The Classical Limit of Representation Theory of the Quantum Plane

TL;DR

This paper demonstrates a classical limit for the representation theory of the quantum plane B_q at |q|=1, connecting quantum and classical group representations through Fourier and Mellin transforms.

Contribution

It establishes a precise analogy between quantum plane representations and classical ax+b group representations, including explicit limits and intertwiners.

Findings

Fourier transform of B_q representation converges to Mellin transform of classical representation.

Intertwiners of tensor product representations have a classical limit.

Explicit multiplicative unitary for the quantum ax+b semigroup is derived.

Abstract

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.