  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=L^2(R) has a limit (in
the dual co-representation) towards the Mellin transform of the unitary
representation of the ax+b group, and furthermore the intertwiners of the
tensor products representation has a limit towards 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 co-representation 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.
