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 ...

We use the categories of representations of finite dimensional quantum groupoids (weak Hopf algebras) to construct ribbon and modular categories that give rise to invariants of knots and 3-manifolds.

We characterize finite index depth 2 inclusions of type II 1 factors in terms of actions of weak Kac algebras and weak C*-Hopf algebras. If N/M/M 1 / M 2 / } } } is the Jones tower constructed from such an inclusion N/M, then B= M$ & M 2 has a natural structure of a weak C*-Hopf algebra and there is a minimal action of B on M 1 such that M is the fixed point subalgebra of M 1 and M 2 is isomorphic to the crossed product of M 1 and B. This extends the well-known results for irreducible depth 2 inclusions. Academic Press

We establish a Galois correspondence for finite quantum groupoid actions on II 1 factors and show that every finite index and finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product. Moreover, any such subfactor is completely and canonically determined by a quantum groupoid and its coideal V-subalgebra. This allows us to express the bimodule category of a subfactor in terms of the representation category of a corresponding quantum groupoid and the principal graph as the Bratteli diagram of an inclusion of certain finite-dimensional C*-algebras related to it. Academic Press

Some examples of quantum groups in literature arise as deformations of a locally compact group by a "dual" 2-cocycle. We make this construction in the framework of Kac algebras; we show that these deformations are still Kac algebras; using this construction, we give new quantizations of the Heisenberg group. From this point of view, we analyse the dimension 8 non-trivial example of Kac and Paljutkin, and give a new example of non-trivial dimension 12 semi-simple *-Hopf algebras (a dimension 12 Kac algebra).