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.