Title: Ergodic actions of the compact quantum group $O_{-1}(2)$,
Abstract: Among the ergodic actions of a compact quantum group $\mathbb{G}$ on possibly
non-commutative spaces, those that are {\it embeddable} are the natural
analogues of actions of a compact group on its homogeneous spaces. These can be
realized as {\it coideal subalgebras} of the function algebra
$\mathcal{O}(\mathbb{G})$ attached to the compact quantum group.
We classify the embeddable ergodic actions of the compact quantum group
$O_{-1}(2)$, basing our analysis on the bijective correspondence between all
ergodic actions of the classical group $O(2)$ and those of its quantum twist
resulting from the monoidal equivalence between their respective tensor
categories of unitary representations.
In the last section we give counterexamples showing that in general we cannot
expect a bijective correspondence between embeddable ergodic actions of two
monoidally equivalent compact quantum groups.