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A complex (finite-dimensional) Hopf algebra is said to be a Kac algebra if it is a ${\rm C^{\star}}$-algebra in such a way that the comultiplication $\Delta$ is a $\star$-homomorphism. Obviously, a (finite-dimensional) Kac algebra is a semisimple Hopf algebra, but what about the converse:

Let $H$ be a complex finite dimensional semisimple Hopf algebra.
Question: Is there a Kac algebra $K$ isomorphic to $H$ as Hopf algebra?
If no, what is the smallest counter-example (for the dimension), and what is the main obstruction?

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  • $\begingroup$ I am not sure if I am not missing something, but I guess that equivalently you need to show that $H$ is a C*-algebra, right? Because it is known that for a semisimple Hopf algebra we have $S^2={\rm id}$. And if $H$ is a C*-algebra, this would mean that it defines a finite (compact) quantum group and for CQGs we have the Haar state and we know that $S^2={\rm id}$ if and only if the Haar state is tracial. $\endgroup$
    – Daniel
    Mar 9, 2021 at 12:01
  • $\begingroup$ @Daniel: a finite-dimensional semisimple algebra over $\mathbb{C}$ is always a C*-algebra. Proof: by the Artin–Wedderburn theorem a finite dimensional semisimple algebra over a field $k$ is a finite product of matrix algebras over division algebras over $k$. Now over an algebraically closed field $k$ (for example the complex numbers $\mathbb{C}$), there are no finite-dimensional (associative) division algebras, except $k$ itself. $\endgroup$
    – Sebastien Palcoux
    Mar 9, 2021 at 13:29
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    $\begingroup$ @Daniel: how does you sentence show that the comultiplication $\Delta$ is a $*$-homomorphism? $\endgroup$
    – Sebastien Palcoux
    Mar 9, 2021 at 13:34
  • $\begingroup$ Oh, yes sure, I meant C*-algebra such that $\Delta$ is a $*$-homomorphism, but that's exactly what you write in the question, I just did not notice, sorry. You can ignore my comment. $\endgroup$
    – Daniel
    Mar 9, 2021 at 13:46
  • $\begingroup$ I opened a similar question here: mathoverflow.net/questions/433176/hopf-algebras-vs-kac-algebras $\endgroup$
    – dm82424
    Nov 2, 2022 at 17:18

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