One-dimensional partial actions of 8-dimensional Hopf algebras and right coideal subalgebras

In this work, we complete the description of the one-dimensional partial actions of 8-dimensional Hopf algebras by computing the remaining cases: the Kac-Paljutkin algebra $\mathcal{A}$ and the unique non-semisimple non-pointed Hopf algebra $\mathcal{K}$. We prove that all these partial actions are symmetric, study their associated partial smash products and we determine all their partial coactions of dimension one. Beyond the 8-dimensional setting, we investigate which right coideal subalgebras of a Hopf algebra $H$ can be realized as partial smash products $\underline{ \Bbbk \# H}$ over the base field. In particular, we show that every right coideal subalgebra of a finite-dimensional cosemisimple Hopf algebra arises in this way.