Towards a classification of simple partial comodules of Hopf algebras

Abstract: 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}).