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I am following the notes on Quantum Groups given by our instructor. There I found that Drinfeld-Jimbo quantum group $U_h (sl_2 (\mathbb C))$ corresponding to the Lie algebra $sl_2 (\mathbb C)$ and the trivial Hopf algebra deformation $U(sl_2 (\mathbb C)) [[h]]$ of $U(sl_2 (\mathbb C))$ are isomorphic as $\mathbb C[[h]]$-algebras but non-isomorphic as Hopf algebras as the former one is non-cocommutative whereas the latter one is cocommutative. Then it claims that the category of modules for both the spaces are the same since the category of modules in an algebra only depends on its algebra structure. But their monoidal structures are different.

Question $:$ What is a monoidal structure and why are they different?

Could anyone give me some suggestion in this regard?

Thanks for your time.

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    $\begingroup$ The "monoidal structure" here is the tensor product of modules. You use the comultiplication map of the Hopf algebra to define its action on a tensor product, so what is probably meant is that these two Hopf algebras have different comultiplication. $\endgroup$
    – Manuel Araújo
    Commented Nov 20, 2022 at 8:19

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