The quantum Gelfand–Kirillov property for the localized quantum group

Let $U$ be the quantum group, $Z$ its center, $U_0$ its Cartan subalgebra, $F_l(U)$ the relevant left fraction algebra, and let $\mathcal O_q[G/H]$ and $T$ be the quantum homogeneous-coordinate algebra and quantum torus occurring in the paper. Write $\operatorname{Frac}$ for the non-commutative fraction field. Quantum Gelfand–Kirillov conjecture. There is an isomorphism of non-commutative fraction fields

$$\operatorname{Frac}\left(F_l(U) \otimes_Z U_0\right) = \operatorname{Frac}\left(\mathcal O_q[G/H] \otimes T\right).$$

In particular, $\operatorname{Frac}\left(F_l(U) \otimes_Z U_0\right)$ coincides with a non-commutative fraction field of a quantum torus algebra, so the quantum Gelfand–Kirillov property holds for $F_l(U) \otimes_Z U_0$. The claim is motivated by the classical isomorphism and the birational-chain result. It identifies the fraction field with that of a quantum torus, but the source supplies no resolution evidence.