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Representation of the quantum plane, its quantum double, and harmonic analysis on \(GL_q^+(2,\mathbb{R })\)

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We give complete detail of the description of the GNS representation of the quantum plane \(\mathcal{A }\) and its dual \({\widehat{\mathcal{A }}}\) 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 \(\mathbf{W}_m\), we give the GNS description of the quantum double \(\mathcal{D }(\mathcal{A })\) which is equivalent to \(GL_q^+(2,\mathbb{R })\). Furthermore, we study the fundamental corepresentation \(T^{\lambda ,t}\) and its matrix coefficients, and show that it can be expressed by the \(b\)-hypergeometric function. We also study the regular corepresentation and representation induced by \(\mathbf{W}_m\) and prove that the space of \(L^2\) functions on the quantum double decomposes into the continuous series representation of \(U_{q\widetilde{q}}(\mathfrak{gl }(2,\mathbb{R }))\) with the quantum dilogarithm \(|S_b(Q+2i\alpha )|^2\) as the Plancherel measure. Finally, we describe certain representation theoretic meaning of integral transforms involving the quantum dilogarithm function.

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Acknowledgments

I would like to thank my advisor Professor Igor Frenkel for suggesting the current project and providing useful insights into the problems. I would also like to thank L. Faddeev for pointing out several useful references, and to J. Teschner and Hyun Kyu Kim for helpful discussions. This work was partially supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.

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  1. Kavli IPMU, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa,  277-8583, Chiba, Japan

    Ivan Chi-Ho Ip

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Ip, I.CH. Representation of the quantum plane, its quantum double, and harmonic analysis on \(GL_q^+(2,\mathbb{R })\) . Sel. Math. New Ser. 19, 987–1082 (2013). https://doi.org/10.1007/s00029-012-0112-4

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