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I take the following quote from an answer to this question

A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. The quantized enveloping algebras and Lusztig's small quantum groups are examples of pointed Hopf algebras.

The finte/restricted Hopf duals of quantized enveloping algebras are all cosemisimple. Does this happen in general, or are the duals of all pointed Hopf algebras cosemisimple?

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No way, doc! Take a finite $p$-group $G$. Let ${\mathbb F}$ be a field of characteristic $p$. The group algebra ${\mathbb F}G$ is as pointed as it gets. But its dual ${\mathbb F}G^{\ast}$ is not cosemisimple because ${\mathbb F}G$ is not semisimple.

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