量子平面表示论的经典极限
The Classical Limit of Representation Theory of the Quantum Plane
摘要：
我们证明了，当|q| =1时，量子平面B_q的表示与经典的ax+b群完全相似.我们证明了H=L^2（R）上B_q表示的傅里叶变换（在对偶余表示中）对ax+b群的酉表示的Mellin变换有一个极限，而且张量积表示的纠缠对经典ax+b表示的Mellin变换的纠缠也有一个极限.我们还明确地写出了定义量子ax+b半群的乘法酉表示，并证明了它定义了与上述B_q表示对偶的余表示，并且精确地对应于ax+b群的经典酉表示族.
英文摘要：
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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