From projective representations to pentagonal cohomology via quantization

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