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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 Kassel, Quantum Groups 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 $\gamma^2 = (S \gamma)^2 = \text{id}_H$.

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

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 Kassel, Quantum Groups could be useful:

A Hopf $H$ 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 $\gamma^2 = (S \gamma)^2 = \text{id}_H$.

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 Kassel, Quantum Groups 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 $\gamma^2 = (S \gamma)^2 = \text{id}_H$.