Let $k$ be the ground field, and $q$ be an invertible element with $q$ not being a root of unity.
Let $\operatorname{SL}_q(2)$ be the quantum coordinate ring of $\operatorname{SL}(2)$ given explicitly as a $k$-algebra generated by symbols $a,b,c,d$ with the relations $ba = qab$, $db = qbd$, $ca=qac$, $dc = qcd$, $cb = bc$, and $ad - da = (q^{-1} - q)bc$, and with the quantum determinant $\det_q = ad - q^{-1}bc = da - q bc = 1$. It is known that $\operatorname{SL}_q(2)$ can be equipped with the comultiplication, the counit, and the antipode such that $\operatorname{SL}_q(2)$ becomes a Hopf algebra.
Let $U_q(\mathfrak{sl}(2))$ be the quantum group of $\mathfrak{sl}(2)$ given explicitly as a $k$-algebra generated by symbols $E,F,K,K^{-1}$ with the relations $K K^{-1} = K^{-1} K = 1$, $KEK^{-1} = q^2 E$, $KFK^{-1} = q^{-2}F$, and $[E,F] = \frac{K-K^{-1}}{q-q^{-1}}$. It is known that $U_q(\mathfrak{sl}(2))$ can be equipped with the comultiplication, the counit, and the antipode such that $U_q(\mathfrak{sl}(2))$becomes a Hopf algebra.
It is also known that there is a Hopf duality between $\operatorname{SL}_q(2)$ and $U_q(\mathfrak{sl}(2))$, that is, a bilnear form on $\operatorname{SL}_q(2) \times U_q(\mathfrak{sl}(2))$ satisfying certain properties. One also establishes the duality between $U_q(\mathfrak{sl}(2))$-modules and $\operatorname{SL}_q(2)$-comodules.
I know that using the FRT construction, one can start with the $R$-matrix, one can construct the quantum coordinate ring $M_q(2)$ which is only one relation short ($\det_q = 1$) from $\operatorname{SL}_q(2)$. By trying to find the antipode, one can find the missing relation and reconstruct $\operatorname{SL}_q(2)$.
Question: How does one construct $U_q(\mathfrak{sl}(2))$? Is there a version of the FRT construction that builds $U_q(\mathfrak{sl}(2))$ instead of $\operatorname{SL}_q(2)$? Or maybe there is a formal way of obtaining the Hopf dual which would help to discover $U_q(\mathfrak{sl}(2))$?
Generally, I am trying to find any way of constructing the quantum group $U_q(\mathfrak{sl}(2))$ without writing down the explicit generators. The FRT construction from the $R$-matrix would do, but I don't know how to do it.
