Does Sklyanin algebra (which is an elliptic extension of the quantum group) admit a bialgebra structure or even Hopf algebraic structure? Or is it proved that it is impossible to have such a structure? 


Note that the algebra is defined as four generators $S\_0, S\_{\alpha = 1,2,3}$ and
\begin{equation}
\begin{aligned}
& \{ S\_0, S\_{\alpha} \} = 2 J\_{\beta \gamma} S\_{\beta} S\_{\gamma}\\[0.5em]
& \{ S\_{\alpha}, S\_{\beta} \} = -2 S\_0 S\_{\gamma}
\end{aligned}
\end{equation}
where $(\alpha, \beta, \gamma)$ is the cyclic permutation of $1,2,3$.


Moreover, if one so far cannot find such bialgebraic structure, is it still possible to define tensor product representation of the algebra?



Bialgebraic structure of Sklyanin algebra
In response to your first question, there is no Hopf structure on any Sklyanin algebra of any dimension. See Corollary 2.8 (i) of the following paper:


<https://arxiv.org/pdf/1601.06687v1.pdf>


