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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?

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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

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