The Classical Limit of Representation Theory of the Quantum Plane
Abstract
We showed that there is a complete analogue of a representation of the quantum plane Bq where |q|=1, with the classical ax+b group. We showed that the Fourier Transform of the representation of Bq on H=L2(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 Bq above, and also correspond precisely to the classical family of unitary representation of the ax+b group.