The classical limit of representation theory of the quantum plane

We showed that there is a complete analogue of a representation of the quantum plane ℬ_q where |q| = 1, with the classical ax+b group. We showed that the Fourier transform of the representation of ℬ-q$ on ℋ = L²(ℝ) has a limit (in the dual corepresentation) toward the Mellin transform of the unitary representation of the ax+b group, and furthermore the intertwiners of the tensor products representation has a limit toward the intertwiners of the Mellin transform of the classical ax+b representation. We also wrote explicitly the multiplicative unitary defining the quantum ax+b semigroup and showed that it defines the corepresentation that is dual to the representation of ℬ_q above, and also correspond precisely to the classical family of unitary representation of the ax+b group.