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We studied the positive representations $P_\lambda$ of split real quantumgroups $U_{q\tilde{q}}(g_R)$ restricted to the Borel subalgebra$U_{q\tilde{q}}(b_R)$. We proved that the restriction is independent of theparameter $\lambda$. Furthermore, we prove that it can be constructed from theGNS-representation of the multiplier Hopf algebra $U_{q\tilde{q}}^{C^*}(b_R)$constructed earlier, which enables us to decompose their tensor product usingthe theory of the "multiplicative unitary". This will be an essentialingredient in the construction of quantum higher Teichm\"{u}ller theory fromthe perspective of representation theory, generalizing earlier work byFrenkel-Kim.

On tensor products of positive representations of split real quantum Borel subalgebra $U_{q\tilde{q}}(b_R)$