Abstract
We study the tensor product decomposition of the split real quantum group \({\mathcal {U}}_{q\widetilde{q}}(\mathfrak {sl}(2,\mathbb {R}))\) from the perspective of finite-dimensional representation theory of compact quantum groups. It is known that the class of positive representations of \({\mathcal {U}}_{q\widetilde{q}}(\mathfrak {sl}(2,\mathbb {R}))\) is closed under taking tensor product. In this paper, we show that one can derive the corresponding Hilbert space decomposition, given explicitly by quantum dilogarithm transformations, from the Clebsch–Gordan coefficients of the tensor product decomposition of finite-dimensional representations of the compact quantum group \({\mathcal {U}}_q(\mathfrak {sl}_2)\) by solving certain functional equations arising from analytic continuation and using normalization arising from tensor products of canonical basis.
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Notes
We used a different coproduct here but the transformation is equivalent up to multiplication by \(e^{2\pi i xy}\).
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The author is supported by the Hong Kong RGC General Research Funds ECS #26303319.
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The author is supported by the Hong Kong RGC General Research Funds ECS #26303319.
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Ip, I.C.H. On tensor product decomposition of positive representations of \({\mathcal {U}}_{q\widetilde{q}}(\mathfrak {sl}(2,\mathbb {R}))\). Lett Math Phys 111, 39 (2021). https://doi.org/10.1007/s11005-021-01381-6
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DOI: https://doi.org/10.1007/s11005-021-01381-6