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Let $H$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
A $\star$-subalgebra $I$ of $H$ is a left coideal if $\Delta(I) \subset H \otimes I$.
$H$ is called maximal if it has no left coideal $\star$-subalgebra other than $\mathbb{C}$ and $H$.

Question: Is there a non-trivial maximal finite dimensional Hopf ${\rm C}^{\star}$-algebra?
(non-trivial means not the Hopf algebra of a group)

Remark: The Hopf algebra of a group is maximal iff the group is cyclic of prime order. There is this paper of Y. Zhu proving that the Hopf algebras of prime dimension are group algebras, so this result answers my question for the prime dimension case, but not in general.

See this post for the Grothendieck ring of what could be the first maximal non-trivial Hopf ${\rm C}^{\star}$-algebra.


Motivation: A finite dimensional Hopf ${\rm C}^{\star}$-algebra $H$ (also called Kac algebra) acts outerly on the hyperfinite ${\rm II}_1$ factor $R$. We can build the subfactor $(R \subset R \rtimes H)$, remembering $H$, and whose isomorphism class is independent of the choice of the action. But there is a $1$-$1$ Galois correspondence between the set of intermediate subfactors of $(R \subset R \rtimes H)$, and the set of left coideal $\star$-subalgebras of $H$. Now a subfactor $(A \subset B)$ without intermediate subfactor other than $A$ and $B$, is called maximal, moreover, the irreducible depth $2$ finite index subfactors are exactly the subfactors given by the Kac algebras. So the motivation here is to find a non-trivial maximal irreducible depth $2$ finite index subfactor.

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  • $\begingroup$ I'm not sure how useful your definition of "quantum group" is or why the specific field $\mathbb{C}$ is required, but the parenthetic remark involving "if appropriate" and "finite dimensional" seems to add confusion. Clarify (or perhaps just omit) this remark? $\endgroup$ – Jim Humphreys Jan 3 '15 at 15:46
  • $\begingroup$ @JimHumphreys: What definition of quantum group do you suggest? Do you suggest a definition as general as possible, as "Hopf algebra over a field $\mathbb{K}$", or do you suggest to just put the specific area I need for my own research (i.e. finite dimensional Hopf ${\rm C}^*$-algebra). I think this question is very natural and interesting in general, but if I get an answer outside of the area of my need, I should post an other question, more specialized. I've edited the post, what do you think? Do you have an example for a field different of $\mathbb{C}$? $\endgroup$ – Sebastien Palcoux Jan 3 '15 at 16:53
  • $\begingroup$ Maybe it's simpler to avoid the term "quantum group" here rather than attempt a precise definition. I think Drinfeld was reluctant to give a general definition in the mid-1980s, since different notions arise in different subject areas. $\endgroup$ – Jim Humphreys Jan 3 '15 at 17:58
  • $\begingroup$ @JimHumphreys: Ok, I've took off the term "quantum group". $\endgroup$ – Sebastien Palcoux Jan 3 '15 at 18:11
  • $\begingroup$ @JimHumphreys: Is "minimal" the good word for this notion? Perhaps "prime" or rather "coprime" is better. Do you know some works about this problem, or a paper in which it is mentioned? $\endgroup$ – Sebastien Palcoux Jan 4 '15 at 4:53

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