Let $\mathcal{F}_h(\operatorname{SL}_2(\mathbb{C}))$ be the $\mathbb{C}[[h]]$-algebra generated by $a, b, c, d$ subject to the following relations:
\begin{align*} & ac = e^{-h}ca, \quad bd = e^{-h}db, \quad ab = e^{-h}ba, \quad cd = e^{-h}dc, \quad bc = cb, \\ & ad - da + (e^h - e^{-h})bc = 0, \quad ad - e^{-h}bc = 1. \end{align*}
The Hopf algebra structure is defined as follows:
\begin{align*} & \Delta_h\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right) = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \otimes \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \\ & S_h\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right) = \begin{pmatrix} d & -e^{h}b \\ -e^{-h}c & a \end{pmatrix}, \\ & \epsilon_h\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right) = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. \end{align*}
Additionally, let $U_h(\mathfrak{sl}_2(\mathbb{C}))$ be the $\mathbb{C}[[h]]$-algebra generated by $H, E, F$ with the following relations:
$$[H, E] = 2E, \quad [H, F] = -2F, \quad [E, F] = \dfrac{e^{hH} - e^{-hH}}{e^h - e^{-h}}.$$
The Hopf algebra structure is given by:
\begin{align*} & \Delta_h(H) = 1 \otimes H + H \otimes 1, \quad \Delta_h(E) = 1 \otimes E + E \otimes e^{hH}, \quad \Delta_h(F) = F \otimes 1 + e^{-hH} \otimes F, \\ & S_h(H) = -H, \quad S_h(E) = -Ee^{-hH}, \quad S_h(F) = -e^{hH}F, \quad \epsilon(H) = \epsilon(E) = \epsilon(F) = 0. \end{align*}
Now, consider a Hopf pairing $\langle- ,- \rangle\colon \mathcal{F}_h(\operatorname{SL}_2(\mathbb{C})) \times U_h(\mathfrak{sl}_2(\mathbb{C}) ) \to \mathbb{C}[[h]]$ defined by:
\begin{align*} & \langle a, H \rangle = 1, \quad \langle b, H \rangle = 0, \quad \langle c, H \rangle = 0, \quad \langle d, H \rangle = -1, \\ & \langle a, E \rangle = 0, \quad \langle b, E \rangle = 1, \quad \langle c, E \rangle = 0, \quad \langle d, E \rangle = 0, \\ & \langle a, F \rangle = 0, \quad \langle b, F \rangle = 0, \quad \langle c, F \rangle = 1, \quad \langle d, F \rangle = 0. \end{align*}
Question
Does there exist a topological basis $\{f_{stu}\}$ of $\mathcal{F}_h(SL_2(\mathbb{C}))$ such that:
$$\langle f_{stu}, E^i F^j H^k \rangle = \delta_{s,i} \delta_{t,j} \delta_{u,k}?$$
If the above holds, does a similar construction work for a simply connected semisimple Lie group $G$ and its Lie algebra $\mathfrak{g}$? Furthermore, if we drop the semisimple condition for $G$, would the same result hold?
