In the paper Modular Double of Quantum Group, Fadeev gives a presentation of $\mathcal{U}_q(\mathfrak{gl}_2)$ in terms of a Weyl algebra $\mathcal{C}_q$ with generators $w_i, i \in \mathbb{Z}/4$ and relations $w_n w_{n+1} = q^2 w_{n+1} w_n$, $w_n$ commuting otherwise.
At the end of the paper, he brings up the problem of giving a coproduct on $\mathcal{C}_q$ which agrees with the coproduct on $\mathcal{U}_q(\mathfrak{gl}_2)$. Has anyone constructed this coproduct? Is it known to not exist?
