Abstract
In this note, we propose a decomposition of the quantum matrix group \( {\textrm{SL}}_q^{+} \)(2, ℝ) as (deformed) exponentiation of the quantum algebra generators of Faddeev’s modular double of Uq(\( \mathfrak{sl} \)(2, ℝ)). The formula is checked by relating hyperbolic representation matrices with the Whittaker function. We interpret (or derive) it in terms of Hopf duality, and use it to explicitly construct the regular representation of the modular double, leading to the Casimir and its modular dual as the analogue of the Laplacian on the quantum group manifold. This description is important for both 2d Liouville gravity, and 3d pure gravity, since both are governed by this algebraic structure. This result builds towards a q-BF formulation of the amplitudes of both of these gravitational models.
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Acknowledgments
We thank A. Blommaert, Y. Fan, J. Simón, G. Wong and S. Yao for discussions and collaborations related to this work. TM acknowledges financial support from the European Research Council (grant BHHQG-101040024). Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
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Mertens, T.G. Quantum exponentials for the modular double and applications in gravity models. J. High Energ. Phys. 2023, 106 (2023). https://doi.org/10.1007/JHEP09(2023)106
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DOI: https://doi.org/10.1007/JHEP09(2023)106