From projective representations to pentagonal cohomology via quantization

Given a locally compact group \(G=Q < imes V\) such that V is Abelian and such that the action of Q on the Pontryagin dual \({\hat{V}}\) has a free orbit of full measure, we construct a family of unitary dual 2-cocycles \(\Omega _\omega \) (aka non-formal Drinfel’d twists) whose equivalence classes \([\Omega _\omega ]\in H^2({\hat{G}},{\mathbb {T}})\) are parametrized by cohomology classes \([\omega ]\in H^2(Q,{\mathbb {T}})\). We prove that the associated locally compact quantum groups are isomorphic to cocycle bicrossed product quantum groups associated with a pair of subgroups of the dual semidirect product \(Q < imes {\hat{V}}\), both isomorphic to Q, and to a pentagonal cocycle \(\Theta _\omega \) explicitly given in terms of the group cocycle \(\omega \).