From projective representations to pentagonal cohomology via quantization

Abstract

Given a locally compact group $G=Q⋉V$ 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 to a pair of subgroups of the dual semidirect product $Q⋉\hat{V}$, both isomorphic to $Q$, and to a pentagonal cocycle ${\Theta }_{\omega }$ explicitly given in terms of the group cocycle $\omega$.

25 pages