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

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- by Ivan C. H. Ip PDF
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**370**(2018), 4177-4200 Request permission

## 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.## References

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## Additional Information

**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
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 4177-4200 - MSC (2010): Primary 81R50, 22D25
- DOI: https://doi.org/10.1090/tran/7110
- MathSciNet review: 3811524