From projective representations to pentagonal cohomology via quantization.

Given a locally compact group G = Q ⋉ V such that V is Abelian and such that the action of Q on the Pontryagin dual V ^ has a free orbit of full measure, we construct a family of unitary dual 2-cocycles Ω ω (aka non-formal Drinfel’d twists) whose equivalence classes [ Ω ω ] ∈ H 2 (G ^ , T) are parametrized by cohomology classes [ ω ] ∈ H 2 (Q , 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 ⋉ V ^ , both isomorphic to Q, and to a pentagonal cocycle Θ ω explicitly given in terms of the group cocycle ω . [ABSTRACT FROM AUTHOR]