2011

Representation of the Quantum Plane, its Quantum Double and Harmonic Analysis on $GL_q^+(2,R)$

Ip, Ivan Chi-Ho

Understand

We give complete detail of the description of the GNS representation of the quantum plane $\cA$ and its dual $\hat{\cA}$ as a von-Neumann algebra.

In particular we obtain a rather surprising result that the multiplicative unitary $W$ is manageable in this quantum semigroup context.
We study the quantum double group construction introduced by Woronowicz, and using Baaj and Vaes' construction of the multiplicative unitary $\bW_m$, we give the GNS description of the quantum double $\cD(\cA)$ which is equivalent to $GL_q^+(2,\R)$.
Furthermore we study the fundamental corepresentation $T^{\l,t}$ and its matrix coefficients, and show that it can be expressed by the $b$-Hypergeometric function.

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