Quantization of subgroups of the affine group

Consider a locally compact group G=Q⋉V such that V is abelian and the action of Q on dual Vˆ has free orbit full measure. We show G can be quantized in three equivalent ways: by reflecting across Galois object defined canonical irreducible representation L2(V); twisting coproduct von Neumann algebra 2-cocycle obtained from G-equivariant Kohn–Nirenberg quantization V×Vˆ; considering bicrossed product matched pair subgroups Q⋉Vˆ both isomorphic to Q. In simplest case ax+b over reals, cocycle (2) an analytic analogue Jordanian twist. It was first found Stachura using different ideas. The equivalence approaches (3) this implies quantum Baaj–Skandalis Stachura. Along way we prove number results for arbitrary groups G. Using recent De Commer class G-Galois objects parametrized certain cohomology classes H2(G;T). This extends Wassermann Davydov finite case. A new phenomenon already unit H2(G;T) correspond nontrivial object. Specifically, any with factor type I admits 2-cocycles corresponding neither commutative nor cocommutative.