Towards a classification of simple partial comodules of Hopf algebras

Making the first steps towards a classification of simple partial comodules, we give a general construction for partial comodules of a Hopf algebra $H$ using central idempotents in right coideal subalgebras and show that any $1$-dimensional partial comodule is of that form. We conjecture that in fact all finite-dimensional simple partial $H$-comodules arise this way. For $H = kG$ for some finite group $G$, we give conditions for the constructed partial comodule to be simple, and we determine when two of them are isomorphic. If $H = kG^*,$ then our construction recovers the work of M. Dokuchaev and N. Zhukavets. We also study the partial modules and comodules of the non-commutative non-cocommutative Kac-Paljutkin algebra $\mathcal{A}$.