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

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.