The Classical Limit of Representation Theory of the Quantum Plane

Details The Classical Limit of Representation Theory of the Quantum Plane The Classical Limit of Representation Theory of the Quantum Plane

2010-12-19

arxiv.org/abs/1012.4145v1

Mathematics

Representation Theory

Scientific paper

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.

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