Extensions of locally compact quantum groups and the bicrossed product construction

Abstract

In the framework of locally compact quantum groups, we study cocycle actions. We develop the cocycle bicrossed product construction, starting from a matched pair of locally compact quantum groups. We define exact sequences and establish a one-to-one correspondence between cocycle bicrossed products and cleft extensions. In this way, we obtain new examples of locally compact quantum groups. PreliminariesWe denote by ⊗ the tensor product of Hilbert spaces (resp., 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 is itself 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 [47] 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 From [26], Theorem 7.14 we know that left invariant weights on (M, ∆) are unique up to a positive scalar and the same holds for right invariant weights.Let us fix a left invariant n.s.f. weight ϕ on (M, ∆) 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 byHere 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 ∆(x) = W * (1 ⊗ x)W for all x ∈ M . Also the von Neumann algebra M can be written in terms of W asNext the l.c. quantum group (M, ∆) 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) * , 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, ∆). From [26], Proposition 5.26 we know that σ(R ⊗ R)∆ = ∆R. So ϕR is a right invariant weight on (M, ∆) and we take ψ := ϕR. Let us denote by (σ t ) the modular automorphism group of ϕ. From [26], Proposition 6.8 we get the existence of a number ν > 0, called the scaling constant, 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 δM , see [26], Definition 7.1. Formally this ...

Measurable Kac cohomology for bicrossed products

“…Suppose next that I is a full coideal in the bicrossed product (M, ∆). We shall produce an element in H 1 (m.p., T) such that I is given by (23). Hence, I will also satisfy the second statement of the proposition.…”

“…Here, we do not explain the notion of such an extension in the framework of locally compact quantum groups (see Definition 3.2 in [23] …”

“…From Proposition 3.8 in [23], we know that two extensions are isomorphic if and only if the associated matched pairs of locally compact groups are the same and there exists a measurable map R : G 2 × G 1 → T such that the pairs of 2-cocycles (U, V) differ by a trivial pair of 2-cocycles (U R , V R ) defined by…”

“…Dividing the group of 2-cocycles by trivial 2-cocycles, we get again a 2-cohomology group. For finite groups G 1 , G 2 , the computational argument in Section 4.4 of [23] allows us to conclude that this 2-cohomology group is isomorphic with the 2-cohomology group defined in Definition 3.1 using pairs (U, V). The more delicate general case will be dealt with below.…”

“…This is explained in detail after Proposition 3.1 in [23]. We only work with left coideals, and so we leave out 'left' from now on.…”