Representation of the quantum plane, its quantum double, and harmonic analysis on GL + q (2,R) Authors Authors and affiliations
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Ivan Chi-Ho Ip [1]
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[1] University of Tokyo
University of Tokyo
Japón
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- Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 19, Nº. 4, 2013, págs. 987-1082
- Idioma: inglés
- Enlaces
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Resumen
We give complete detail of the description of the GNS representation of the quantum plane AA and its dual AˆA^ as a von Neumann algebra. In particular, we obtain a rather surprising result that the multiplicative unitary WW 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 WmWm, we give the GNS description of the quantum double D(A)D(A) which is equivalent to GL+q(2,R)GLq+(2,R). Furthermore, we study the fundamental corepresentation Tλ,tTλ,t and its matrix coefficients, and show that it can be expressed by the bb-hypergeometric function. We also study the regular corepresentation and representation induced by WmWm and prove that the space of L2L2 functions on the quantum double decomposes into the continuous series representation of Uqq˜(gl(2,R))Uqq~(gl(2,R)) with the quantum dilogarithm |Sb(Q+2iα)|2|Sb(Q+2iα)|2 as the Plancherel measure. Finally, we describe certain representation theoretic meaning of integral transforms involving the quantum dilogarithm function.
