What is an example of a complex Hopf algebra $H$, different from $\mathbb{C}$, which does not admit a non zero coderivation? Is there a complete classification of all Hopf algebras with this property?


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    $\begingroup$ Consider a cyclic group $Z_n$ of order $n > 0$. The group algebra $A := \mathbb{C}\left[Z_n\right]$ has no nonzero derivations. (This is easily checked: Let $g$ be a generator of $Z_n$; then, any derivation $d$ of $A$ would satisfy $d\left(g^k\right) = k g^{k-1} d\left(g\right)$ for all $k \geq 0$; applying this to $k = n$, we would quickly obtain $d\left(g\right) = 0$ because of $d\left(1\right) = 0$.) Thus, dualizing, we conclude that its dual Hopf algebra $A^*$ has no nonzero coderivation. (Note that $A^* \cong A$ as Hopf algebras, via the discrete Fourier transform.) $\endgroup$ – darij grinberg Feb 11 at 22:54

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