4
$\begingroup$

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?

$\endgroup$

1 Answer 1

4
+100
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.