From projective representations to pentagonal cohomology via quantization

Abstract

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 $\omega$.