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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On tensor products of positive representations of split real quantum Borel subalgebra $ \mathcal{U}_{q\widetilde{q}}(\mathfrak{b}_\mathbb{R})$


Author: Ivan C. H. Ip
Journal: Trans. Amer. Math. Soc. 370 (2018), 4177-4200
MSC (2010): Primary 81R50, 22D25
DOI: https://doi.org/10.1090/tran/7110
Published electronically: December 27, 2017
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Abstract: We study the positive representations $ \mathcal {P}_\lambda $ of split real quantum groups $ \mathcal {U}_{q\widetilde {q}}(\mathfrak{g}_\mathbb{R})$ restricted to the Borel subalgebra $ \mathcal {U}_{q\widetilde {q}}(\mathfrak{b}_\mathbb{R})$. We prove that the restriction is independent of the parameter $ \lambda $. Furthermore, we prove that it can be constructed from the GNS-representation of the multiplier Hopf algebra $ \mathcal {U}_{q\widetilde {q}}^{C^*}(\mathfrak{b}_\mathbb{R})$ defined earlier, which allows us to decompose their tensor product using the theory of the ``multiplicative unitary''. In particular, the quantum mutation operator can be constructed from the multiplicity module, which will be an essential ingredient in the construction of quantum higher Teichmüller theory from the perspective of representation theory, generalizing earlier work by Frenkel-Kim.


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Ivan C. H. Ip
Affiliation: Center for the Promotion of Interdisciplinary Education and Research Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan
Email: ivan.ip@math.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/tran/7110
Keywords: Positive representations, split real quantum groups, modular double, GNS-representation, higher Teichm\"uller theory, quantum dilogarithm
Received by editor(s): May 29, 2014
Received by editor(s) in revised form: February 19, 2016, and October 26, 2016
Published electronically: December 27, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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