Suppose that a finite-dimesnional Hopf $C^\*$-algebra $H$ acts on a type $II\_1$ factor $N$ minimally (that is, $N^{\prime}\cap (N\rtimes H)=\mathbb{C}$). Is it true that there always exists a minimal action of the dual Hopf algebra $H^\*$ on $N$?



Action of a dual Hopf algebra on a factor
No, there might be no minimal action at all of $H^\*$ on $N$. By [Theorem A in this paper of Falguières and Raum](https://arxiv.org/abs/1112.4088) (see also [this paper](https://arxiv.org/abs/0811.1764) from which other examples may be deduced), for any rigid C$^\*$-tensor category $\mathcal{C}$ with finitely many irreducible objects, there exists a $II\_1$ factor $N$ such that the bimodule category of $N$ is isomorphic with $\mathcal{C}$. In particular, given any finite, nonabelian group $G$, you can find a $II\_1$ factor $N$ and an outer action $\alpha$ of $G$ on $N$ such that the only irreducible finite index $N$-$N$-bimodules are $1$-dimensional and given by the automorphisms $(\alpha\_g)\_{g \in G}$. Letting $H$ be the Hopf algebra associated with $G$, there is no minimal action of its dual $H^\*$ on the $II\_1$ factor $N$. Indeed, since $G$ is nonabelian, $G$ has irreducible representations of dimension larger than one and a minimal action of $H^\*$ on $N$ would give irreducible finite index $N$-$N$-bimodules of dimension larger than $1$.


