Quantization of locally compact groups associated with essentially bijective 1-cocycles

Given an extension Formula: see text of locally compact groups, with Formula: see text abelian, and a compatible essentially bijective Formula: see text-cocycle Formula: see text, we define a dual unitary Formula: see text-cocycle on Formula: see text and show that the associated deformation of Formula: see text is a cocycle bicrossed product defined by a matched pair of subgroups of Formula: see text. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang–Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of Formula: see text on Formula: see text and a unitary quantization map Formula: see text of Kohn–Nirenberg type.