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I recently came across Kac algebra. They are roughly Hopf algebras and $C^*$-algebras with compatible structures. It follows from Artin–Wedderburn theorem that every semisimple complex Hopf algebra can be seen as a $C^*$-algebra structure, but I am unsure whether this structure will be compatible with the comultiplication. I guess that this is not the case, as I think that there are dimensions where Kac algebras have been classified, but semisimple Hopf algebras have not.

Question: Is every semisimple/finite-dimensional complex Hopf algebra a Kac algebra?

Something similar has been previously asked here, but the question is unanswered. If this is unknown in general, it would also be helpful to know if this holds under some stronger assumptions (eg. for a given dimension).

EDIT: Lemma IV.8.2 in Quantum Groups by Kassel could be useful:

A Hopf algebra $H$ has a Hopf $*$-algebra structure if and only if there exists an antilinear automorphism $\gamma$ of $H$ such that

$(i)$ the map $\gamma$ is a morphism of real algebras and an antimorphism of real coalgebras, and

$(ii)$ we have $\gamma^2 = (S \gamma)^2 = \text{id}_H$.

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    $\begingroup$ Anything here: The set of types of n-dimensional semisimple and cosemisimple Hopf algebras is finite, J. Algebra 193 (1997), 571–580. $\endgroup$
    – JP McCarthy
    Commented Nov 4, 2022 at 8:33
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    $\begingroup$ Also Compact involutions of semisimple quantum groups. Czech. J. Phys., vol. 44 (1994) pp. 963-972. link.springer.com/article/10.1007/BF01690448 $\endgroup$
    – JP McCarthy
    Commented Nov 4, 2022 at 8:33
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    $\begingroup$ I think we can deduce from SOME ADVANCES ABOUT THE EXISTENCE OF COMPACT INVOLUTIONS IN SEMISIMPLE HOPF ALGEBRAS that the question is open. In this paper it is shown that for dimensions up to 23 semisimple and finite dimensional Hopf algebras are finite quantum groups and thus Kac algebras. The general problem is left open. $\endgroup$
    – JP McCarthy
    Commented Nov 4, 2022 at 15:19
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    $\begingroup$ Apologies if the two references first given were not relevant: I had hoped they might have been. $\endgroup$
    – JP McCarthy
    Commented Nov 4, 2022 at 15:23
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    $\begingroup$ Okey, I just wanted to check that I wasn't missing anything. I agree that it looks like 24 is the smallest dimension, where this seems to be open. I previously had a look at "Some advances about the existence of compact involutions in semisimple Hopf algebra" by Abella and "On the Semisolvability of Semisimple Hopf Algebras of Low Dimension" by Natale. In the latter, she gives a table on the classification of s.s. Hopf algebras. The smallest dimension where this open is 24. However, Kac algebras of dimension 24 have been classified. This dimension was the original motivation for my question. $\endgroup$
    – dm82424
    Commented Nov 4, 2022 at 15:39

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As pointed out in the comments all semisimple Hopf algebras of dimension up to $23$ are Kac algebras. So it seems like dimension $24$ is the smallest one where this is open.

Corollary 9.7 of "Weakly group-theoretical and solvable fusion categories" by Etingof, Nikshych and Ostrik says that a semisimple Hopf algebra of dimension $p q^2$ is either a Kac algebra or a twisted group algebra (by a twist corresponding to the subgroup $(Z/qZ)^2$) or the dual of a twisted group algebra.

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