_{(3)}} ⊗ ι)α(A) B)

where we used that α is an action, and equation (5.1). Identifying again with H ⊗ A, we get

(H ⊗ A)(G ⊗ B) = ∑ H_{(1)}G_{(1)} ⊗ χ(H_{(2)} ⊗ G_{(2)}) (G_{(3)} ▷ A) B  (5.2)

where

▷ : H ⊗ A → A : H ▷ A = (ω_{H} ⊗ ι)α(A)
χ : H ⊗ H → A : χ(H ⊗ G) = (ω_{H} ⊗ ω_{G} ⊗ ι)(U^{∗}) .

More specifically, we get that (H ▷ A)(r) = H_{e}[A(α_{···} (r))] and χ(H ⊗ G)(r) = (H_{e} ⊗ G_{e})[U^{∗}(·, ·, r)] for all r ∈ G_{2}. We see that, up to some left/right conventions, the algebra A becomes this way a cocycle H-module algebra, and that H ⊗ A is the cocycle cross product algebra (compare (5.2) with the formulas appearing in [30], Proposition 6.3.7). We can go further and describe generators and relations for the algebra H ⊗ A.

One can check that for X, Y ∈ 𝔤_{1} we have

(X ⊗ 1)(Y ⊗ 1) = XY ⊗ 1 + 1 ⊗ χ(X ⊗ Y)

and for X ∈ 𝔤_{1} and A ∈ A we have

(X ⊗ 1)(1 ⊗ A) = X ⊗ A and (1 ⊗ A)(X ⊗ 1) = X ⊗ A + 1 ⊗ (X ▷ A) .

For this we use the relations χ(X ⊗ 1) = χ(1 ⊗ X) = 0 for all X ∈ 𝔤_{1} and χ(1 ⊗ 1) = 1 which follow from U(g, e, s) = U(e, g, s) = 1 for all g ∈ G_{1} and s ∈ G_{2}. So the algebra H ⊗ A is the algebra generated by

{A | A ∈ A} and {X | X ∈ 𝔤_{1}}

with commutation relations

[A, B] = 0 when A, B ∈ A  (5.3)
[A, X] = X ▷ A when X ∈ 𝔤_{1}, A ∈ A
[X, Y] = [X, Y]_{𝔤_{1}} + χ(X ⊗ Y − Y ⊗ X) when X, Y ∈ 𝔤_{1}

where [X, Y]_{𝔤_{1}} denotes the Lie bracket of the Lie algebra 𝔤_{1}.

Next we want to compute the coproduct on the algebra H ⊗ A inherited from the quantum group (M, Δ). Using Propositions 2.4 and 2.5 we get, for all H ∈ H and A ∈ A,

Δ^{op}((ω_{H} ⊗ ι ⊗ ι)(W̃) α(A))
= (ω_{H} ⊗ ι ⊗ ι ⊗ ι ⊗ ι)((W̃ ⊗ 1 ⊗ 1) ((ι ⊗ α)β ⊗ ι ⊗ ι)(W̃) (ι ⊗ α ⊗ α)(V)) (α ⊗ α)Δ_{2}^{op}(A)
= ∑((ω_{H_{(1)}} ⊗ ι ⊗ ι)(W̃) ⊗ 1 ⊗ 1) (α ⊗ ι ⊗ ι)((ω_{H_{(2)}} ⊗ ι)β ⊗ ι ⊗ ι)(W̃)
(α ⊗ α)(ω_{H_{(3)}} ⊗ ι ⊗ ι)(V) (α(A_{(2)}) ⊗ α(A_{(1)})) .

Using the chain rule one sees that it is possible to define a linear map

β̂ : H → H ⊗ A : β̂(H) = ∑ H^{(¯1)} ⊗ H^{(¯2)}

such that for all smooth functions B on G_{1}, r ∈ G_{2} and H ∈ H we have

((ω_{H} ⊗ ι)β(B))(r) = H_{e}[B(β_{r}(·))] = ∑ H_{e}^{(¯1)}[B] H^{(¯2)}(r) .

Fix some coordinates in the neighbourhood of the unit element e of G_{1} and let X^{i} be the vector field in this neighbourhood which takes the derivative to the i-th coordinate. Let Y^{i} ∈ 𝔤_{1} be such that Y^{i} and X^{i} agree at e. If now X ∈ 𝔤_{1} then

β̂(X) = ∑_{i} Y^{i} ⊗ β̂_{i}(X) where β̂_{i}(X)(r) = X_{e}[β_{r}^{i}(·)]

where β_{r}^{i}(g) denotes the i-th coordinate of β_{r}(g). Also define

Ψ : H → A ⊗ A : Ψ(H) = σ(ω_{H} ⊗ ι ⊗ ι)(V)

where σ is the flip. Observe that Ψ(H)(r, s) = H_{e}[V(·, s, r)] for all r, s ∈ G_{2}. Using the notation

Ψ(H) = ∑ Ψ(H)^{(1)} ⊗ Ψ(H)^{(2)}

and the Sweedler notation for Δ_{2}(A) we get

Δ(H ⊗ A) = ∑(H_{(2)}^{(¯1)} ⊗ Ψ(H_{(3)})^{(1)}A_{(1)}) ⊗ (H_{(1)} ⊗ H_{(2)}^{(¯2)}Ψ(H_{(3)})^{(2)}A_{(2)}) .

Comparing this formula with the formula in [30], Proposition 6.3.8, we get that this coproduct turns H ⊗ A into a cocycle cross coproduct coalgebra (again, up to some left/right conventions).

In terms of generators A ∈ A and X ∈ 𝔤_{1} for the algebra H ⊗ A we get more specifically

Δ(A) = Δ_{2}(A) for all A ∈ A  (5.4)
Δ(X) = 1 ⊗ X + β̂(X) + Ψ(X) for all X ∈ 𝔤_{1} .

To obtain these formulas we use that Ψ(1) = 1 ⊗ 1 which follows from V(e, r, s) = 1 for all r, s ∈ G_{2}.

Remark 5.1. When G_{1} and G_{2} is a matched pair of Lie groups, it is helpful to look also at the corresponding matched pair of Lie algebras 𝔤_{1}, 𝔤_{2} (two Lie algebras form a matched pair if the direct sum of their linear spaces is again a Lie algebra - see [30], 8.3). On the other hand, a Lie algebra (resp., dual vector space to a Lie algebra) can be viewed as a Lie bialgebra with zero cobracket (resp., bracket). Recall that the notion of a Lie bialgebra is due to Drinfel’d [11]. Now for any pair of Lie bialgebras of the form 𝔤_{1}, 𝔤_{2}^{∗} (where 𝔤_{1}, 𝔤_{2} is a matched pair of Lie algebras) all their Lie bialgebra extensions, i.e., exact sequences 0 → 𝔤_{2}^{∗} → 𝔥 → 𝔤_{1} → 0 of Lie bialgebras, are described in [33], 1.12 in terms of compatible actions of 𝔤_{1} and 𝔤_{2} on each other, as on vector spaces, and a pair of compatible 2-cocycles for these actions. Here 2-cocycles are defined as linear maps 𝔤_{1} ⋀ 𝔤_{1} → 𝔤_{2}^{∗} (resp., 𝔤_{2} ⋀ 𝔤_{2} → 𝔤_{1}^{∗}) verifying certain cocycle identities [30], 2.3.

We consider this theory of Lie bialgebra extensions as an infinitesimal version of our theory and, even if there is no any strict claim in this direction, we believe it helps to understand better some concrete situations. For instance, if one of Lie groups forming a matched pair is ℝ, the corresponding cocycle on Lie bialgebra level must be cohomologous to zero, according to the above definition. From [33], Theorem 4.11 it follows that the same picture holds for Hopf algebras. This suggests that the situation is similar for l.c. quantum groups. If both G_{1} = G_{2} = ℝ, there should exist only split extension.

5.3 Example 1

Consider the group

G = {(a, b) | a ∈ ℝ \ {0}, b ∈ ℝ} with (a, b)(c, d) = (ac, d + cb) .

Define G_{1} = G_{2} = ℝ \ {0}, with multiplication as group operation. Then define

i : G_{1} → G : i(g) = (g, g − 1) j : G_{2} → G : j(s) = (s, 0) .

This way (G_{1}, G_{2}) is a matched pair of groups in the sense of Definition 4.7. A direct computation then gives, for g, s ∈ ℝ \ {0} and s(g − 1) + 1 ≠ 0,

α_{g}(s) = \frac{gs}{s(g − 1) + 1} β_{s}(g) = s(g − 1) + 1.

Taking U = V = 1, we can construct the bicrossed product l.c. quantum group (M, Δ) having the following nice property.

Proposition 5.2. The l.c. quantum group (M, Δ) is self-dual, i.e., (M̂, Δ̂) ≅ (M, Δ).

Proof. Proposition 2.9 and Theorem 2.13 show that the dual (M̂, Δ̂) is again a bicrossed product, obtained by interchanging α and β. Now defining an isomorphism u : G_{1} → G_{2} : u(g) = g^{−1}, one verifies that

u(β_{s}(g)) = α_{u^{−1}(s)}(u(g))

for all g, s ∈ ℝ \ {0}. Hence, interchanging α and β gives an isomorphic matched pair and so an isomorphic l.c. quantum group.

Proposition 5.3. The l.c. quantum group (M, Δ) is not a Kac algebra, it is non-compact, non-discrete and non-unimodular. The scaling group is non-trivial and the left and right Haar weights are not traces.

Proof. Proposition 2.17 shows that (M, Δ) is non-compact and non-discrete. The modular functions of G_{1} and G_{2} are trivial and δ(a, b) = |a|. Using the notation introduced in Propositions 4.14 and 4.15 we get

P (g, s) = |\frac{g}{s(g − 1) + 1}| and ∇(g, s) = |\frac{1}{s(g − 1) + 1}| .

Because P is non-trivial, the scaling group of (M, Δ) is non-trivial, so (M, Δ) is not a Kac algebra. Because ∇ is non-trivial, the modular automorphism group of the left Haar weight φ is non-trivial. Hence φ is not a trace. Because the right Haar weight ψ on (M, Δ) is given by ψ = φR, also ψ is not a trace. Finally, Proposition 4.15 gives

δ_{M}(g, s) = |\frac{s(g − 1) + 1}{gs}|

and hence (M, Δ) is non-unimodular.

Let us now describe, following the scheme of the previous subsection, the infinitesimal Hopf algebra of the l.c. quantum group (M, Δ). Let A be the algebra of polynomials on G_{2} in the variable s, generated by A(s) = s for all s ∈ G_{2}. Then Δ_{2}(A) = A ⊗ A. Let X denote the generator of the Lie algebra 𝔤_{1} of G_{1} given by

X_{x}[B] = \frac{d}{dy} B(xy)|_{y=1} .

From (5.3) and (5.4) we see that this Hopf algebra is generated by elements A and X, where A is invertible and

[A, X] = X ▷ A , Δ(A) = A ⊗ A and Δ(X) = 1 ⊗ X + β̂(X) .

Now we get that

(X ▷ A)(r) = \frac{d}{dx}α_{x}(r)|_{x=1} = r(1 − r) and β̂(X) = X ⊗ (r ↦ \frac{d}{dx}β_{r}(x)|_{x=1}) = X ⊗ A .

Hence finally this Hopf algebra is generated by elements A and X, where A is invertible and

[A, X] = A(1 − A) , Δ(A) = A ⊗ A and Δ(X) = X ⊗ A + 1 ⊗ X .

Remark 5.4. a) From Remark 5.1 it follows that, on the level of Lie bialgebras, in the above situation there is no nontrivial extension. The same is true on the level of Hopf algebras (see [33], 4.11).

b) The above example is closely related to [30], 6.2.17, where G_{1} = G_{2} = (ℝ, +) and the mutual actions are only defined in the neighbourhood of the origin (0, 0). This suffices to compute the corresponding infinitesimal Hopf algebra [30], 6.2.18; the last one coincides with our infinitesimal Hopf algebra.

5.4 Example 2: split extension

Define H = SL_{2}(ℝ). Consider ℤ/2ℤ as a normal subgroup K of H by identifying it with {1, −1} and define G = H/K. Then we define

G_{1} = \{(a, b) | a > 0, b ∈ ℝ\} with (a, b)(c, d) = (ac, ad + \frac{b}{c})
G_{2} = (ℝ, +)

and

i(a, b) = (\begin{matrix}a & b \\ 0 & \frac{1}{a} \end{matrix}) mod K j(x) = (\begin{matrix}1 & 0 \\ x & 1 \end{matrix}) mod K .

This way (G_{1}, G_{2}) is a matched pair of groups. A direct computation gives that, whenever a + bx ≠ 0

α_{(a,b)}(x) = \frac{x}{a(a + bx)} β_{x}(a, b) = \begin{cases}(a + bx, b) if a + bx > 0 \\ (−a − bx, −b) if a + bx < 0 \end{cases} .

Taking U = V = 1, we can construct the bicrossed product l.c. quantum group (M, Δ) and its dual (M̂, Δ̂). The following result is similar to Proposition 5.3.

Proposition 5.5. The l.c. quantum groups (M, Δ) and (M̂, Δ̂) are not Kac algebras, they are non-compact, and non-discrete. The scaling groups are non-trivial. The left and right Haar weights of (M, Δ) and (M̂, Δ̂) are not traces. Finally (M, Δ) is unimodular, but (M̂, Δ̂) is non-unimodular.

Proof. The proof is completely similar to the one of Proposition 5.3 and uses again Propositions 4.14 and 4.15. Let us mention only that the groups G and G_{2} are unimodular, δ_{1}(a, b) = \frac{1}{a^{2}} and we get that

ˆ∇(a, b, s) = P (a, b, s) = \frac{a^{2}}{(a + bs)^{2}} , ∇(a, b, s) = \frac{1}{a^{2}(a + bs)^{2}} , δ_{M} = 1 and δM̂(a, b, s) = (a + bs)^{4} .

Let us now describe the infinitesimal Hopf algebras for (M, Δ) and (M̂, Δ̂), following again the strategy of Subsection 5.2. Let A be the algebra of polynomials on G_{2} in the variable x generated by A(x) = x. Take generators X, Y for the Lie algebra 𝔤_{1} of G_{1} given by

X_{(a,b)}[B] = \frac{d}{dx}B((a, b)(x, 0))|_{x=1} Y_{(a,b)}[B] = \frac{d}{dx}B((a, b)(1, x))|_{x=0}

whenever (a, b) ∈ G_{1} and when B is a smooth function on G_{1}. So we observe that [X, Y]_{𝔤_{1}} = 2Y. To determine the infinitesimal Hopf algebra for (M, Δ) we compute

(X ▷ A)(x) = \frac{d}{da}α_{(a,0)}(x)|_{a=1} = −2x (Y ▷ A)(x) = \frac{d}{db}α_{(1,b)}(x)|_{b=0} = −x^{2}

and taking the obvious coordinates on G_{1} we see that

β̂_{1}(X)(x) = \frac{d}{da}β_{x}^{1}(a, 0)|_{a=1} = 1  β̂_{2}(X)(x) = \frac{d}{da}β_{x}^{2}(a, 0)|_{a=1} = 0
β̂_{1}(Y)(x) = \frac{d}{db}β_{x}^{1}(1, b)|_{b=0} = x  β̂_{2}(Y)(x) = \frac{d}{db}β_{x}^{2}(1, b)|_{b=0} = 1 .

So we get that

X ▷ A = −2A , Y ▷ A = −A^{2} , β̂(X) = X ⊗ 1 and β̂(Y) = X ⊗ A + Y ⊗ 1 .

So this Hopf algebra has generators A, X and Y with relations

[X, A] = 2A , [Y, A] = A^{2} and [X, Y] = 2Y
Δ(A) = A ⊗ 1 + 1 ⊗ A
Δ(X) = X ⊗ 1 + 1 ⊗ X
Δ(Y) = Y ⊗ 1 + X ⊗ A + 1 ⊗ Y .

To obtain the infinitesimal Hopf algebra for the dual l.c. quantum group (M̂, Δ̂) interchange α and β. We take as A the algebra of polynomials on G_{1} in the variables a and b generated by A(a, b) = a and B(a, b) = b. We denote with X the generator of the Lie algebra 𝔤_{2} of G_{2} given by

X_{x}[B] = \frac{d}{dy} B(x + y)|_{y=0}

whenever x ∈ G_{1} and when B is a smooth function on G_{1}. Similar computations as above yield that

X ▷ A = B , X ▷ B = 0 and β̂(X) = X ⊗ A^{−2} .

So this Hopf algebra is generated by elements X, A and B, with A invertible and

[A, X] = B , [A, B] = 0 and [B, X] = 0
Δ(A) = A ⊗ A
Δ(B) = A ⊗ B + B ⊗ A^{−1}
Δ(X) = X ⊗ A^{−2} + 1 ⊗ X .

5.5 Example 2: nontrivial extensions

Let (G_{1}, G_{2}) and α, β be as in the previous subsection. We look for a cocycle U and Remark 5.1 suggests to take V = 1. Referring to Lemma 4.10 we look for a measurable function U on G_{1} × G_{1} × G_{2} such that

U(g, h, α_{k}(s)) U(gh, k, s) = U(h, k, s) U(g, hk, s),
U(g, h, s) U(β_{α_{h}(s)}(g), β_{s}(h), t) = U(g, h, t + s)

almost everywhere. Define a function A(·) by

U(g, h, s) = exp(iA(g, h, s)) .

So A(·) should satisfy

A(g, h, α_{k}(s)) + A(gh, k, s) = A(h, k, s) + A(g, hk, s) (mod2π),  (5.5)
A(g, h, s) + A(β_{α_{h}(s)}(g), β_{s}(h), t) = A(g, h, t + s) (mod2π)  (5.6)

almost everywhere. For s ∈ G_{2} and g, h ∈ G_{1} let us denote

ϕ_{s}(g, h) = (β_{α_{h}(s)}(g), β_{s}(h)) .

Then ϕ_{t+s} = ϕ_{t}ϕ_{s} almost everywhere, and equation (5.6) becomes

A(g, h, s) + A(ϕ_{s}(g, h), t) = A(g, h, t + s) (mod2π) .  (5.7)

It is natural to look for a solution of this equation in the form

A(g, h, s) = P ∫_{0}^{s} f(ϕ_{r}(g, h)) dr .  (5.8)

with a function f on G_{1} × G_{1} such that for almost all g, h ∈ G_{1} the function

r ↦ f(ϕ_{r}(g, h))

has a principal value integral over any interval in ℝ. So in order to fulfill equation (5.5), our function f should satisfy, for almost all g, h, k ∈ G_{1} and s ∈ G_{2}

P ∫_{0}^{α_{k}(s)} f(ϕ_{r}(g, h)) dr + P ∫_{0}^{s} f(ϕ_{r}(gh, k)) dr = P ∫_{0}^{s} f(ϕ_{r}(h, k)) dr + P ∫_{0}^{s} f(ϕ_{r}(g, hk)) dr (mod2π) . (5.9)

Differentiating to s we should look for a function f satisfying

\frac{d}{ds}α_{k}(s) f(g̃, h̃) + f(g̃h̃, k̃) = f(h̃, k̃) + f(g̃, h̃k̃)

where g̃ = β_{α_{hk}(s)}(g), h̃ = β_{α_{k}(s)}(h), k̃ = β_{s}(k). Observing that

α_{β_{s}(k)}(t) + α_{k}(s) = α_{k}(t + s)

and differentiating to t at t = 0 we get that

\frac{d}{ds}α_{k}(s) = \frac{d}{dt}αk̃(t)|_{t=0} .

Hence, f should satisfy

\frac{d}{dt}α_{k}(t)|_{t=0} f(g, h) + f(gh, k) = f(h, k) + f(g, hk)

for almost all g, h, k ∈ G_{1}. Putting g = (a, b), h = (c, d) and k = (l, m) we see that f should satisfy

\frac{1}{l^{2}} f(a, b, c, d) + f(ac, ad + \frac{b}{c}, l, m) = f(c, d, l, m) + f(a, b, cl, cm + \frac{d}{l})  (5.10)

where f is defined on ℝ_{0}^{+} × ℝ × ℝ_{0}^{+} × ℝ.

Proposition 5.6. A general sufficiently smooth solution f(a, b, c, d) of (5.10) is given by

f(a, b, c, d) = λ\frac{b log c}{ac^{2}} + \frac{1}{c^{2}} B(a, b) + B(c, d) − B(ac, ad + \frac{b}{c})

where λ ∈ ℝ and B is a smooth function on G_{1}.

Proof. First, one can compute directly that

f(a, b, c, d) = \frac{1}{c^{2}} B(a, b) + B(c, d) − B(ac, ad + \frac{b}{c})  (5.11)

where B is an arbitrary function on G_{1}, is a solution of equation (5.10). We will call this solution trivial. Because the solutions of (5.10) form a linear space, we will only determine a general solution of (5.10) modulo a trivial solution.

In the course of the proof we use the subscript notations f_{2}, f_{13}, … to denote partial derivative of f to the first variable and to the first and third variable respectively. Take the derivative to b of a sufficiently smooth solution f of equation (5.10) and evaluate it in a = 1 and b = 0. Then we get

f_{2}(c, d, l, m) = c h(cl, cm + \frac{d}{l}) − \frac{c}{l^{2}} h(c, d)

where h(c, d) = f_{2}(1, 0, c, d). If H(·) is a smooth function such that H_{2}(c, d) = h(c, d), this gives

f(c, d, l, m) = cl H(cl, cm + \frac{d}{l}) − \frac{c}{l^{2}} H(c, d) + k(c, l, m)

where k is some smooth function. Hence, modulo a trivial solution,

f(c, d, l, m) = l H(l, m) + k(c, l, m),

i.e., f does not depend on its second variable:

f(a, b, c, d) = g(a, c, d) .

Now equation (5.10) gives

\frac{1}{l^{2}} g(a, c, d) + g(ac, l, m) = g(c, l, m) + g(a, cl, cm + \frac{d}{l}) .

Taking the derivative to a and evaluating it in a = 1 we get

g_{1}(c, l, m) = \frac{1}{c} g_{1}(1, cl, cm + \frac{d}{l}) − \frac{1}{cl^{2}} g_{1}(1, c, d) .  (5.12)

Taking derivative to d it follows that

\frac{1}{cl} g_{13}(1, cl, cm + \frac{d}{l}) = \frac{1}{cl^{2}} g_{13}(1, c, d) .  (5.13)

With l = 1 we get that g_{13}(1, c, cm + d) = g_{13}(1, c, d) and hence g_{13}(1, c, d) = q(c) for some smooth function q(·). Plugging this into equation (5.13) we get

\frac{1}{cl} q(cl) = \frac{1}{cl^{2}} q(c)

and so there exists a number λ ∈ ℝ such that q(c) = \frac{λ}{c}. Then it follows that

g_{1}(1, c, d) = \frac{λd}{c} + r(c)

for some smooth function r(·). Using equation (5.12) we have

g_{1}(c, l, m) = \frac{λm}{cl} + \frac{1}{c} r(cl) − \frac{1}{cl^{2}} r(c) .

If R(·) is a primitive for the function \frac{1}{x}r(x) we get

g(c, l, m) = \frac{λm}{l} log c + R(cl) − \frac{1}{l^{2}} R(c) + t(l, m)

for some smooth function t(·). Since f(a, b, c, d) = g(a, c, d), modulo a trivial solution we have

f(c, d, l, m) = \frac{λm}{l} log c + t(l, m) + R(l)

or

f(c, d, l, m) = \frac{λm}{l} log c + v(l, m)

for some smooth function v(·). Plugging this into equation (5.10) we get

\frac{1}{l^{2}} v(c, d) = v(cl, cm + \frac{d}{l}) .

With l = 1 it follows that v does not depend on its second variable. Then there is a number ν ∈ ℝ such that v(c, d) = \frac{ν}{c^{2}} . But also f(a, b, c, d) = \frac{ν}{c^{2}} is a trivial solution with B(a, b) = ν. Hence we may conclude that

f(a, b, c, d) = λ\frac{d}{c} log a

modulo a trivial solution, where λ ∈ ℝ. One can check directly that this is indeed a solution.

It will be more convenient, taking B_{0}(a, b) = λ\frac{b}{a} log a, to get a general sufficiently smooth solution of equation (5.10) in the form

f(a, b, c, d) = λ\frac{b log c}{ac^{2}} + \frac{1}{c^{2}} B(a, b) + B(c, d) − B(ac, ad + \frac{b}{c})

where λ ∈ ℝ and B(·) is an arbitrary smooth function.

Since two solutions differing on trivial solution give rise to cohomologous cocycles, we can work on with the solution

f_{λ}(a, b, c, d) = λ\frac{b log c}{ac^{2}}

where λ ∈ ℝ is some parameter. Making an easy computation one observes that

f_{λ}(ϕ_{r}(a, b, c, d)) = \frac{λb}{(c + dr)(ac + (ad + \frac{b}{c})r)} log |c + dr| .

As a function of r we indeed have a principal value integral over any interval in ℝ, for almost all a, b, c, d. One verifies that

P ∫_{−∞}^{+∞} f_{λ}(ϕ_{r}(a, b, c, d)) dr = \begin{cases}\frac{λ}{2}π^{2} if \frac{d}{b}(ad + \frac{b}{c}) > 0 \\ −\frac{λ}{2}π^{2} if \frac{d}{b}(ad + \frac{b}{c}) < 0 \end{cases} .

Define A_{λ}(·) with f_{λ}(·) by (5.8) and check equation (5.9). When s = 0 this equation is trivially true. When s grows, the equation remains true, because the differentiated equation is fulfilled. Also, passing a pole is not a problem because the substitution rule can be applied to this principal value integrals. The only problem is when s passes the pole where α_{k}(s) grows to infinity. Then the principal value integral

P ∫_{0}^{α_{k}(s)} f_{λ}(ϕ_{r}(g, h)) dr

changes to

P ∫_{0→+∞,−∞→α_{k}(s)} f_{λ}(ϕ_{r}(g, h)) dr = P ∫_{0}^{α_{k}(s)} f_{λ}(ϕ_{r}(g, h)) dr + P ∫_{−∞}^{+∞} f_{λ}(ϕ_{r}(g, h)) dr

= P ∫_{0}^{α_{k}(s)} f_{λ}(ϕ_{r}(g, h)) dr ± \frac{λ}{2}π^{2} .

Hence, equation (5.9) remains true iff λ has the form

λ = \frac{4n}{π} with n ∈ ℤ .

Thus, for any n ∈ ℤ, the functions

U_{n}(g, h, s) = exp(iA_{\frac{4n}{π}} (g, h, s))

and V = 1 satisfy (4.2), so they give the necessary data to perform the cocycle bicrossed product construction. Let us describe the corresponding l.c. quantum group (M_{n}, Δ_{n}) and its dual (M̂_{n}, Δ̂_{n}).

Clearly (M_{0}, Δ_{0}) and (M̂_{0}, Δ̂_{0}) are precisely the examples studied in the previous subsection, because U_{0} = 1. Thanks to Propositions 4.14 and 4.15, Proposition 5.5 remains valid for (M_{n}, Δ_{n}) and (M̂_{n}, Δ̂_{n}), also when n ≠ 0. Let us mention the relations between (M_{n}, Δ_{n}) and (M_{0}, Δ_{0}), and between (M̂_{n}, Δ̂_{n}) and (M̂_{0}, Δ̂_{0}). Represent M̂_{n} on L^{2}(G_{1} × G_{2}) as in Proposition 2.9. Because V = 1, we get M̂_{n} = M̂_{0} for all n ∈ ℤ. Further we have

Ŵ_{n}^{∗} = C_{n}Ŵ_{0}^{∗} where C_{n} = (1 ⊗ (ι ⊗ α)(Ŵ_{2}^{∗})) (β ⊗ ι ⊗ ι)(U_{n}) (1 ⊗ (ι ⊗ α)(Ŵ_{2})) .

An easy computation shows that C_{n} is the multiplication operator with the function

C_{n}(g, s, h, t) = U_{n}(β_{α_{h}(t)s}(g), h, t) .

It is also clear that

Δ̂_{n}(z) = C_{n}Δ̂_{0}(z)C_{n}^{∗} for all z ∈ M̂_{n} = M̂_{0} .

From Proposition 2.9 it follows that the left Haar weight φ̂_{n} equals φ̂_{0}. Propositions 4.14 and 4.15 imply that Ĵ_{n} = Ĵ_{0} for all n ∈ ℤ and they also give us that the operators ∇_{n}, ˆ∇_{n}, P_{n}, δ_{M_{n}} and δM̂_{n} do not depend on n. In particular it follows that τ̂_{n} = τ̂_{0}, but we see that J_{n} ≠ J_{0} and then also R̂_{n} ≠ R̂_{0} for n ≠ 0.

As in the previous examples we want to compute the infinitesimal Hopf algebras of (M_{n}, Δ_{n}) and (M̂_{n}, Δ̂_{n}). We start with (M_{n}, Δ_{n}) and we follow the strategy of Subsection 5.2. In the previous Subsection we already defined the algebra A of polynomials on G_{2}, with generator A, and we defined the generators X and Y for the Lie algebra 𝔤_{1} of G_{1}. If H denotes the universal enveloping algebra of 𝔤_{1}, we have to look at the map

χ_{n} : H ⊗ H → A : χ_{n}(H ⊗ G)(r) = (H_{e} ⊗ G_{e})[U_{n}^{∗}(·, ·, r)] .  (5.14)

An easy computation yields that, for all λ ∈ ℝ and r ∈ G_{2}, we have

(X_{e} ⊗ Y_{e})[f_{λ}(ϕ_{r}(·, ·))] = 0 and (Y_{e} ⊗ X_{e})[f_{λ}(ϕ_{r}(·, ·))] = λ1

and then it follows that

χ_{n}(X ⊗ Y − Y ⊗ X) = −i\frac{4n}{π}A .

So, the Hopf algebra underlying (M_{n}, Δ_{n}) has generators X, Y and A and relations

[X, A] = 2A , [Y, A] = A^{2} and [X, Y] = 2Y − i\frac{4n}{π}A

Δ(A) = A ⊗ 1 + 1 ⊗ A Δ(X) = X ⊗ 1 + 1 ⊗ X Δ(Y) = Y ⊗ 1 + X ⊗ A + 1 ⊗ Y .

To obtain the infinitesimal Hopf algebra of (M̂_{n}, Δ̂_{n}) we have to interchange α and β and we have to use the cocycle V_{n}(s, g, h) = U_{n}(h, g, s) and take U = 1. Now we take for A the algebra of polynomials on G_{1} in the variables a, log a and b generated by A(a, b) = a, (log A)(a, b) = log a and B(a, b) = b. As before we denote with X the generator of the Lie algebra 𝔤_{2} of G_{2}. If H denotes the universal enveloping algebra of 𝔤_{2} we have to look at the map

Ψ_{n} : H → A ⊗ A : Ψ_{n}(H)(g, h) = H_{e}[V_{n}(·, h, g)] .  (5.15)

So we immediately get that

Ψ_{n}(X)(g, h) = \frac{d}{ds}U_{n}(g, h, s)|_{s=0} = if_{\frac{4n}{π}} (g, h)

and hence

Ψ_{n}(X) = i\frac{4n}{π} A^{−1}B ⊗ A^{−2} log A .

The Hopf algebra underlying (M̂_{n}, Δ̂_{n}) has generators A, log A, B and X with relations

B is central , [A, log A] = 0 , [A, X] = B and [log A, X] = A^{−1}B Δ(A) = A ⊗ A , Δ(log A) = log A ⊗ 1 + 1 ⊗ log A

Δ(B) = A ⊗ B + B ⊗ A^{−1}

Δ(X) = X ⊗ A^{−2} + 1 ⊗ X + i\frac{4n}{π} A^{−1}B ⊗ A^{−2} log A .

Remark 5.7. One can show, using the Lie bialgebra version of the G.I. Kac’ exact sequence [18], (3.14) obtained by Masuoka (see [33], 2.10), that the group of extensions for the above example, on the level of Lie bialgebras or Hopf algebras, is exactly ℝ. In equations (5.14) and (5.15) above we see that the maps χ and Ψ can be defined for arbitrary real parameter λ, and also the infinitesimal Hopf algebras make sense for arbitrary real parameter λ instead of \frac{4n}{π}. But, as we have seen, on the level of l.c. quantum groups only for the extensions with λ = \frac{4n}{π} (n ∈ ℤ), the cocycles make sense on the operator algebra level.

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